The Distribution of the Domination Number of a Family of Random Interval Catch Digraphs
aa r X i v : . [ m a t h . C O ] M a r Technical Report
Elvan Ceyhan ∗ August 10, 2018
Abstract
We study a new kind of proximity graphs called proportional-edge proximity catch digraphs (PCDs) in arandomized setting. PCDs are a special kind of random catch digraphs that have been developed recentlyand have applications in statistical pattern classification and spatial point pattern analysis. PCDs are alsoa special type of intersection digraphs; and for one-dimensional data, the proportional-edge PCD familyis also a family of random interval catch digraphs. We present the exact (and asymptotic) distributionof the domination number of this PCD family for uniform (and non-uniform) data in one dimension. Wealso provide several extensions of this random catch digraph by relaxing the expansion and centralityparameters, thereby determine the parameters for which the asymptotic distribution is non-degenerate.We observe sudden jumps (from degeneracy to non-degeneracy or from a non-degenerate distribution toanother) in the asymptotic distribution of the domination number at certain parameter values.
Keywords: asymptotic distribution; class cover catch digraph; degenerate distribution; exact distribution;intersection digraph; proximity catch digraph; proximity map; random graph; uniform distribution
AMS 2000 Subject Classification: ∗ Address: Department of Mathematics, Ko¸c University, 34450 Sarıyer, Istanbul, Turkey. e-mail: [email protected], tel:+90(212) 338-1845, fax: +90 (212) 338-1559. Introduction
The proximity catch digraphs (PCDs) were motivated by their applications in pattern classification and spatialpattern analysis, hence they have become focus of considerable attention recently. The PCDs are a specialtype of proximity graphs which were introduced by Toussaint (1980). A digraph is a directed graph with vertexset V and arcs (directed edges) each of which is from one vertex to another based on a binary relation. Thenthe pair ( p, q ) ∈ V × V is an ordered pair which stands for an arc from vertex p to vertex q in V . For example, nearest neighbor (di)graph which is defined by placing an arc between each vertex and its nearest neighbor isa proximity digraph where vertices represent points in some metric space (Paterson and Yao (1992)). PCDsare data-random digraphs in which each vertex corresponds to a data point and arcs are defined in terms ofsome bivariate relation on the data.The PCDs are closely related to the class cover problem of Cannon and Cowen (2000). Let (Ω , M ) be ameasurable space and X n = { X , X , . . . , X n } and Y m = { Y , Y , . . . , Y m } be two sets of Ω-valued randomvariables from classes X and Y , respectively, with joint probability distribution F X,Y . Let d ( · , · ) : Ω × Ω → [0 , ∞ ) be any distance function. The class cover problem for a target class, say X , refers to finding a collectionof neighborhoods, N i around X i such that (i) X n ⊆ (cid:0) ∪ i N i (cid:1) and (ii) Y m ∩ (cid:0) ∪ i N i (cid:1) = ∅ . A collection ofneighborhoods satisfying both conditions is called a class cover . A cover satisfying (i) is a proper cover of X n while a cover satisfying (ii) is a pure cover relative to Y m . This article is on the cardinality of smallestclass covers ; that is, class covers satisfying both (i) and (ii) with the smallest number of neighborhoods. SeeCannon and Cowen (2000) and Priebe et al. (2001) for more on the class cover problem.The first type of PCD was class cover catch digraph (CCCD) introduced by Priebe et al. (2001) whogave the exact distribution of its domination number for uniform data from two classes in R . DeVinney et al.(2002), Marchette and Priebe (2003), Priebe et al. (2003a), Priebe et al. (2003b), DeVinney and Priebe (2006)extended the CCCDs to higher dimensions and demonstrated that CCCDs are a competitive alternative tothe existing methods in classification. Furthermore, DeVinney and Wierman (2003) proved a SLLN result forthe one-dimensional class cover problem; Wierman and Xiang (2008) provided a generalized SLLN result andXiang and Wierman (2009) provided a CLT result for CCCD based on one-dimensional data. However, CCCDshave some disadvantages in higher dimensions; namely, finding the minimum dominating set for CCCDs isan NP-hard problem in general, although a simple linear time algorithm is available for one dimensional data(Priebe et al. (2001)); and the exact and the asymptotic distributions of the domination number of the CCCDsare not analytically tractable in multiple dimensions. Ceyhan and Priebe (2003, 2005) introduced the centralsimilarity proximity maps and proportional-edge proximity maps for data in R d with d > R and then the dominationnumber is used as a statistic for testing bivariate spatial patterns (Ceyhan and Priebe (2005), Ceyhan (2010)).The relative density of these two PCD families is also calculated and used for the same purpose (Ceyhan et al.(2006) and Ceyhan et al. (2007)). Moreover, the distribution of the domination number of CCCDs is derivedfor non-uniform data (Ceyhan (2008)).In this article, we provide the exact (and asymptotic) distribution of the domination number of proportional-edge PCDs for uniform (and non-uniform) one-dimensional data. First, some special cases and bounds for thedomination number of proportional-edge PCDs is presented, then the domination number is investigated foruniform data in one interval (in R ) and the analysis is generalized to uniform data in multiple intervals andto non-uniform data in one and multiple intervals. These results can be seen as generalizations of the resultsof Ceyhan (2008). Some trivial proofs are omitted, shorter proofs are given in the main body of the article;while longer proofs are deferred to the Appendix.We define the proportional-edge PCDs and their domination number in Section 2, provide the exact andasymptotic distributions of the domination number of proportional-edge PCDs for uniform data in one intervalin Section 3, discuss the distribution of the domination number for data from a general distribution in Section4. We extend these results to multiple intervals in Section 5, and provide discussion and conclusions in Section6. For convenience in notation and presentation, we resort to non-standard extended (perhaps abused) forms of2ernoulli and Binomial distributions, denoted BER( p ) and BIN( n, p ), respectively, where p is the probabilityof success and n is the number of trials. Throughout the article, we take p ∈ [0 ,
1] (unlike p ∈ (0 , X ∼ BER( p ), then P ( X = 1) = p and P ( X = 0) = 1 − p . If Y ∼ BIN( n, p ), then P ( Y = k )=( nk ) p k (1 − p ) n − k for p ∈ (0 , k ∈ { , , , . . . , n } and P ( Y = n ) = 1 for p = 1 and P ( Y = 0) = 1 for p = 0. Consider the map N : Ω → ℘ (Ω) where ℘ (Ω) represents the power set of Ω. Then given Y m ⊆ Ω, the proximitymap N ( · ) : Ω → ℘ (Ω) associates with each point x ∈ Ω a proximity region N ( x ) ⊆ Ω. For B ⊆ Ω, the Γ -regionis the image of the map Γ ( · , N ) : ℘ (Ω) → ℘ (Ω) that associates the region Γ ( B, N ) := { z ∈ Ω : B ⊆ N ( z ) } with the set B . For a point x ∈ Ω, we denote Γ ( { x } , N ) as Γ ( x, N ). Notice that while the proximity regionis defined for one point, a Γ -region is defined for a set of points. The data-random PCD has the vertex set V = X n and arc set A defined by ( X i , X j ) ∈ A iff X j ∈ N ( X i ).Let Ω = R and Y ( i ) be the i th order statistic (i.e., i th smallest value) of Y m for i = 1 , , . . . , m with theadditional notation for i ∈ { , m + 1 } as −∞ =: Y (0) < Y (1) < . . . < Y ( m ) < Y ( m +1) := ∞ . Then Y ( i ) partition R into ( m + 1) intervals which is called the intervalization of R by Y m . Let also that I i := (cid:0) Y ( i ) , Y ( i +1) (cid:1) for i ∈ { , , , . . . , m } and M c,i := Y ( i ) + c ( Y ( i +1) − Y ( i ) ) (i.e., M c,i ∈ I i such that c × I i is to the left of M c,i ). We define the proportional-edge proximity region with the expansionparameter r ≥ c ∈ [0 ,
1] for two one-dimensional data sets, X n and Y m , from classes X and Y , respectively, as follows. For x ∈ I i with i ∈ { , , . . . , m − } N ( x, r, c ) = ((cid:0) Y ( i ) , Y ( i ) + r (cid:0) x − Y ( i ) (cid:1)(cid:1) ∩ I i if x ∈ ( Y ( i ) , M c,i ), (cid:0) Y ( i +1) − r (cid:0) Y ( i +1) − x (cid:1) , Y ( i +1) (cid:1) ∩ I i if x ∈ (cid:0) M c,i , Y ( i +1) (cid:1) , (1)Additionally, for x ∈ I i with i ∈ { , m } N ( x, r, c ) = ((cid:0) Y (1) − r (cid:0) Y (1) − x (cid:1) , Y (1) (cid:1) if x < Y (1) , (cid:0) Y ( m ) , Y ( m ) + r (cid:0) x − Y ( m ) (cid:1)(cid:1) if x > Y ( m ) . (2)Notice that for i ∈ { , m } , the proportional-edge proximity region does not depend on the centrality param-eter c . For x ∈ Y m , we define N ( x, r, c ) = { x } for all r ≥ x = M c,i , then in Equation (1), wearbitrarily assign N ( x, r, c ) to be one of (cid:0) Y ( i ) , Y ( i ) + r (cid:0) x − Y ( i ) (cid:1)(cid:1) ∩ I i or (cid:0) Y ( i +1) − r (cid:0) Y ( i +1) − x (cid:1) , Y ( i +1) (cid:1) ∩ I i .For c = 0, we have (cid:0) M c,i , Y ( i +1) (cid:1) = I i and for c = 1, we have ( Y ( i ) , M c,i ) = I i . So, we set N ( x, r,
0) := (cid:0) Y ( i +1) − r (cid:0) Y ( i +1) − x (cid:1) , Y ( i +1) (cid:1) ∩ I i and N ( x, r,
1) := (cid:0) Y ( i ) , Y ( i ) + r (cid:0) x − Y ( i ) (cid:1)(cid:1) ∩ I i . For r >
1, we have x ∈ N ( x, r, c ) for all x ∈ I i . Furthermore, lim r →∞ N ( x, r, c ) = I i for all x ∈ I i , so we define N ( x, ∞ , c ) = I i for all such x .For X i iid ∼ F , with the additional assumption that the non-degenerate one-dimensional probability densityfunction (pdf) f exists with support S ( F ) ⊆ I i and f is continuous around M c,i and around the end pointsof I i , implies that the special cases in the construction of N ( · , r, c ) — X falls at M c,i or the end points of I i — occurs with probability zero. For such an F , the region N ( X i , r, c ) is an interval a.s.The data-random proportional-edge PCD has the vertex set X n and arc set A defined by ( X i , X j ) ∈ A iff X j ∈ N ( X i , r, c ). We call such digraphs D n,m ( r, c )-digraphs. A D n,m ( r, c )-digraph is a pseudo digraph according some authors, if loops are allowed (see, e.g., Chartrand and Lesniak (1996)). The D n,m ( r, c )-digraphsare closely related to the proximity graphs of Jaromczyk and Toussaint (1992) and might be considered as aspecial case of covering sets of Tuza (1994) and intersection digraphs of Sen et al. (1989). Our data-randomproximity digraph is a vertex-random digraph and is not a standard random graph (see, e.g., Janson et al.32000)). The randomness of a D n,m ( r, c )-digraph lies in the fact that the vertices are random with the jointdistribution F X,Y , but arcs ( X i , X j ) are deterministic functions of the random variable X j and the random set N ( X i , r, c ). In R , the data-random PCD is a special case of interval catch digraphs (see, e.g., Sen et al. (1989)and Prisner (1994)). Furthermore, when r = 2 and c = 1 / M c,i = (cid:0) Y ( i ) + Y ( i +1) (cid:1) /
2) we have N ( x, r, c ) = B ( x, r ( x )) where B ( x, r ( x )) is the ball centered at x with radius r ( x ) = d ( x, Y m ) = min y ∈Y m d ( x, y ). Theregion N ( x, , /
2) corresponds to the proximity region which gives rise to the CCCD of Priebe et al. (2001). D n,m ( r, c ) -digraphs In a digraph D = ( V , A ) of order |V| = n , a vertex v dominates itself and all vertices of the form { u : ( v, u ) ∈A} . A dominating set , S D , for the digraph D is a subset of V such that each vertex v ∈ V is dominatedby a vertex in S D . A minimum dominating set , S ∗ D , is a dominating set of minimum cardinality; and the domination number , denoted γ ( D ), is defined as γ ( D ) := | S ∗ D | , where | · | is the set cardinality functional (West(2001)). If a minimum dominating set consists of only one vertex, we call that vertex a dominating vertex .The vertex set V itself is always a dominating set, so γ ( D ) ≤ n .Let F (cid:0) R d (cid:1) := { F X,Y on R d with P ( X = Y ) = 0 } . As in Priebe et al. (2001) and Ceyhan (2008), weconsider D n,m ( r, c )-digraphs for which X n and Y m are random samples from F X and F Y , respectively, and thejoint distribution of X, Y is F X,Y ∈ F (cid:0) R d (cid:1) . We call such digraphs F (cid:0) R d (cid:1) -random D n,m ( r, c ) -digraphs andfocus on the random variable γ ( D ). To make the dependence on sample sizes n and m , the distribution F ,and the parameters r and c explicit, we use γ n,m ( F, r, c ) instead of γ ( D ). For n ≥ m ≥
1, it is trivial tosee that 1 ≤ γ n,m ( F, r, c ) ≤ n , and 1 ≤ γ n,m ( F, r, c ) < n for nontrivial digraphs. F ( R ) -random D n,m ( r, c ) -digraphs Let X n and Y m be two samples from F ( R ), X [ i ] := X n ∩ I i , and Y [ i ] := { Y ( i ) , Y ( i +1) } for i = 0 , , , . . . , m .This yields a disconnected digraph with subdigraphs each of which might be null or itself disconnected. Let D [ i ] be the component of the random D n,m ( r, c )-digraph induced by the pair X [ i ] and Y [ i ] for i = 0 , , , . . . , m , n i := (cid:12)(cid:12) X [ i ] (cid:12)(cid:12) , and F i be the density F X restricted to I i , and γ ni, ( F i , r, c ) be the domination number of D [ i ] . Letalso that M c,i ∈ I i be the point that divides the interval I i in ratios c and 1 − c (i.e., length of the subintervalto the left of M c,i is c ×
100 % of the length of I i ). Then γ n,m ( F, r, c ) = P mi =0 ( γ ni, ( F i , r, c ) I ( n i > I ( · ) is the indicator function. We study the simpler random variable γ ni, ( F i , r, c ) first. The following lemmafollows trivially. Lemma 2.1.
For i ∈ { , m } , we have γ ni, ( F i , r, c ) = I ( n i > for all r ≥ . Let Γ ( B, r, c ) be the Γ -region for set B associated with the proximity map on N ( · , r, c ). Lemma 2.2.
The Γ -region for X [ i ] in I i with r ≥ and c ∈ [0 , is Γ (cid:0) X [ i ] , r, c (cid:1) = Y ( i ) + max (cid:0) X [ i ] (cid:1) r , M c,i M c,i , Y ( i +1) − Y ( i +1) − min (cid:0) X [ i ] (cid:1) r ! with the understanding that the intervals ( a, b ) , ( a, b ] , and [ a, b ) are empty if a ≥ b . Proof:
By definition, Γ (cid:0) X [ i ] , r, c (cid:1) = { x ∈ I i : X [ i ] ⊂ N ( x, r, c ) } . Suppose r ≥ c ∈ [0 , x ∈ ( Y ( i ) , M c,i ], we have X [ i ] ⊂ N ( x, r, c ) iff Y ( i ) + r ( x − Y ( i ) ) > max (cid:0) X [ i ] (cid:1) iff x > Y ( i ) + max ( X [ i ] ) r . Likewise for x ∈ [ M c,i , Y ( i +1) ), we have X [ i ] ⊂ N ( x, r, c ) iff Y ( i +1) − r ( Y ( i +1) − x ) < min (cid:0) X [ i ] (cid:1) iff x < Y ( i +1) − Y ( i +1) − min ( X [ i ] ) r .Therefore Γ (cid:0) X [ i ] , r, c (cid:1) = (cid:16) Y ( i ) + max ( X [ i ] ) r , M c,i i Sh M c,i , Y ( i +1) − Y ( i +1) − min ( X [ i ] ) r (cid:17) . (cid:4) X [ i ] ∩ Γ (cid:0) X [ i ] , r, c (cid:1) = ∅ , we have γ ni, ( F i , r, c ) = 1, hence the name Γ -region and the notationΓ ( · ). For i = 1 , , , . . . , ( m −
1) and n i >
0, we prove that γ ni, ( F i , r, c ) = 1 or 2 with distribution dependentprobabilities. Hence, to find the distribution of γ ni, ( F i , r, c ), it suffices to find P ( γ ni, ( F i , r, c ) = 1) or p ni ( F i , r, c ) := P (cid:0) γ ni, ( F i , r, c ) = 2 (cid:1) . For computational convenience, we employ the latter in our calculations,henceforth. Theorem 2.3.
For i = 1 , , , . . . , ( m − , let the support of F i have a positive Lebesgue measure. Then for n i > , r ∈ (1 , ∞ ) , and c ∈ (0 , , we have γ ni, ( F i , r, c ) ∼ (cid:0) p ni ( F i , r, c ) (cid:1) . Furthermore, γ , ( F i , r, c ) =1 for all r ≥ and c ∈ [0 , ; γ ni, ( F i , r,
0) = γ ni, ( F i , r,
1) = 1 for all n i ≥ and r ≥ ; and γ ni, ( F i , ∞ , c ) = 1 for all n i ≥ and c ∈ [0 , . Proof:
Let X − i := argmin x ∈X [ i ] ∩ ( Y ( i ) ,M c,i ) d ( x, M c,i ) provided that X [ i ] ∩ (cid:0) Y ( i ) , M c,i (cid:1) = ∅ , and X + i :=argmin x ∈X [ i ] ∩ ( M c,i ,Y ( i +1) ) d ( x, M c,i ) provided that X [ i ] ∩ (cid:0) M c,i , Y ( i +1) (cid:1) = ∅ . That is, X − i and X + i are clos-est class X points (if they exist) to M c,i from left and right, respectively. Notice that since n i >
0, at leastone of X − i and X + i exists a.s. If X [ i ] ∩ (cid:0) Y ( i ) , M c,i (cid:1) = ∅ , then X [ i ] ⊂ N (cid:0) X + i , r, c (cid:1) ; so γ ni, ( F i , r, c ) = 1. Sim-ilarly, if X [ i ] ∩ ( M c,i , Y ( i ) ) = ∅ , then X [ i ] ⊂ N (cid:0) X − i , r, c (cid:1) ; so γ ni, ( F i , r, c ) = 1. If both of X [ i ] ∩ (cid:0) Y ( i ) , M c,i (cid:1) and X [ i ] ∩ ( M c,i , Y ( i ) ) are nonempty, then X [ i ] ⊂ N (cid:0) X − i , r, c (cid:1) ∪ N (cid:0) X + i , r, c (cid:1) , so γ ni, ( F i , r, c ) ≤
2. Since n i >
0, we have 1 ≤ γ ni, ( F i , r, c ) ≤
2. The desired result follows, since the probabilities 1 − p ni ( F, r, c )) = P ( γ ni, ( F i , r, c ) = 1) and p ni ( F, r, c )) = P ( γ ni, ( F i , r, c ) = 2) are both positive. The special cases in thetheorem follow by construction. (cid:4) The probability p ni ( F, r, c )) = P (cid:0) X [ i ] ∩ Γ (cid:0) X [ i ] , r, c (cid:1) = ∅ (cid:1) depends on the conditional distribution F X | Y and the interval Γ (cid:0) X [ i ] , r, c (cid:1) , which, if known, will make possible the calculation of p ni ( F i , r, c ). As animmediate result of Lemma 2.1 and Theorem 2.3, we have the following upper bound for γ n,m ( F, r, c ). Theorem 2.4.
Let D n,m ( r, c ) be an F ( R ) -random D n,m ( r, c ) -digraph and k , k , and k be three naturalnumbers defined as k := P m − i =1 I ( n i > , k := P m − i =1 I ( n i = 1) , and k := P i ∈{ ,m } I ( n i > . Then for n ≥ , m ≥ , r ≥ , and c ∈ [0 , , we have ≤ γ n,m ( F, r, c ) ≤ k + k + k ≤ min( n, m ) . Furthermore, γ ,m ( F, r, c ) = 1 for all m ≥ , r ≥ , and c ∈ [0 , ; γ n, ( F, r, c ) = P i ∈{ , } I ( n i > for all n ≥ and r ≥ ; γ , ( F, r, c ) = 1 for all r ≥ ; γ n,m ( F, r,
0) = γ n,m ( F, r,
1) = k + k + k for all m > , n ≥ , and r ≥ ; and γ n,m ( F, ∞ , c ) = k + k + k for all m > , n ≥ , and c ∈ [0 , . Proof:
Suppose n ≥ , m ≥ r ≥
1, and c ∈ [0 , i = 1 , , . . . , ( m − γ ni, ( F i , r, c ) ∈ { , } provided that n i >
1, and γ , ( F i , r, c ) = 1. For i ∈ { , m } , by Lemma 2.1, we have γ ni, ( F i , r, c ) = I ( n i > γ n,m ( F, r, c ) = P mi =0 ( γ ni, ( F i , r, c ) I ( n i > (cid:4) For r = 1, the distribution of γ ni, ( F i , r, c ) is simpler and the distribution of γ n,m ( F i , r, c ) has simpler upperbounds. Theorem 2.5.
Let D n,m (1 , c ) be an F ( R ) -random D n,m (1 , c ) -digraph, k be defined as in Theorem 2.4, and k be a natural number defined as k := P m − i =1 (cid:2) I (cid:0)(cid:12)(cid:12) X [ i ] ∩ (cid:0) Y ( i ) , M c,i (cid:1)(cid:12)(cid:12) > (cid:1) + I (cid:0)(cid:12)(cid:12) X [ i ] ∩ (cid:0) M c,i , Y ( i +1) (cid:1)(cid:12)(cid:12) > (cid:1)(cid:3) .Then for n ≥ , m > , and c ∈ [0 , , we have ≤ γ n,m ( F, , c ) = k + k ≤ min( n, m ) . Proof:
Suppose n ≥ , m >
1, and c ∈ [0 ,
1] and let X − i and X + i be defined as in the proof of The-orem 2.3. Then by construction, X [ i ] ∩ (cid:0) Y ( i ) , M c,i (cid:1) ⊂ N (cid:0) X − i , , c (cid:1) , but N (cid:0) X − i , , c (cid:1) ⊆ (cid:0) Y ( i ) , M c,i (cid:1) . So (cid:2) X [ i ] ∩ (cid:0) M c,i , Y ( i +1) (cid:1)(cid:3) ∩ N (cid:0) X − i , , c (cid:1) = ∅ . Similarly X [ i ] ∩ (cid:0) M c,i , Y ( i +1) (cid:1) ⊂ N (cid:0) X + i , , c (cid:1) and (cid:2) X [ i ] ∩ (cid:0) Y ( i ) , M c,i (cid:1)(cid:3) ∩ N (cid:0) X + i , , c (cid:1) = ∅ . Then γ ni, ( F i , , c ) = 1, if X [ i ] ⊂ (cid:0) Y ( i ) , M c,i (cid:1) or X [ i ] ⊂ (cid:0) M c,i , Y ( i +1) (cid:1) , and γ n,m ( F, , c ) = 2,if X [ i ] ∩ (cid:0) Y ( i ) , M c,i (cid:1) = ∅ and X [ i ] ∩ (cid:0) M c,i , Y ( i +1) (cid:1) = ∅ . Hence for i = 1 , , , . . . , ( m − γ n,m ( F, , c ) = I (cid:0)(cid:12)(cid:12) X [ i ] ∩ (cid:0) Y ( i ) , M c,i (cid:1)(cid:12)(cid:12) > (cid:1) + I (cid:0)(cid:12)(cid:12) X [ i ] ∩ (cid:0) M c,i , Y ( i +1) (cid:1)(cid:12)(cid:12) > (cid:1) , and for i ∈ { , m } , we have γ ni, ( F i , , c ) = I ( n i > γ n,m ( F, , c ) = P mi =0 ( γ ni, ( F i , , c ) I ( n i > (cid:4) P ( γ ni, ( F, , c ) = 1) = P ( X [ i ] ⊂ (cid:0) Y ( i ) , M c,i (cid:1) ) + P ( X [ i ] ⊂ (cid:0) M c,i , Y ( i +1) (cid:1) )and P ( γ ni, ( F, , c ) = 2) = P ( X [ i ] ∩ (cid:0) Y ( i ) , M c,i (cid:1) = ∅ , X [ i ] ∩ (cid:0) M c,i , Y ( i +1) (cid:1) = ∅ ). In the special case of fixed Y = { y , y } with −∞ < y < y < ∞ and X n = { X , X , . . . , X n } a random samplefrom U ( y , y ), the uniform distribution on ( y , y ), we have a D n, ( r, c )-digraph for which F X = U ( y , y ).We call such digraphs as U ( y , y ) -random D n, ( r, c ) -digraphs and provide the exact distributions of theirdomination number for the whole range of r and c . Let γ n, ( U , r, c ) be the domination number of the PCDbased on N ( · , r, c ) and X n and p n ( U , r, c ) := P ( γ n, ( U , r, c ) = 2), and p ( U , r, c ) := lim n →∞ p n ( U , r, c ). Wepresent a “scale invariance” result for N ( · , r, c ). This invariance property will simplify the notation andcalculations in our subsequent analysis by allowing us to consider the special case of the unit interval, (0 , Proposition 3.1. (Scale Invariance Property) Suppose X n is a random sample (i.e., a set of iid randomvariables) from U ( y , y ) . Then for any r ∈ [1 , ∞ ] the distribution of γ n, ( U , r, c ) is independent of Y andhence the support interval ( y , y ) . Proof:
Let X n be a random sample from U ( y , y ) distribution. Any U ( y , y ) random variable can betransformed into a U (0 ,
1) random variable by the transformation φ ( x ) = ( x − y ) / ( y − y ), which mapsintervals ( t , t ) ⊆ ( y , y ) to intervals (cid:0) φ ( t ) , φ ( t ) (cid:1) ⊆ (0 , X ∼ U ( y , y ), then we have φ ( X ) ∼ U (0 ,
1) and P ( X ∈ ( t , t )) = P ( φ ( X ) ∈ (cid:0) φ ( t ) , φ ( t ) (cid:1) for all ( t , t ) ⊆ ( y , y ). So, without loss ofgenerality, we can assume X n is a random sample from the U (0 ,
1) distribution. Therefore, the distribution of γ n, ( U , r, c ) does not depend on the support interval ( y , y ). (cid:4) Note that scale invariance of γ n, ( F, ∞ , c ) follows trivially for all X n from any F with support in ( y , y ),since for r = ∞ , we have γ n, ( F, ∞ , c ) = 1 a.s. for all n > c ∈ (0 , γ , ( F, r, c )holds for n = 1 for all r ≥ c ∈ [0 , γ n, ( F, r, c ) with c ∈ { , } holds for n ≥ r ≥
1, as well. Based on Proposition 3.1, for uniform data, we may assume that ( y , y ) is the unit interval(0 ,
1) for N ( · , r, c ) with general c . Then the proportional-edge proximity region for x ∈ (0 ,
1) with parameters r ≥ c ∈ [0 ,
1] becomes N ( x, r, c ) = ( (0 , r x ) ∩ (0 ,
1) if x ∈ (0 , c ),(1 − r (1 − x ) , ∩ (0 ,
1) if x ∈ ( c, N ( c, r, c ) is arbitrarily taken to be one of (0 , r x ) ∩ (0 ,
1) or (1 − r (1 − x ) , ∩ (0 , N (0 , r, c ) := { } and N (1 , r, c ) := { } for all r ≥ c ∈ [0 , X i iid ∼ U ( y , y ), the special cases in theconstruction of N ( · , r, c ) — X falls at c or the end points of ( y , y ) — occur with probability zero. Moreover,the region N ( x, r, c ) is an interval a.s.The Γ -region, Γ ( X n , r, c ), depends on X (1) , X ( n ) , r , and c . If Γ ( X n , r, c ) = ∅ , then we have Γ ( X n , r, c ) =( δ , δ ) where at least one end points δ , δ is a function of X (1) and X ( n ) . For U (0 ,
1) data, given X (1) = x and X ( n ) = x n , the probability of p n ( U , r, c ) is (1 − ( δ − δ ) / ( x n − x )) ( n − provided that Γ ( X n , r, c ) = ∅ ;and if Γ ( X n , r, c ) = ∅ , then γ n, ( U , r, c ) = 2 holds. Then P ( γ n, ( U , r, c ) = 2 , Γ ( X n , r, c ) = ∅ ) = Z Z S f n ( x , x n ) (cid:18) − δ − δ x n − x (cid:19) ( n − dx n dx (4)where S = { < x < x n < x , x n Γ ( X n , r, c ) and Γ ( X n , r, c ) = ∅} and f n ( x , x n ) = n ( n − x n − x ] ( n − I (0 < x < x n < P ( γ n, ( U , r, c ) = 2 , Γ ( X n , r, c ) = ∅ ) = Z Z S n ( n − x n − x + δ − δ ] ( n − dx n dx . (5)6f Γ ( X n , r, c ) = ∅ , then γ n, ( U , r, c ) = 2. So P ( γ n, ( U , r, c ) = 2 , Γ ( X n , r, c ) = ∅ ) = Z Z S f n ( x , x n ) dx n dx (6)where S = { < x < x n < ( X n , r, c ) = ∅} .The probability p n ( U , r, c ) is the sum of the probabilities in Equations (5) and (6). U ( y , y ) -random D n, (2 , / -digraphs For r = 2 and c = 1 /
2, we have N ( x, , /
2) = B ( x, r ( x )) where r ( x ) = min( x, − x ) for x ∈ (0 , N ( x, , /
2) is equivalent to the CCCD of Priebe et al. (2001). Moreover,Γ ( X n , , /
2) = (cid:0) X ( n ) / , (cid:0) X (1) (cid:1) / (cid:1) . It has been shown that p n ( U , , /
2) = 4 / − (16 /
9) 4 − n (Priebe et al.(2001)). Hence, for U ( y , y ) data with n ≥
1, we have γ n, ( U , , /
2) = (cid:26) / /
9) 4 − n , / − (16 /
9) 4 − n , (7)where w.p. stands for “with probability”. Then as n → ∞ , γ n, ( U , , /
2) converges in distribution to1 + BER(4 / m >
2, Priebe et al. (2001) computed the exact distribution of γ n,m ( U , , /
2) also.However, the scale invariance property does not hold for general F ; that is, for X i iid ∼ F with support S ( F ) ⊆ ( y , y ), the exact and asymptotic distribution of γ n, ( F, , /
2) depends on F and Y (Ceyhan(2008)). U ( y , y ) -random D n, (2 , c ) -digraphs For r = 2, c ∈ (0 , y , y ) = (0 , -region is Γ ( X n , , c ) = ( X ( n ) / , c ] ∪ [ c, (1 + X (1) ) / X ( n ) / , c ] or [ c, (1 + X (1) ) /
2) could be empty, but not simultaneously.
Theorem 3.2.
For U ( y , y ) data and n ≥ , we have γ n, ( U , , c ) ∼ p n ( U , , c )) where p n ( U , , c ) = ν ,n ( c ) I ( c ∈ (0 , /
3] + ν ,n ( c ) I ( c ∈ (1 / , /
2] + ν ,n ( c ) I ( c ∈ (1 / , /
3] + ν ,n ( c ) I ( c ∈ (2 / , with ν ,n ( c ) = 23 (cid:18) c + 12 (cid:19) n −
89 4 − n − (cid:18) − c (cid:19) n + 19 (1 − c ) n − (cid:18) c − (cid:19) n ,ν ,n ( c ) = 23 (cid:18) c + 12 (cid:19) n −
89 4 − n − (cid:18) − c (cid:19) n − (cid:18) c − (cid:19) n − (cid:18) c − (cid:19) n ,ν ,n ( c ) = ν ,n (1 − c ) , and ν ,n ( c ) = ν ,n (1 − c ) . Furthermore, γ n, ( U , ,
0) = γ n, ( U , ,
1) = 1 for all n ≥ . Observe that the parameter p n ( U , , c ) is continuous in c ∈ (0 ,
1) for fixed n < ∞ , but there are jumps(hence discontinuities) in p n ( U , , c ) at c ∈ { , } . In particular, lim c → p n ( U , , c ) = lim c → p n ( U , , c ) =lim c → ν ,n ( c ) = lim c → ν ,n ( c ) = − ( − n − − n , but p n ( U , ,
0) = p n ( U , ,
1) = 0 for all n ≥
1. For c = 1 /
2, we have p n ( U , , c ) = 4 / − (16 /
9) 4 − n , hence the distribution of γ n, ( U , , c = 1 /
2) is same as inEquation (7).In the limit as n → ∞ , for c ∈ [0 , γ n, ( U , , c ) ∼ (cid:26) / , for c = 1 / , for c = 1 / γ n, ( U , , c ) around c = 1 /
2. Theparameter p ( U , , c ) is continuous in c ∈ [0 , \ { / } (in fact it is unity), but there is a jump (hence disconti-nuity) in p ( U , , c ) at c = 1 /
2, since p ( U , , /
2) = 4 / p ( U , , c ) = 0 for c = 1 /
2. Hence for c = 1 /
2, theasymptotic distribution is non-degenerate, and for c = 1 /
2, the asymptotic distribution is degenerate. That is,for c = 1 / ± ε with ε > U ( y , y ) -random D n, ( r, / -digraphs For r ≥ c = 1 /
2, and ( y , y ) = (0 , -region is Γ ( X n , r, /
2) = ( X ( n ) /r, / ∪ [1 / , ( r − X (1) ) /r )where ( X ( n ) /r, /
2] or [1 / , ( r − X (1) ) /r ) could be empty, but not simultaneously. Theorem 3.3.
For U ( y , y ) data with n ≥ , we have γ n, ( U , r, / ∼ p n ( U , r, / where p n ( U , r, /
2) = r ( r +1) (cid:16)(cid:0) r (cid:1) n − − (cid:0) r − r (cid:1) n − (cid:17) for r ≥ , − r n − (2 r ) n − ( r +1) + ( r − n ( r +1) (cid:16) − (cid:0) r − r (cid:1) n − (cid:17) for ≤ r < . Notice that for fixed n < ∞ , the parameter p n ( U , r, /
2) is continuous in r ≥
2. In particular, for r = 2, wehave p n ( U , , /
2) = 4 / − (16 /
9) 4 − n , hence the distribution of γ n, ( U , r = 2 , /
2) is same as in Equation (7).Furthermore, lim r → p n ( U , r, /
2) = p n ( U , , /
2) = 1 − − n and lim r →∞ p n ( U , r, /
2) = p n ( U , ∞ , /
2) = 0.In the limit, as n → ∞ , we have γ n, ( U , r, / ∼ r > /
9) for r = 2,2 for 1 ≤ r < γ n, ( U , r, /
2) around r = 2. The parameter p ( U , r, /
2) is continuous (in fact piecewise constant) for r ∈ [1 , ∞ ) \ { } . Hence for r = 2, the asymptoticdistribution is degenerate, as p ( U , r, /
2) = 1 for r > p ( U , r, /
2) = 2 w.p. 1 for r <
2. That is, for r = 2 ± ε with ε > U ( y , y ) -random D n, ( r, c ) -digraphs For r ≥ c ∈ (0 , -region is Γ ( X n , r, c ) = ( X ( n ) /r, c ] ∪ [ c, ( r − X (1) ) /r ) where ( X ( n ) /r, c ] or[ c, ( r − X (1) ) /r ) could be empty, but not simultaneously. Theorem 3.4.
Main Result 1:
For U ( y , y ) data with n ≥ , r ≥ , and c ∈ ((3 − √ / , / , we have γ n, ( U , r, c ) ∼ p n ( U , r, c )) where p n ( U , r, c ) = π ,n ( r, c ) I ( r ≥ /c ) + π ,n ( r, c ) I (1 / (1 − c ) ≤ r < /c ) + π ,n ( r, c ) I ((1 − c ) /c ≤ r < / (1 − c )) + π ,n ( r, c ) I (1 ≤ r < (1 − c ) /c ) with π ,n ( r, c ) = 2 r ( r + 1) (cid:18) r (cid:19) n − − (cid:18) r − r (cid:19) n − ! ,π ,n ( r, c ) = 1( r + 1) r n − (cid:20) (1 + c r ) n − (1 − c ) n − r + 1 ( c r − r + c r + 1) n − ( r − n − r + 1 (cid:18) r n − + ( c r − c ) n (cid:19)(cid:21) , ,n ( r, c ) = 1+ ( r − n − ( r + 1) (cid:20) ( r − − r n − (( c r − c ) n + ( r − c r − c ) n ) (cid:21) − r + 1 [ c n +(1 − c ) n ] (cid:18) r n + 1 r n − (cid:19) , and π ,n ( r, c ) = 1 + ( r − n − ( r + 1) (1 − c r − c ) n r − (cid:18) − r (cid:19) n − ( r − ! + ( r − n ( r + 1) (cid:18) − r (cid:18) r − c r − cr (cid:19) n (cid:19) − r + 1 [ c n + (1 − c ) n ] (cid:18) r n − r n − (cid:19) . And for c ∈ (0 , (3 − √ / , we have p n ( U , r, c ) = ϑ ,n ( r, c ) I ( r ≥ /c ) + ϑ ,n ( r, c ) I ((1 − c ) /c ≤ r < /c ) + ϑ ,n ( r, c ) I (1 / (1 − c ) ≤ r < (1 − c ) /c ) + ϑ ,n ( r, c ) I (1 ≤ r < / (1 − c )) where ϑ ,n ( r, c ) = π ,n ( r, c ) , ϑ ,n ( r, c ) = π ,n ( r, c ) , ϑ ,n ( r, c ) = π ,n ( r, c ) , and ϑ ,n ( r, c ) = r ( r + 1) " ( r − n − (1 − c r − c ) n (cid:18) r + ( r − (cid:18) − r (cid:19) n (cid:19) − (cid:18) r − r (cid:19) n − − (cid:18) c r − c + c r + 1 r (cid:19) n + r + 1 r n [(1 + c r ) n − (1 − c ) n ] . Furthermore, we have γ n, ( U , r,
0) = γ n, ( U , r,
1) = 1 for all n ≥ . Some remarks are in order for the Main Result 1. The partitioning of c ∈ (0 , /
2) as c ∈ (0 , (3 − √ / c ∈ ((3 − √ / , /
2) is due to the relative positions of 1 / (1 − c ) and (1 − c ) /c . For c ∈ ((3 − √ / , / / (1 − c ) > (1 − c ) /c and for c ∈ (0 , (3 − √ / / (1 − c ) < (1 − c ) /c . At c = (3 − √ / / (1 − c ) = (1 − c ) /c = ( √ / π ,n ( r, (3 − √ /
2) = ϑ ,n ( r, (3 − √ / π ,n ( r, (3 − √ /
2) = ϑ ,n ( r, (3 − √ / π ,n ( r, (3 − √ /
2) = ϑ ,n ( r, (3 − √ /
2) terms survive. Also, notice the ( − n termsin π ,n ( r, c ) and ϑ ,n ( r, c ) which might suggest fluctuations of these probabilities as n changes (increases).However, as n increases, π ,n ( r, c ) strictly increases towards 1 (see Figure 1), and ϑ ,n ( r, c ) decreases (strictlydecreases for n ≥
3) towards 0 (see Figure 2). . . . . . . n p ( r , c ) Figure 1: The probability π ,n ( r, c ) in Main Result 1 with r = 1 . c = 0 . n = 2 , , . . . , Remark . By symmetry, in Theorem 3.4, for c ∈ (1 / , ( √ − / p n ( U , r, c ) = π ,n ( r, − c ) I ( r ≥ / (1 − c )) + π ,n ( r, − c ) I (1 /c ≤ r < / (1 − c )) + π ,n ( r, − c ) I ( c/ (1 − c ) ≤ r < /c ) + π ,n ( r, − c ) I (1 ≤ r 10 15 20 25 . . . . . . n P S f r ag r e p l a ce m e n t s ϑ , n ( r , c ) Figure 2: The probability ϑ ,n ( r, c ) in Main Result 1 with r = 2 and c = 0 . n = 2 , , . . . , r → p n ( U , r, c ) = lim r → π ,n ( r, c ) = 1 as expected. For fixed 1 < n < ∞ , the probability p n ( U , r, c ) is continuous in ( r, c ) ∈ { ( r, c ) ∈ R : r ≥ , < c < } . In particular, for c ∈ ((3 − √ / , / r, c ) → (2 , / 2) in { ( r, c ) ∈ R : r ≥ /c } , p n ( U , r, c ) = π ,n ( r, c ) → / − (16 / 9) 4 − n ; as ( r, c ) → (2 , / { ( r, c ) ∈ R : 1 / (1 − c ) ≤ r < /c } , p n ( U , r, c ) = π ,n ( r, c ) → / − (16 / 9) 4 − n ; and as ( r, c ) → (2 , / { ( r, c ) ∈ R : (1 − c ) /c ≤ r < / (1 − c ) } , p n ( U , r, c ) = π ,n ( r, c ) → / − (16 / 9) 4 − n . The limit ( r, c ) → (2 , / 2) is not possible for { ( r, c ) ∈ R : 1 ≤ r < (1 − c ) /c } . For c ∈ (0 , (3 − √ / r, c ) → (2 , / 2) cannot occur either. And for ( r, c ) = (2 , / γ n, ( U , r, c ) is 1 + BER( p n ( U , , / p n ( U , , / 2) = 4 / − (16 / 9) 4 − n as in Equation (7). Therefore for fixed 1 < n < ∞ , as ( r, c ) → (2 , / 2) in S = { ( r, c ) ∈ R : r ≥ , < c < / } , we have p n ( U , r, c ) → / − (16 / 9) 4 − n . Hence as ( r, c ) → (2 , / 2) in S , γ n, ( U , r, c ) converges in distribution to γ n, ( U , , / p n ( U , r, c ) has jumps (hence discontinuities)at c ∈ { , } . As c → + (which implies we should consider c ∈ (0 , (3 − √ / /c → ∞ , (1 − c ) /c → ∞ , and1 / (1 − c ) → + . Hence lim c → + ϑ ,n ( r, c ) = ϑ ,n ( ∞ , 0) = 0, lim c → + ϑ ,n ( r, c ) = ϑ ,n ( ∞ , 0) = 0. Moreover,lim c → + ϑ ,n ( r, c ) = r ( r − n − ( r +1) [ r + ( − r ) − n (1 − r ) − r − n ]; lim c → + ϑ ,n ( r, c ) = 1 + r +1 [( r − n (1 − ( − r ) − n ) − r n − r − n ]. But p n ( U , r, 0) = 0 for all r ≥ 1. Similar results can be obtained as c → − . Observe also thatlim r → p n ( U , r, c ) = lim r → π ,n ( r, c ) = 1. Theorem 3.6. Main Result 2: Let D n, ( r, c ) be based on U ( y , y ) data with c ∈ (0 , and τ = max ( c, − c ) .Then the domination number γ n, ( U , r, c ) of the PCD has the following asymptotic distribution. As n → ∞ , γ n, ( U , r, c ) ∼ r/ ( r + 1)) , for r = 1 /τ , , for r > /τ , , for ≤ r < /τ . (8)Notice the interesting behavior of the asymptotic distribution of γ n, ( U , r, c ) around r = 1 /τ for any given c ∈ (0 , r = 1 /τ . For r > /τ , lim n →∞ γ n, ( U , r, c )) =1 w.p. 1, and for 1 ≤ r < /τ , lim n →∞ γ n, ( U , r, / r = 1 /τ correspondsto c = ( r − /r , if c ∈ (0 , / 2) (i.e., τ = 1 − c ) and c = 1 /r , if c ∈ (1 / , 1) (i.e., τ = c ) and these are onlypossible for r ∈ (1 , r = (1 /τ ) ± ε for ε arbitrarily small, although the exact distribution isnon-degenerate, the asymptotic distribution is degenerate. The parameter p ( U , r, c ) is continuous in r and c for ( r, c ) ∈ S \ { /τ, c } and there is a jump (hence discontinuity) in the probability p ( U , r, c ) at r = 1 /τ , since p ( U , /τ, c ) = 1 / (1 + τ ) = r/ ( r + 1). Therefore, given a centrality parameter c ∈ (0 , r for which the asymptotic distribution is non-degenerate, and vice versa.There is yet another interesting behavior of the asymptotic distribution around ( r, c ) = (2 , / p ( U , r, c ) has jumps at c = 1 /r and ( r − /r for r ∈ [1 , 2] with p ( U , r, /r ) = p ( U , r, ( r − /r ) = r/ ( r + 1). That is, for fixed ( r, c ) ∈ S , lim n →∞ p n ( U , r, ( r − /r ) = lim n →∞ p n ( U , r, /r ) = r/ ( r + 1). Letting10 r, c ) → (2 , / 2) (i.e., r → 2) we get p ( U , r, ( r − /r ) → / p ( U , r, /r ) → / 3, but p ( U , , / 2) = 4 / r ∈ [1 , 2) the distributions of γ n, ( U , r, ( r − /r ) and γ n, ( U , r, /r ) are identical and both convergeto 1 + BER( r/ ( r + 1)), but the distribution of γ n, ( U , , / 2) converges to 1 + BER(4 / 9) as n → ∞ . In otherwords, p ( U , r, ( r − /r ) = p ( U , r, /r ) has another jump at r = 2 (which corresponds to ( r, c ) = (2 , / c = 1 / 2. Because for c ∈ (0 , / r = 1 / (1 − c ),for sufficiently large n , a point X i in ( c, 1) can dominate all the points in X n (implying γ n, ( U , r, ( r − /r ) = 1),but no point in (0 , c ) can dominate all points a.s. Likewise, for c ∈ (1 / , 1) with r = 1 /c , for sufficiently large n , a point X i in (0 , c ) can dominate all the points in X n (implying γ n, ( U , r, /r ) = 1), but no point in ( c, c = 1 / r = 2, for sufficiently large n , points to the left or rightof c can dominate all other points in X n . F ( R ) -random D n, ( r, c ) -digraphs Let F ( y , y ) be a family of continuous distributions with support in S F ⊆ ( y , y ). Consider a distributionfunction F ∈ F ( y , y ). For simplicity, assume y = 0 and y = 1. Let X n be a random sample from F , Γ ( X n , r, c ) = ( δ , δ ), p n ( F, r, c ) := P ( γ n, ( F, r, c ) = 2), and p ( F, r, c ) := lim n →∞ P ( γ n, ( F, r, c ) = 2).The exact (i.e., finite sample) and asymptotic distributions of γ n, ( F, r, c ) are 1 + BER ( p n ( F, r, c )) and 1 +BER ( p ( F, r, c )), respectively. That is, for finite n > r ∈ [1 , ∞ ), and c ∈ (0 , γ n, ( F, r, c ) = (cid:26) − p n ( F, r, c ) , p n ( F, r, c ) . (9)Moreover, γ , ( F, r, c ) = 1 for all r ≥ c ∈ [0 , γ n, ( F, r, 0) = γ , ( F, r, 1) = 1 for all n ≥ r ≥ γ n, ( F, ∞ , c ) = 1 for all n ≥ c ∈ [0 , γ n, ( F, , c ) = k for all n ≥ c ∈ (0 , k is as in Theorem 2.5 with m = 2. The asymptotic distribution is similar with p n ( F, r, c ) beingreplaced by p ( F, r, c ). The special cases are similar in the asymptotics with the exception that p ( F, , c ) =1 for all c ∈ (0 , γ n, ( F, r, c ) are given by 1 + p n ( F, r, c ) and p n ( F, r, c ) (1 − p n ( F, r, c )), respectively; and the asymptotic mean and variance of γ n, ( F, r, c ) are given by1 + p ( F, r, c ) and p ( F, r, c ) (1 − p ( F, r, c )), respectively.Given X (1) = x and X ( n ) = x n , the probability of γ n, ( F, r, c ) = 2 (i.e., p n ( F, r, c )) is (1 − [ F ( δ ) − F ( δ )] / [ F ( x n ) − F ( x )]) ( n − provided that Γ ( X n , r, c ) = ( δ , δ ) = ∅ ; if Γ ( X n , r, c ) = ∅ , then γ n, ( F, r, c ) = 2.Then P ( γ n, ( F, r, c ) = 2 , Γ ( X n , r, c ) = ∅ ) = Z Z S f n ( x , x n ) (cid:18) − F ( δ ) − F ( δ ) F ( x n ) − F ( x ) (cid:19) ( n − dx n dx (10)where S = { < x < x n < x , x n ) Γ ( X n , r, c ) , Γ ( X n , r, c ) = ∅} and f n ( x , x n ) = n ( n − f ( x ) f ( x n ) (cid:2) F ( x n ) − F ( x ) (cid:3) ( n − I (0 < x < x n < 1) which is the joint pdf of X (1) , X ( n ) . The integralin (10) becomes P ( γ n, ( F, r, c ) = 2 , Γ ( X n , r, c ) = ∅ ) = Z Z S H ( x , x n ) dx n dx , (11)where H ( x , x n ) := n ( n − f ( x ) f ( x n ) (cid:2) F ( x n ) + F ( δ ) − ( F ( δ ) + F ( x )) (cid:3) n − . (12)If Γ ( X n , r, c ) = ∅ , then γ n, ( F, r, c ) = 2. So P ( γ n, ( F, r, c ) = 2 , Γ ( X n , r, c ) = ∅ ) = Z Z S f n ( x , x n ) dx n dx (13)where S = { < x < x n < ( X n , r, c ) = ∅} . 11he probability p n ( F, r, c ) is the sum of the probabilities in Equations (11) and (13).For Y = { y , y } ⊂ R with −∞ < y < y < ∞ , a quick investigation shows that the Γ -region isΓ ( X n , r, c ) = ( y + ( X ( n ) − y ) /r, M c ] ∪ [ M c , y − ( y − X (1) ) /r ). Notice that for a given c ∈ [0 , M c ∈ [ y , y ] is M c = y + c ( y − y ).Let F be a continuous distribution with support S ( F ) ⊆ (0 , U (0 , γ n, ( F, r, c ) with ( r, c ) = (1 , / X ∼ F , then byprobability integral transform, F ( X ) ∼ U (0 , F , we can construct a proximity mapdepending on F for which the distribution of the domination number of the associated digraph has the samedistribution as that of γ n, ( U , r, c ). Proposition 4.1. Let X i iid ∼ F which is an absolutely continuous distribution with support S ( F ) = (0 , and X n = { X , X , . . . , X n } . Define the proximity map N F ( x, r, c ) := F − ( N ( F ( x ) , r, c )) . Then the dominationnumber of the digraph based on N F , X n , and Y = { , } has the same distribution as γ n, ( U , r, c ) . Proof: Let U i := F ( X i ) for i = 1 , , . . . , n and U n := { U , U , . . . , U n } . Hence, by probability integraltransform, U i iid ∼ U (0 , U ( i ) be the i th order statistic of U n for i = 1 , , . . . , n . Furthermore, anabsolutely continuous F preserves order; that is, for x ≤ y , we have F ( x ) ≤ F ( y ). So the image of N F ( x, r, c )under F is F ( N F ( x, r, c )) = N ( F ( x ) , r, c ) for (almost) all x ∈ (0 , F ( N F ( X i , r, c )) = N ( F ( X i ) , r, c ) = N ( U i , r, c ) for i = 1 , , . . . , n . Since U i iid ∼ U (0 , N ( · , r, c ), U n , and { , } is given in Theorem 3.4. Observe that for any j , X j ∈ N F ( X i , r, c ) iff X j ∈ F − ( N ( F ( X i ) , r, c )) iff F ( X j ) ∈ N ( F ( X i ) , r, c ) iff U j ∈ N ( U i , r, c ) for i = 1 , , . . . , n . Hence P ( X n ⊂ N F ( X i , r, c )) = P ( U n ⊂ N ( U i , r, c ) for all i = 1 , , . . . , n . Therefore, X n ∩ Γ ( X n , N F ( r, c )) = ∅ iff U n ∩ Γ ( U n , r, c ) = ∅ . Hence the desired result follows. (cid:4) There is also a stochastic ordering between γ n, ( F, r, c ) and γ n, ( U , r, c ) provided that F satisfies someregularity conditions. Proposition 4.2. Let X n = { X , X , . . . , X n } be a random sample from an absolutely continuous distribution F with S ( F ) ⊆ (0 , . If F (cid:0) X ( n ) /r (cid:1) < F (cid:0) X ( n ) (cid:1) /r and F (cid:0) X (1) (cid:1) < r F (cid:0)(cid:0) X (1) + r − (cid:1) /r (cid:1) + 1 − r hold a.s., (14) then γ n, ( F, r, c ) < ST γ n, ( U , r, F ( c )) where < ST stands for “stochastically smaller than”. If < ’s in (14) arereplaced with > ’s, then γ n, ( U , r, F ( c )) < ST γ n, ( F, r, c ) . If < ’s in expression (14) are replaced with = ’s, then γ n, ( F, r, c ) d = γ n, ( U , r, F ( c )) where d = stands for equality in distribution. Proof: Let U i and U ( i ) be as in proof of Proposition 4.1. Then the parameter c for N ( · , r, c ) with X n in (0 , 1) corresponds to F ( c ) for U n . Then the Γ -region for U n based on N ( · , r, F ( c )) is Γ ( U n , r, F ( c )) =( U ( n ) /r, F ( c )] ∪ [ F ( c ) , (cid:0) U (1) + r − (cid:1) /r ); likewise, Γ ( X n , r, c ) = ( X ( n ) /r, M c ] ∪ [ M c , (cid:0) X (1) + r − (cid:1) /r ). Butthe conditions in (14) imply that Γ ( U n , r, F ( c )) ( F (Γ ( X n , r, c )), since such an F preserves order. So U n ∩ F (Γ ( X n , r, c )) = ∅ implies that U n ∩ Γ ( U n , r, F ( c )) = ∅ and U n ∩ F (Γ ( X n , r, c )) = ∅ iff X n ∩ Γ ( X n , r, F ( c )) = ∅ . Hence p n ( F, r, c ) = P ( X n ∩ Γ ( X n , r, c ) = ∅ ) < P ( U n ∩ Γ ( U n , r, F ( c )) = ∅ ) = p n ( U , r, F ( c )) . Then γ n, ( F, r, c ) < ST γ n, ( U , r, F ( c )) follows. The other cases follow similarly. (cid:4) Remark . We can also find the exact distribution of γ n, ( F, r, c ) for F whose pdf is piecewise constant withsupport in (0 , 1) as in Ceyhan (2008). Note that the simplest of such distributions is the uniform distribution U (0 , γ n, ( F, r, c ) for (piecewise) polynomial f ( x ) with at least one piece is ofdegree 1 or higher and support in (0 , 1) can be obtained using the multinomial expansion of the term ( · ) n − inEquation (12) with careful bookkeeping. However, the resulting expression for p n ( F, r, c ) is extremely lengthyand not that informative (see Ceyhan (2008)). 12or fixed n , one can obtain p n ( F, r, c ) for F (omitted for the sake of brevity) by numerical integration ofthe below expression. p n ( F, r, c ) = P (cid:0) γ n, ( F, r, c ) = 2 (cid:1) = Z Z S ( F ) \ ( δ ,δ ) H ( x , x n ) dx n dx , where H ( x , x n ) is given in Equation (12). (cid:3) Recall the F ( R d )-random D n,m ( r, c )-digraphs. We call the digraph which obtains in the special case of Y = { y , y } and support of F X in ( y , y ), F ( y , y ) -random D n, ( r, c ) -digraph . Below, we provide asymptoticresults pertaining to the distribution of such digraphs. F ( y , y ) -random D n, ( r, c ) -digraphs Although the exact distribution of γ n, ( F, r, c ) is not analytically available in a simple closed form for F whosedensity is not piecewise constant, the asymptotic distribution of γ n, ( F, r, c ) is available for larger familiesof distributions. First, we present the asymptotic distribution of γ n, ( F, r, c ) for D n, ( r, c )-digraphs with Y = { y , y } ⊂ R with y < y for general F with support S ( F ) ⊆ ( y , y ). Then we will extend this to thecase with Y m ⊂ R with m > c ∈ (0 , / 2) and r ∈ (1 , c = ( r − /r , i.e., M c = y + ( r − y − y ) /r , we define thefamily of distributions F (cid:0) y , y (cid:1) := n F : ( y , y + ε ) ∪ (cid:0) M c , M c + ε (cid:1) ⊆ S ( F ) ⊆ ( y , y ) for some ε ∈ (0 , c ) with c = ( r − /r o . Similarly, let c ∈ (1 / , 1) and r ∈ (1 , c = 1 /r , i.e., M c = y + ( y − y ) /r with r ∈ (1 , F (cid:0) y , y (cid:1) := n F : ( y − ε, y ) ∪ (cid:0) M c − ε, M c (cid:1) ⊆ S ( F ) ⊆ ( y , y ) for some ε ∈ (0 , − c ) with c = 1 /r o . Let the k th order right (directed) derivative at x be defined as f ( k ) ( x + ) := lim h → + f ( k − ( x + h ) − f ( k − ( x ) h for all k ≥ u be defined as f ( u + ) := lim h → + f ( u + h ). The left derivatives and limitsare defined similarly with +’s being replaced by − ’s. Theorem 4.4. Main Result 3: Let Y = { y , y } ⊂ R with −∞ < y < y < ∞ , X n = { X , X , . . . , X n } with X i iid ∼ F ∈ F ( y , y ) , and c ∈ (0 , / . Let D n, be the F ( y , y ) -random D n, ( r, c ) -digraph based on X n and Y .(i) Then for n > , r ∈ (1 , ∞ ) , and c = ( r − /r we have γ n, ( F, r, ( r − /r ) ∼ (cid:0) p n ( F, r, ( r − /r ) (cid:1) .Note also that γ , ( F, r, ( r − /r ) = 1 for all r ≥ ; for r = 1 , we have γ n, ( F, , 0) = 1 for all n ≥ and for r = ∞ , we have γ n, ( F, ∞ , 1) = 1 for all n ≥ .(ii) Furthermore, suppose k ≥ is the smallest integer for which F ( · ) has continuous right derivatives up toorder ( k + 1) at y , y + ( r − y − y ) /r , and f ( k ) ( y +1 ) + r − ( k +1) f ( k ) (cid:16) (( r − y − y ) /r ) + (cid:17) = 0 and f ( i ) ( y +1 ) = f ( i ) (cid:16) ( y + ( r − y − y ) /r ) + (cid:17) = 0 for all i = 0 , , , . . . , ( k − and suppose also that F ( · ) has a continuous left derivative at y . Then for bounded f ( k ) ( · ) , c = ( r − /r , and r ∈ (1 , , we havethe following limit p ( F, r, ( r − /r ) = lim n →∞ p n ( F, r, ( r − /r ) = f ( k ) ( y +1 ) f ( k ) ( y +1 ) + r − ( k +1) f ( k ) (cid:16) ( y + ( r − y − y ) /r ) + (cid:17) . • with ( y , y ) = (0 , p ( F, r, ( r − /r ) = f ( k ) (0 + ) f ( k ) (0 + )+ r − ( k +1) f ( k ) ( (( r − /r ) + ) , • if f ( k ) ( y +1 ) = 0 and f ( k ) (cid:16) ( y + ( r − y − y ) /r ) + (cid:17) = 0, then p n ( F, r, ( r − /r ) → n → ∞ , atrate O (cid:0) κ ( f ) · n − ( k +2) / ( k +1) (cid:1) where κ ( f ) is a constant depending on f and • if f ( k ) ( y +1 ) = 0 and f ( k ) (cid:16) ( y + ( r − y − y ) /r ) + (cid:17) = 0, then p n ( F, r, ( r − /r ) → n → ∞ , atrate O (cid:0) κ ( f ) · n − ( k +2) / ( k +1) (cid:1) . Theorem 4.5. Main Result 4: Let Y = { y , y } ⊂ R with −∞ < y < y < ∞ , X n = { X , X , . . . , X n } with X i iid ∼ F ∈ F ( y , y ) , and c ∈ (1 / , . Let D n, be the F ( y , y ) -random D n, ( r, c ) -digraph based on X n and Y .(i) Then for n > , r ∈ (1 , ∞ ) , and c = 1 /r we have γ n, ( F, r, /r ) ∼ (cid:0) p n ( F, r, /r ) (cid:1) . Note alsothat γ , ( F, r, /r ) = 1 for all r ≥ ; for r = 1 , we have γ n, ( F, , 1) = 1 for all n ≥ and for r = ∞ , wehave γ n, ( F, ∞ , 0) = 1 for all n ≥ .(ii) Furthermore, suppose ℓ ≥ is the smallest integer for which F ( · ) has continuous left derivatives upto order ( ℓ + 1) at y , and y + ( y − y ) /r , and f ( ℓ ) ( y − ) + r − ( ℓ +1) f ( ℓ ) (cid:16) ( y + ( y − y ) /r ) − (cid:17) = 0 and f ( i ) ( y − ) = f ( i ) (cid:16) ( y + ( y − y ) /r ) − (cid:17) = 0 for all i = 0 , , , . . . , ( ℓ − and suppose also that F ( · ) has acontinuous right derivative at y . Additionally, for bounded f ( ℓ ) ( · ) , c = 1 /r , and r ∈ (1 , we have thefollowing limit p ( F, r, /r ) = lim n →∞ p n ( F, r, /r ) = f ( ℓ ) ( y − ) f ( ℓ ) ( y − ) + r − ( ℓ +1) f ( ℓ ) (cid:16) ( y + ( y − y ) /r ) − (cid:17) . Note that in Theorem 4.5 • with ( y , y ) = (0 , p ( F, r, /r ) = f ( ℓ ) (1 − ) f ( ℓ ) (1 − )+ r − ( ℓ +1) f ( ℓ ) ( (1 /r ) − ) , • if f ( ℓ ) ( y − ) = 0 and f ( ℓ ) (cid:16) ( y + ( y − y ) /r ) − (cid:17) = 0, then p n ( F, r, /r ) → n → ∞ , at rate O (cid:0) κ ( f ) · n − ( ℓ +2) / ( ℓ +1) (cid:1) where κ ( f ) is a constant depending on f and • if f ( ℓ ) ( y − ) = 0 and f ( ℓ ) (cid:16) ( y + ( y − y ) /r ) − (cid:17) = 0, then p n ( F, r, /r ) → n → ∞ , at rate O (cid:0) κ ( f ) · n − ( ℓ +2) / ( ℓ +1) (cid:1) . Remark . In Theorems 4.4 and 4.5, we assume that f ( k ) ( · ) and f ( ℓ ) ( · ) are bounded on ( y , y ), respectively.If f ( k ) ( · ) is not bounded on ( y , y ) for k ≥ 0, in particular at y , and y + ( r − y − y ) /r , for example,lim x → y +1 f ( k ) ( x ) = ∞ , then we have p ( F, r, ( r − /r ) = lim δ → + f ( k ) ( y + δ ) (cid:2) f ( k ) ( y + δ ) + r − ( k +1) f ( k ) (( y + ( r − y − y ) /r ) + δ ) (cid:3) . If f ( ℓ ) ( · ) is not bounded on ( y , y ) for ℓ ≥ 0, in particular at y + ( y − y ) /r , and y , for example,lim x → y − f ( ℓ ) ( x ) = ∞ , then we have p ( F, r, /r ) = lim δ → + f ( ℓ ) ( y − δ ) (cid:2) f ( ℓ ) ( y − δ ) + r − ( ℓ +1) f ( ℓ ) (( y + ( y − y ) /r ) − δ ) (cid:3) . (cid:3) emark . The rates of convergence in Theorems 4.4 and 4.5 depends on f . From the proofs of Theorems4.4 and 4.5, it follows that for sufficiently large n , p n ( F, r, ( r − /r ) ≈ p ( F, r, ( r − /r ) + κ ( f ) n − ( k +2) / ( k +1) and p n ( F, r, /r ) ≈ p ( F, r, /r ) + κ ( f ) n − ( ℓ +2) / ( ℓ +1) , where κ ( f ) = s s k +13 + s Γ ( k +2 k +1 ) ( k +1) s k +2 k +13 with Γ( x ) = R ∞ e − t t ( x − dt , s = n k +1 k ! f ( k ) ( y +1 ), s = n ( k +1)! f ( k +1) ( y +1 ),and s = k +1)! p ( F, r, ( r − /r ), κ ( f ) = q Γ ( ℓ +2 ℓ +1 ) + q q ℓ +13 ( ℓ +1) q ℓ +2 ℓ +13 , q = ( − ℓ +1 n ( ℓ +1)! f ( ℓ +1) ( y − ), q = ( − ℓ n ℓ +1 ℓ ! f ( ℓ ) ( y − ), and q = ( − ℓ +1 ( ℓ +1)! p ( F, r, /r ) provided the derivatives exist. (cid:3) The conditions of the Theorems 4.4 and 4.5 might seem a bit esoteric. However, most of the well knownfunctions that are scaled and properly transformed to be pdf of some random variable with support in ( y , y )satisfy the conditions for some k or ℓ , hence one can compute the corresponding limiting probabilities p ( F, r, ( r − /r ) and p ( F, r, /r ). Example 4.8. (a) For example, with F = U ( y , y ), in Theorem 4.4, we have k = 0 and f ( y +1 ) = f (( y + ( r − y − y ) /r ) + ) =1 / ( y − y ), and in Theorem 4.5, we have ℓ = 0 and f ( y − ) = f (( y + ( y − y ) /r ) − ) = 1 / ( y − y ). Thenlim n →∞ p n ( F, r, ( r − /r ) = lim n →∞ p n ( F, r, /r ) = r/ ( r + 1), which agrees with the result given in Equation( ?? ).(b) For F with pdf f ( x ) = (cid:0) x +1 / (cid:1) I (cid:0) < x < (cid:1) , we have k = 0, f (0 + ) = 1 / 2, and f (cid:0) ( r − r ) + (cid:1) = 3 / − /r in Theorem 4.4. Then p ( F, r, ( r − /r ) = r r +3 r − . As for Theorem 4.5, we have ℓ = 0, f (1 − ) = 3 / f (cid:0) ( r ) − (cid:1) = 1 /r + 1 / 2. Then p ( F, r, /r ) = r r + r +2 .(c) For F with pdf f ( x ) = ( π/ | sin(2 πx ) | I (0 < x < 1) = ( π/ πx ) I (0 < x ≤ / − sin(2 πx ) I (1 / 2, we have k = 1, f ′ (0 + ) = ( πr ) r − , and f ′ (cid:0) ( r − r ) + (cid:1) = ( πr ) r − in Theorem 4.4. Then p ( F, r, ( r − /r ) = r / ( r − ν ,n , ν ,n , denoted β ( ν ,n , ν ,n ), where ν ,n , ν ,n ≥ 1, the pdfis given by f ( x, ν ,n , ν ,n ) = x ν ,n − (1 − x ) ν ,n − β ( ν ,n , ν ,n ) I (0 < x < 1) where β ( ν ,n , ν ,n ) = Γ( ν ,n ) Γ( ν ,n )Γ( ν ,n + ν ,n ) . Then in Theorem 4.4 we have k = 0, f (0 + ) = 0, and f (cid:0) ( r − r ) + (cid:1) = ( r − ν ,n − r ν ,n + ν ,n − . So p ( β ( ν ,n , ν ,n ) , r, ( r − /r ) =0. As for Theorem 4.5, we have ℓ = 0, f (1 − ) = 0, and f (cid:0) ( r ) − (cid:1) = ( r − ν ,n − r ν ,n + ν ,n − . Then p ( β ( ν ,n , ν ,n ) , r, /r ) = 0.(f) Consider F with pdf f ( x ) = (cid:16) π p x (1 − x ) (cid:17) − I (0 < x < f ( x ) is unbounded at x ∈ { , } . Using Remark 4.6, it follows that p ( F, r, ( r − /r ) = p ( F, r, /r ) = 1. (cid:3) Remark . In Theorem 4.4, if we have f ( k ) (0 + ) = f ( k ) (cid:0) ( r − r ) + (cid:1) , then lim n →∞ p n ( F, r, ( r − /r ) = r − ( k +1) . In particular, if k = 0, then lim n →∞ p n ( F, r, ( r − /r ) = r/ ( r +1). Hence γ n, ( F, r, ( r − /r ) and γ n, ( U , r, ( r − /r ) have the same asymptotic distribution.In Theorem 4.5, if we have f ( ℓ ) (1 − ) = f ( ℓ ) (cid:0) ( r ) − (cid:1) , then lim n →∞ p n ( F, r, /r ) = r − ( ℓ +1) . In particular, if ℓ = 0, then lim n →∞ p n ( F, r, /r ) = r/ ( r +1). Hence γ n, ( F, r, /r ) and γ n, ( U , r, /r ) have the same asymptoticdistribution. (cid:3) γ n, ( F, r, c ) for r = 2 and c = 1 / Theorem 4.10. Let F (cid:0) y , y (cid:1) := n F : ( y , y + ε ) ∪ ( y − ε, y ) ∪ (cid:0) ( y + y ) / − ε, ( y + y ) / ε (cid:1) ⊆ S ( F ) ⊆ ( y , y ) for some ε ∈ (0 , ( y + y ) / o . Let Y = { y , y } ⊂ R with −∞ < y < y < ∞ , X n = { X , . . . , X n } with X i iid ∼ F ∈ F ( y , y ) , and D n, be the random D n, -digraph based on X n and Y .(i) Then for n > , we have γ n, ( F, , / ∼ (cid:0) p n ( F, , / (cid:1) . Note also that γ , ( F, , / 2) = 1 .(ii) Furthermore, suppose k ≥ is the smallest integer for which F ( · ) has continuous right derivatives up toorder ( k +1) at y , ( y + y ) / , f ( k ) ( y +1 )+2 − ( k +1) f ( k ) (cid:16)(cid:0) y + y (cid:1) + (cid:17) = 0 and f ( i ) ( y +1 ) = f ( i ) (cid:16)(cid:0) y + y (cid:1) + (cid:17) = 0 for all i = 0 , , . . . , k − ; and ℓ ≥ is the smallest integer for which F ( · ) has continuous left derivatives upto order ( ℓ +1) at y , ( y + y ) / , f ( ℓ ) ( y − )+2 − ( ℓ +1) f ( ℓ ) (cid:16)(cid:0) y + y (cid:1) − (cid:17) = 0 and f ( i ) ( y − ) = f ( i ) (cid:16)(cid:0) y + y (cid:1) − (cid:17) =0 for all i = 0 , , . . . , ℓ − . Additionally, for bounded f ( k ) ( · ) and f ( ℓ ) ( · ) , we have the following limit p ( F, , / 2) = lim n →∞ p n ( F, , / 2) = f ( k ) ( y +1 ) f ( ℓ ) ( y − ) h f ( k ) ( y +1 ) + 2 − ( k +1) f ( k ) (cid:16)(cid:0) y + y (cid:1) + (cid:17)i h f ( ℓ ) ( y − ) + 2 − ( ℓ +1) f ( ℓ ) (cid:16)(cid:0) y + y (cid:1) − (cid:17)i . Notice the interesting behavior of p ( F, r, c ) around ( r, c ) = (2 , / p ( F, r, ( r − /r ) and in p ( F, r, /r ) at r = 2. D n,m ( r, c ) -digraphs We now consider the more challenging case of m > 2. For ω < ω in R , define the family of distributions H ( R ) := (cid:8) F X,Y : ( X i , Y i ) ∼ F X,Y with support S ( F X,Y ) = ( ω , ω ) ( R , X i ∼ F X and Y i iid ∼ F Y (cid:9) . We provide the exact distribution of γ n,m ( F, r, c ) for H ( R )-random digraphs in the following theorem. Let[ m ] := (cid:8) , , , . . . , m − (cid:9) and Θ Sa,b := (cid:8) ( u , u , . . . u b ) : P bi =1 u i = a : u i ∈ S, ∀ i (cid:9) . If Y i have acontinuous distribution, then the order statistics of Y m are distinct a.s. Given Y ( i ) = y ( i ) for i = 1 , , . . . , m ,let ~n be the vector of numbers n i , f ~Y ( ~ y ) be the joint distribution of the order statistics of Y m , i.e., f ~Y ( ~ y ) = m ! Q mi =1 f ( y i ) I ( ω < y < y < . . . < y m < ω ), and f i,j ( y i , y j ) be the joint distribution of Y ( i ) , Y ( j ) . Then wehave the following theorem. Theorem 5.1. Let D be an H ( R ) -random D n,m ( r, c ) -digraph with n > , m > , r ∈ [1 , ∞ ) and c ∈ (0 , .Then the probability mass function of the domination number of D is given by P ( γ n,m ( F, r, ( r − /r ) = q ) = Z S X ~n ∈ Θ [ n +1] n, ( m +1) X ~q ∈ Θ [3] q, ( m +1) P ( ~N = ~n ) ζ ( q , n ) ζ ( q m +1 , n m +1 ) m Y j =2 η ( q i , n i ) f ~Y ( ~ y ) d y . . . d y m where P ( ~N = ~n ) is the joint probability of n i points falling into intervals I i for i = 0 , , , . . . , m , q i ∈ { , , } , q = P mi =0 q i and ζ ( q i , n i ) = max (cid:0) I ( n i = q i = 0) , I ( n i ≥ q i = 1) (cid:1) for i = 1 , ( m + 1) , and η ( q i , n i ) = max (cid:0) I ( n i = q i = 0) , I ( n i ≥ q i ≥ (cid:1) · p ni ( F i , r, ( r − /r )) I ( q i =2) (cid:0) − p ni ( F i , r, ( r − /r )) (cid:1) I ( q i =1) for i = 1 , , , . . . , ( m − , and the region of integration is given by S := (cid:8) ( y , y , . . . , y m ) ∈ ( ω , ω ) : ω < y < y < . . . < y m < ω (cid:9) . The special cases of n = 1 , m = 1 , r ∈ { , ∞} and c ∈ { , } are as in Theorem 2.4. c = 1 /r .This exact distribution for finite n and m has a simpler form when X and Y points are both uniform in abounded interval in R . Define U ( R ) as follows U ( R ) := (cid:8) F X,Y : X and Y are independent X i iid ∼ U ( ω , ω ) and Y i iid ∼ U ( ω , ω ) , with −∞ < ω < ω < ∞ (cid:9) . Clearly, U ( R ) ( H ( R ). Then we have the following corollary to Theorem 5.1. Corollary 5.2. Let D be a U ( R ) -random D n,m ( r, c ) -digraph with n > , m > , r ∈ [1 , ∞ ) and c ∈ (0 , .Then the probability mass function of the domination number of D is given by P ( γ n,m ( r, ( r − /r ) = q ) = n ! m !( n + m )! X ~n ∈ Θ [ n +1] n, ( m +1) X ~q ∈ Θ [3] q, ( m +1) ζ ( q , n ) ζ ( q m +1 , n m +1 ) m Y j =2 η ( q i , n i ) . The special cases of n = 1 , m = 1 , r ∈ { , ∞} and c ∈ { , } are as in Theorem 2.4. Proof is as in Theorem 2 of Priebe et al. (2001). A similar construction is available for c = 1 /r . For n, m < ∞ , the expected value of domination number is E [ γ n,m ( F, r, c )] = P (cid:0) X (1) < Y (1) (cid:1) + P (cid:0) X ( n ) > Y ( m ) (cid:1) + m − X i =1 n X k =1 P ( N i = k ) E [ γ ni, ( F i , r, c )] (15)where P ( N i = k ) = Z ω ω Z ω y ( i ) f i − ,i (cid:0) y ( i ) , y ( i +1) (cid:1) h F X (cid:0) y ( i +1) (cid:1) − F X (cid:0) y ( i ) (cid:1)i k h − (cid:0) F X (cid:0) y ( i +1) (cid:1) − F X (cid:0) y ( i ) (cid:1)(cid:1)i n − k d y ( i +1) d y ( i ) and E [ γ ni, ( F i , r, c )] = 1 + p n ( F i , r, c ). Then as in Corollary 6.2 of Ceyhan (2008), we have Corollary 5.3. For F X,Y ∈ H ( R ) with support S ( F X ) ∩ S ( F Y ) of positive measure with r ∈ [1 , ∞ ) and c ∈ (0 , , we have lim n →∞ E [ γ n,n ( F, r, c )] = ∞ . Theorem 5.4. Main Result 5: Let D n,m ( r, c ) be an H ( R ) -random D n,m ( r, c ) -digraph. Then(i) for fixed n < ∞ , lim m →∞ γ n,m ( F, r, c ) = n a.s. for all r ≥ and c ∈ [0 , .For fixed m < ∞ , and(ii) for r = 1 and c ∈ (0 , , lim n →∞ P ( γ n,m ( F, , c ) = 2 m ) = 1 and lim n →∞ P ( γ n,m ( F, , 0) = m + 1) =lim n →∞ P ( γ n,m ( F, , 1) = m + 1) = 1 (iii) for r > and c ∈ (0 , , lim n →∞ P ( γ n,m ( F, r, c ) = m + 1) = 1 ,(iv) for r = 2 , if c = 1 / , then lim n →∞ P ( γ n,m ( F, , c ) = m + 1) = 1 ;if c = 1 / , then lim n →∞ γ n,m ( F, , / d = m + 1 + BIN( m, p ( F i , , / ,(v) for r ∈ (1 , , if r = τ = max( c, − c ) , then lim n →∞ γ n,m ( F, r, c ) is degenerate; otherwise, it is non-degenerate. That is, for r ∈ [1 , , as n → ∞ , γ n,m ( F, r, c ) ∼ m + 1 + BIN( m, p ( F i , r, c )) , for r = 1 /τ , m + 1 , for r > /τ , m + 1 , for r < /τ . (16) Proof: n i → ∞ , we have X [ i ] = ∅ a.s.for all i .Part (iii) follows from Theorem 3.6, since for c ∈ (0 , r > /τ implies r > n i → ∞ , we have γ ni, ( F i , r, c ) → i .In part (iv), for r = 2 and c = 1 / 2, based on Theorem 3.2, as n i → ∞ , we have γ ni, ( F i , r, c ) → i . The result for r = 2 and c = 1 / (cid:4) Remark . Extension to Higher Dimensions: Let Y m = { y , y , . . . , y m } be m points in general position in R d and T i be the i th Delaunay cell in theDelaunay tessellation (assumed to exist) based on Y m for i = 1 , , . . . , J m . Let X n be a random sample froma distribution F in R d with support S ( F ) ⊆ C H ( Y m ) where C H ( Y m ) stands for the convex hull of Y m . In R a Delaunay tessellation is an intervalization (i.e., partitioning of R by intervals), provided that no two pointsin Y m are concurrent.We define the proportional-edge proximity region in R . The extension to R d with d > T ( Y ) be the triangle (including the interior)with vertices Y = { y , y , y } , e i be the edge opposite vertex y i , and M i be the midpoint of edge e i for i = 1 , , 3. We first construct the vertex regions based on a point M ∈ R \ Y called M -vertex regions , bythe lines joining M to a point on each of the edges of T ( Y ). See Ceyhan (2005) for a more general definitionof vertex regions. Preferably, M is selected to be in the interior of the triangle T ( Y ). For such an M ,the corresponding vertex regions can be defined using a line segment joining M to e i . With center of mass M CM , the lines joining M CM and Y are the median lines which cross edges at midpoints M i for i = 1 , , r ∈ [1 , ∞ ], define N ( · , r, M ) to bethe (parametrized) proportional-edge proximity map with M -vertex regions as follows (see also Figure 3 with M = M CM and r = 2). Let R M ( v ) be the vertex region associated with vertex v and M . For x ∈ T ( Y ) \ Y ,let v ( x ) ∈ Y be the vertex whose region contains x ; i.e., x ∈ R M ( v ( x )). If x falls on the boundary of two M -vertex regions, we assign v ( x ) arbitrarily. Let e ( x ) be the edge of T ( Y ) opposite of v ( x ), ℓ ( v ( x ) , x ) bethe line parallel to e ( x ) and passes through x , and d ( v ( x ) , ℓ ( v ( x ) , x )) be the Euclidean distance from v ( x ) to ℓ ( v ( x ) , x ). For r ∈ [1 , ∞ ), let ℓ r ( v ( x ) , x ) be the line parallel to e ( x ) such that d ( v ( x ) , ℓ r ( v ( x ) , x )) = r d ( v ( x ) , ℓ ( v ( x ) , x )) and d ( ℓ ( v ( x ) , x ) , ℓ r ( v ( x ) , x )) < d ( v ( x ) , ℓ r ( v ( x ) , x )) . Let T r ( x ) be the triangle similar to and with the same orientation as T ( Y ) having v ( x ) as a vertex and ℓ r ( v ( x ) , x ) as the opposite edge. Then the proportional-edge proximity region N ( x, r, M ) is defined to be T r ( x ) ∩ T ( Y ). Notice that ℓ ( v ( x ) , x ) divides the two edges of T r ( x ) (other than the one lies on ℓ r ( v ( x ) , x ))proportionally with the factor r . Hence the name proportional-edge proximity region .Notice that in R , M is the center parametrized by c , e.g., the center of mass M CM corresponds to c = 1 / M ∈ T ( Y ), there is no direct counterpart in R . The vertex regions in R with Y = { y , y } are( y , M c ) and ( M c , y ). Observe that N ( x, r, c ) in R is an open interval, while in R d , the region N ( x, r, M ) isa closed region. However, the interiors of N ( x, r, M ) satisfy the class cover problem of Cannon and Cowen(2000). (cid:3) In this article, we present the distribution of the domination number of a random digraph family calledproportional-edge proximity catch digraph (PCD) which is based on two classes of points. Points from one ofthe classes constitute the vertices of the PCDs, while the points from the other class are used in the binaryrelation that assigns the arcs of the PCDs. 18 = v ( x ) x M CM ℓ ( v ( x ) , x ) ℓ ( v ( x ) , x ) y e ( x ) y d d Figure 3: Construction of proportional-edge proximity region, N ( x, , M CM ) (shaded region) for an x in the CM-vertex region for y , R CM ( y ) where d = d ( v ( x ) , ℓ ( v ( x ) , x )) and d = d ( v ( x ) , ℓ ( v ( x ) , x )) =2 d ( v ( x ) , ℓ ( v ( x ) , x )).We introduce the proximity map which is the one-dimensional version of N ( · , r, c ) of Ceyhan and Priebe(2005) and Ceyhan and Priebe (2007). This proximity map can also be viewed as an extension of the proximitymap of Priebe et al. (2001) and Ceyhan (2008). The PCD we consider is based on a parametrized proximitymap in which there is an expansion parameter r and a centrality parameter c . We provide the exact andasymptotic distributions of the domination number for proportional-edge PCDs for uniform data and computethe asymptotic distribution for non-uniform data for the entire range of ( r, c ). The results in this article canalso be viewed as generalizations of the main results of Priebe et al. (2001) and Ceyhan (2008) in severaldirections. Priebe et al. (2001) provided the exact distribution of the domination number of class covercatch digraphs (CCCDs) based on X n and Y m both of which were sets of iid random variables from uniformdistribution on ( ω , ω ) ⊂ R with −∞ < ω < ω < ∞ and the proximity map N ( x ) := B ( x, r ( x )) where r ( x ) := min y ∈Y m d ( x, y ). Ceyhan (2008) investigates the distribution of the domination number of CCCDs fornon-uniform data and provides the asymptotic distribution for a large family of (non-uniform) distributions.Furthermore, this article will form the foundation of the generalizations and calculations for uniform and non-uniform cases in multiple dimensions. As in Ceyhan and Priebe (2005), we can use the domination number intesting one-dimensional spatial point patterns and our results will help make the power comparisons possiblefor a large family of distributions.We demonstrate an interesting behavior of the domination number of proportional-edge PCD for one-dimensional data. For uniform data or data from a distribution which satisfies some regularity conditions(see Section 4.1) and fixed 1 < n < ∞ , the distribution of the domination number is a translated form of(extended) binomial distribution BIN( m, p n ( F, r, c )) where m is the number of (inner) intervals and p n ( F, r, c )is the probability that the domination number of the proportional-edge PCD is 2. Here p n ( F, r, c ) is allowedto take 0 or 1 also. For finite n > 1, the parameter, p n ( U , r, c ), of distribution of the domination numberof the proportional-edge PCD based on uniform data is continuous in r and c for all r ≥ c ∈ (0 , c = 0 , 1. For fixed ( r, c ) ∈ [1 , ∞ ) × (0 , p ( U , r, c ),of the asymptotic distribution exhibits some discontinuities. For c ∈ (0 , 1) the asymptotic distribution isnondegenerate at r = 1 / max( c, − c ). The asymptotic distribution of the domination number is degeneratefor the expansion parameter r > 2. For r ∈ (1 , c for which the asymptotic distribution is non-degenerate. In particular, for c ∈ { ( r − /r, /r } with r ∈ (1 , m, p ( U , r, c ) where p ( F, r, c )is continuous in r . Additionally, by symmetry, we have p ( U , r, ( r − /r ) = p ( U , r, /r ) for r ∈ (1 , r > / max( c, − c ) the domination number converges in probability to 1, and for r < / max( c, − c ) thedomination number converges in probability to 2. On the other hand, at ( r, c ) = (2 , / m, p ( U , , / r = 2, as p ( U , , / 2) = 4 / r → p ( U , r, ( r − /r ) = lim r → p ( U , r, /r ) = 2 / 3. This second jump might be due19o the symmetry for the domination number at c = 1 / Acknowledgments Supported by TUBITAK Kariyer Project Grant 107T647. References Cannon, A. and Cowen, L. (2000). Approximation algorithms for the class cover problem. In Proceedings ofthe 6th International Symposium on Artificial Intelligence and Mathematics .Ceyhan, E. (2005). 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The cases in which b < a and b < c are not possible, since c < / b > / 2, and a < / Case (1) Γ ( X n , , c ) = ( a, b ), i.e., a < c < b : For c ∈ [1 / , / P ( γ n, ( U , , c ) = 2 , Γ ( X n , , c ) = ( a, b ) , c ∈ [1 / , / Z / Z c (1+ x ) / + Z c / Z c x ! n ( n − f ( x ) f ( x n ) (cid:2) F ( x n ) − F ( x ) + F ( a ) − F ( b ) (cid:3) ( n − dx n dx =49 (cid:18) c − (cid:19) n − 89 4 − n − (cid:18) c − (cid:19) n . (17)21or c ∈ (0 , / P ( γ n, ( U , , c ) = 2 , Γ ( X n , , c ) = ( a, b ) , c ∈ (0 , / Z c − Z c (1+ x ) / n ( n − f ( x ) f ( x n ) (cid:2) F ( x n ) − F ( x ) + F ( a ) − F ( b ) (cid:3) ( n − dx n dx =49 (1 − c ) n + 49 (cid:18) c − (cid:19) n − 89 4 − n . (18) Case (2) Γ ( X n , , c ) = [ c, b ), i.e., c < a < b : For c ∈ [1 / , / P ( γ n, ( U , , c ) = 2 , Γ ( X n , , c ) = [ c, b ) , c ∈ [1 / , / Z c − Z c (1+ x ) / n ( n − f ( x ) f ( x n ) (cid:2) F ( x n ) − F ( x ) + F ( c ) − F ( b ) (cid:3) ( n − dx n dx =23 (cid:18) c − (cid:19) n − (cid:18) − c (cid:19) n − (cid:18) c − (cid:19) n + 23 (cid:18) c + 12 (cid:19) n . (19)For c ∈ (0 , / P ( γ n, ( U , , c ) = 2 , Γ ( X n , , c ) = [ c, b ) , c ∈ (0 , / Z c − Z c + Z c c − Z x ) / ! n ( n − f ( x ) f ( x n ) (cid:2) F ( x n ) − F ( x ) + F ( c ) − F ( b ) (cid:3) ( n − dx n dx =23 (cid:18) c + 12 (cid:19) n − 13 (1 − c ) n − (cid:18) c − (cid:19) n − (cid:18) − c (cid:19) n (20)The probability P ( γ n, ( U , , c ) = 2 , c ∈ [1 / , / P ( γ n, ( U , , c ) =2 , c ∈ (0 , / P ( γ n, ( U , , c ) = 2 , c ∈ [1 / , P ( γ n, ( U , , − c ) = 2 , − c ∈ (0 , / c = { , } follow by construction. (cid:4) Proof of Theorem 3.3 Given X (1) = x and X ( n ) = x n , let a = x n /r and b = ( x + r − /r . For r ≥ c = 1 / 2, the Γ -region isΓ ( X n , r, / 2) = ( a, / ∪ [1 / , b ), so we have four cases for Γ -region: case (1) Γ ( X n , r, / 2) = ( a, b ) whichoccurs when a < / < b , case (2) Γ ( X n , r, / 2) = ( a, / 2] which occurs when a < b < / b < a < / 2, case(3) Γ ( X n , r, / 2) = [1 / , b ) which occurs when 1 / < a < b or 1 / < b < a , and case (4) Γ ( X n , r, / 2) = ∅ which occurs when b < / < a . Cases (2) and (3) are symmetric, so they yield the same probabilities. Case (1) Γ ( X n , r, / 2) = ( a, b ), i.e., a < / < b : For r ≥ 2, we have P ( γ n, ( U , r, / 2) = 2 , Γ ( X n , r, / 2) = ( a, b ) , r ≥ 2) = Z / ( r +1)0 Z x + r − /r + Z /r / ( r +1) Z r x ! n ( n − f ( x ) f ( x n ) (cid:2) F ( x n ) − F ( x ) + F ( a ) − F ( b ) (cid:3) ( n − dx n dx =2 r ( r + 1) "(cid:18) r (cid:19) n − − (cid:18) r − r (cid:19) n − r . (21)22or 1 ≤ r < 2, we have P ( γ n, ( U , r, / 2) = 2 , Γ ( X n , r, / 2) = ( a, b ) , ≤ r < 2) = Z / ( r +1)1 − r/ Z r/ x + r − /r + Z / / ( r +1) Z r/ r x ! n ( n − f ( x ) f ( x n ) (cid:2) F ( x n ) − F ( x ) + F ( a ) − F ( b ) (cid:3) ( n − dx n dx = r ( r − n ( r + 1) " − (cid:18) r − r (cid:19) n − . (22) Case (2) Γ ( X n , r, / 2) = ( a, / a < b < / b < a < / 2: Here, r ≥ x < − r/ 2. For 1 ≤ r < 2, we have P ( γ n, ( U , r, / 2) = 2 , Γ ( X n , r, / 2) = ( a, / , ≤ r < 2) = Z − r/ Z r/ / n ( n − f ( x ) f ( x n ) (cid:2) F ( x n ) − F ( x ) + F ( a ) − F (1 / (cid:3) ( n − dx n dx = r ( r + 1) (cid:20)(cid:16) r (cid:17) n − ( r − n − (cid:18) r (cid:19) n + (cid:18) ( r − r (cid:19) n (cid:21) . (23)By symmetry, Case (3) yields the same result as Case (2) . Case (4) Γ ( X n , r, / 2) = ∅ , i.e., b < / < a : Here, r ≥ x n − x > r − 1. For1 ≤ r < 2, we have P ( γ n, ( U , r, / 2) = 2 , Γ ( X n , r, / 2) = ∅ , < r < 2) = Z − r/ Z / n ( n − f ( x ) f ( x n ) (cid:2) F ( x n ) − F ( x ) (cid:3) ( n − dx n dx = 1 + ( r − n − (cid:16) r (cid:17) n . (24)The probability P ( γ n, ( U , r, / 2) = 2 , r ≥ 2) is as in (21), and P ( γ n, ( U , r, / 2) = 2 , r ∈ [1 , ∞ )) is thesum of probabilities in (22), (24), and twice the probability in (23). (cid:4) Proof of Theorem 3.4 Given X (1) = x and X ( n ) = x n , let a = x n /r and b = ( x + r − /r and assume c ∈ (0 , / c , namely, Case I- c ∈ ((3 − √ / , / 2) and Case II- c ∈ (0 , (3 − √ / Case I- For r ≥ c ∈ ((3 − √ / , / -region is Γ ( X n , r, c ) = ( a, c ] ∪ [ c, b ). So we have fourcases for Γ -region: case (1) Γ ( X n , r, c ) = ( a, b ) which occurs when a < c < b , case (2) Γ ( X n , r, c ) = ( a, c ]which occurs when a < b < c or b < a < c , case (3) Γ ( X n , r, c ) = [ c, b ) which occurs when c < a < b or c < b < a , and case (4) Γ ( X n , r, c ) = ∅ which occurs when b < c < a . Case (1) Γ ( X n , r, c ) = ( a, b ), i.e., a < c < b : In this case, for r ≥ /c , we have P ( γ n, ( U , r, c ) = 2 , Γ ( X n , r, c ) = ( a, b ) , r ≥ /c ) = Z / ( r +1)0 Z x + r − /r + Z /r / ( r +1) Z r x ! n ( n − f ( x ) f ( x n ) (cid:2) F ( x n ) − F ( x ) + F ( a ) − F ( b ) (cid:3) ( n − dx n dx =2 r ( r + 1) (cid:18) r (cid:19) n − − (cid:18) r − r (cid:19) n − ! (25)23or 1 / (1 − c ) ≤ r < /c , we have P ( γ n, ( U , r, c ) = 2 , Γ ( X n , r, c ) = ( a, b ) , / (1 − c ) ≤ r < /c ) = Z / ( r +1)0 Z c r ( x + r − /r + Z c / ( r +1) Z c rr x ! n ( n − f ( x ) f ( x n ) (cid:2) F ( x n ) − F ( x ) + F ( a ) − F ( b ) (cid:3) ( n − dx n dx = r ( r + 1) "(cid:18) c ( r + 1) − r − r (cid:19) n − (cid:18) r − r (cid:19) n − (cid:18) ( c r + c − n − r n (cid:19) . (26)For (1 − c ) /c ≤ r < / (1 − c ), we have P ( γ n, ( U , r, c ) = 2 , Γ ( X n , r, c ) = ( a, b ) , (1 − c ) /c ≤ r < / (1 − c )) = Z / ( r +1) r ( c − Z c r ( x + r − /r + Z c / ( r +1) Z c rr x ! n ( n − f ( x ) f ( x n ) (cid:2) F ( x n ) − F ( x ) + F ( a ) − F ( b ) (cid:3) ( n − dx n dx = r ( r − n − ( r + 1) (cid:20) ( r − − r n − [( r − c r − c ) n − ( c r + c − n ] (cid:21) . (27)For 1 ≤ r < (1 − c ) /c , we have P ( γ n, ( U , r, c ) = 2 , Γ ( X n , r, c ) = ( a, b ) , ≤ r < (1 − c ) /c ) = Z c r − r +1 r ( c − Z c r ( x + r − /r n ( n − f ( x ) f ( x n ) (cid:2) F ( x n ) − F ( x ) + F ( a ) − F ( b ) (cid:3) ( n − dx n dx = r ( r − n − ( r + 1) (cid:20) r − − c r − c ) n + ( r − c r − c ) n r n − (cid:21) . (28) Case (2) Γ ( X n , r, c ) = ( a, c ], i.e., a < b < c or b < a < c : Here r ≥ / (1 − c ) is not possible, since r ( c − 1) + 1 > 0. Hence r ≥ /c is not possible either. For 1 ≤ r < / (1 − c ), we have P ( γ n, ( U , r, c ) = 2 , Γ ( X n , r, c ) = ( a, c ] , ≤ r < / (1 − c )) = Z r ( c − Z c rc n ( n − f ( x ) f ( x n )[ F ( x n ) − F ( x ) + F ( a ) − F ( c )] ( n − dx n dx = rr + 1 (cid:20) c n (cid:18) r n − r n (cid:19) − ( r − n (cid:18) − r − c r − cr (cid:19) n (cid:21) . (29) Case (3) Γ ( X n , r, c ) = [ c, b ), i.e., c < a < b or c < b < a : Here r ≥ /c is not possible, since x n > c r . For1 / (1 − c ) ≤ r < /c , we have P ( γ n, ( U , r, c ) = 2 , Γ ( X n , r, c ) = [ c, b ) , / (1 − c ) ≤ r < /c ) = Z c Z c r n ( n − f ( x ) f ( x n ) (cid:2) F ( x n ) − F ( x ) + F ( c ) − F ( b ) (cid:3) ( n − dx n dx =1( r + 1) r n − (cid:2) ( r − n ( c r − c ) n + (1 + c r ) n − ( c r + c r − r + 1) n − (1 − c ) n (cid:3) . (30)For 1 ≤ r < / (1 − c ), we have P ( γ n, ( U , r, c ) = 2 , Γ ( X n , r, c ) = [ c, b ) , ≤ r < / (1 − c )) = Z cr ( c − Z c r n ( n − f ( x ) f ( x n ) (cid:2) F ( x n ) − F ( x ) + F ( c ) − F ( b ) (cid:3) ( n − dx n dx = rr + 1 (cid:20) ( r − n (cid:18)(cid:18) c r − cr (cid:19) n − (cid:19) + (1 − c ) n (cid:18) r n − r n (cid:19)(cid:21) . (31)24 ase (4) Γ ( X n , r, c ) = ∅ , i.e., b < c < a : Here, r ≥ /c is not possible, since x n > c r ; and 1 / (1 − c ) ≤ r < /c is not possible, since x < r ( c − 1) + 1. For 1 ≤ r < / (1 − c ), we have P ( γ n, ( U , r, c ) = 2 , Γ ( X n , r, c ) = ∅ , ≤ r < / (1 − c )) = Z r ( c − Z c r n ( n − f ( x ) f ( x n )( F ( x n ) − F ( x )) ( n − dx n dx = 1 + ( r − n − r n [ c n + (1 − c ) n ] . (32)The probability P ( γ n, ( U , r, c ) = 2 , r ≥ /c ) is the same as in (25); P ( γ n, ( U , r, c ) = 2 , / (1 − c ) ≤ r < /c )is the sum of probabilities in (26) and (30); P ( γ n, ( U , r, c ) = 2 , (1 − c ) /c ≤ r < / (1 − c )) is the sum ofprobabilities in (27), (29), (31), and (32); P ( γ n, ( U , r, c ) = 2 , ≤ r < (1 − c ) /c ) is the sum of probabilities in(28), (29), (31), and (32). Case II- For r ≥ c ∈ (0 , (3 − √ / -region is as above. Case (1) : For r ≥ /c , the probability P ( γ n, ( U , r, c ) = 2 , Γ ( X n , r, c ) = ( a, b ) , r ≥ /c ) as in (25).For (1 − c ) /c ≤ r < /c , the probability P ( γ n, ( U , r, c ) = 2 , Γ ( X n , r, c ) = ( a, b ) , (1 − c ) /c ≤ r < /c ) is asin (26).For 1 / (1 − c ) ≤ r < (1 − c ) /c , we have P ( γ n, ( U , r, c ) = 2 , Γ ( X n , r, c ) = ( a, b ) , / (1 − c ) ≤ r < (1 − c ) /c ) = Z c r − r +10 Z c r ( x + r − /r n ( n − f ( x ) f ( x n ) (cid:2) F ( x n ) − F ( x ) + F ( a ) − F ( b ) (cid:3) ( n − dx n dx = r ( r + 1) (cid:20) ( r − n − (cid:18) (1 − c r − c ) n − r n − (cid:19) + (cid:18) c r + c r − r + 1 r (cid:19) n (cid:21) . (33)For 1 ≤ r < / (1 − c ), the probability P ( γ n, ( U , r, c ) = 2 , Γ ( X n , r, c ) = ( a, b ) , ≤ r < / (1 − c )) is as in(28). Cases (2) and (4) are as before. Case (3) : For (1 − c ) /c ≤ r < /c , the probability P ( γ n, ( U , r, c ) = 2 , Γ ( X n , r, c ) = [ c, b ) , (1 − c ) /c ≤ r < /c )is as in (30).For 1 / (1 − c ) ≤ r < (1 − c ) /c , the probability P ( γ n, ( U , r, c ) = 2 , Γ ( X n , r, c ) = [ c, b ) , / (1 − c ) ≤ r < (1 − c ) /c ) is as in (30).For 1 ≤ r < / (1 − c ), the probability P ( γ n, ( U , r, c ) = 2 , Γ ( X n , r, c ) = [ c, b ) , ≤ r < / (1 − c )) is as in(31).The probability P ( γ n, ( U , r, c ) = 2 , r ≥ /c ) is the same as in (25); P ( γ n, ( U , r, c ) = 2 , (1 − c ) /c ≤ r < /c )is the sum of probabilities in (26) and (30); P ( γ n, ( U , r, c ) = 2 , / (1 − c ) ≤ r < (1 − c ) /c ) is the sum ofprobabilities in (30) and (33); P ( γ n, ( U , r, c ) = 2 , ≤ r < / (1 − c )) is the sum of probabilities in (28), (29),(31), and (32).By symmetry, P ( γ n, ( U , r, c ) = 2 , c ∈ [1 / , P ( γ n, ( U , r, − c ) = 2 , − c ∈ (0 , / c ∈ { , } follows trivially by construction. (cid:4) Proof of Theorem 3.6 Let c ∈ (0 , / τ = 1 − c . We first consider Case I : c ∈ ((3 − √ / , / r ≥ /c > 2, it follows that lim n →∞ p n ( U , r, c ) = lim n →∞ π ,n ( r, c ) = 0, since 2 /r < r − r < / (1 − c ) < r < /c , we have c rr < r > / (1 − c )), − cr < ( c r − r + c r +1) r < r − r < r − < r < r ), and ( r − c r − c ) r < 1. Hence for 1 / (1 − c ) < r < /c lim n →∞ p n ( U , r, c ) = lim n →∞ π ,n ( r, c ) =0 For (1 − c ) /c < r < / (1 − c ), we have r − < r < / (1 − c ) < ( r − c r − c ) r < ( r − r − c r − c ) r < c/r < c < r ), (1 − c ) /r < − c < r ), c r < − c ) r < r < / (1 − c ) < /c ). Hence lim n →∞ p n ( U , r, c ) = lim n →∞ π ,n ( r, c ) = 1.For 1 ≤ r < (1 − c ) /c , we have r − < r < / (1 − c ) < r − − c r − c ) < ( r − − c r − c ) r < ( r − r − c r − c ) r < c/r < c < r ), (1 − c ) /r < c r < − c ) r < 1. Hence lim n →∞ p n ( U , r, c ) =lim n →∞ π ,n ( r, c ) = 1,But for r = 1 / (1 − c ) or c = ( r − /r , we have p n ( U , r, ( r − /r ) = r ( r + 1) " ( r + 1) − ( r − n − (cid:18) r − r − r (cid:19) n + ( r − n − r + 1 r n − (cid:18) r − r (cid:19) n − . (34)Letting n → ∞ , we get p n ( U , r, ( r − /r ) → r/ ( r + 1) for r ∈ (1 , r − r < r − < r < 1, and ( r − r − r − r < Case II: c ∈ (0 , (3 − √ / r ≥ /c > 2, it follows thatlim n →∞ p n ( U , r, c ) = lim n →∞ ϑ ,n ( r, c ) = 0, since 2 /r < r − r < − c ) /c < r < /c , we have c rr < r > / (1 − c )), − cr < ( c r − r + c r +1) r < r − r < r − < r < r ), and ( r − c r − c ) r < 1. Hence lim n →∞ p n ( U , r, c ) = lim n →∞ ϑ ,n ( r, c ) = 0For 1 / (1 − c ) < r < (1 − c ) /c , we have ( r − − c r − c ) < ( r − − c r − c ) r < r − r < r − r < cr ) /r < c r − c + c r +1 r < c/r < c < r ), (1 − c ) /r < − c < r ), c r < − c ) r < r < / (1 − c ) < /c ). Hence lim n →∞ p n ( U , r, c ) = lim n →∞ ϑ ,n ( r, c ) = 0.For 1 ≤ r < / (1 − c ), we have r − < r < / (1 − c ) < r − − c r − c ) < ( r − − c r − c ) r < ( r − r − c r − c ) r < c/r < c < r ), (1 − c ) /r < c r < − c ) r < 1. Hence lim n →∞ p n ( U , r, c ) =lim n →∞ ϑ ,n ( r, c ) = 1,But for r = 1 / (1 − c ) or c = ( r − /r , we have p n ( U , r, ( r − /r ) = r ( r + 1) " ( r + 1) + ( r + 1) (cid:18) ( r − r − r − r (cid:19) n − ( r − n +( − n (cid:18) r − r (cid:19) n − ( r − r − n − r − r n + (cid:18) r − r (cid:19) n − . (35)Letting n → ∞ , we get p n ( U , r, ( r − /r ) → r/ ( r + 1) for r ∈ (1 , r − < r < ( r − r − r − r < r − r − < c ∈ (1 / , τ = c . By symmetry, the above results follow with c being replaced by 1 − c andas n → ∞ , we get p n ( U , r, /r ) → r/ ( r + 1) for r ∈ (1 , (cid:4) Proof of Theorem 4.4 Case (i) follows trivially from Theorem 2.3. The special cases for n = 1 and r = { , ∞} follow by construction.26ase (ii): Suppose ( y , y ) = (0 , 1) and c ∈ (0 , / ( X n , r, c ) = ( X ( n ) /r, M c ] S [ M c , ( X (1) + r − /r ) ⊂ (0 , 1) and γ n, ( F, r, c ) = 2 iff X n ∩ Γ ( X n , r, c ) = ∅ . Then for finite n , p n ( F, r, c ) = P (cid:0) γ n, ( F, r, c ) = 2 (cid:1) = Z S ( F ) \ ( δ ,δ ) H ( x , x n ) dx n dx , where ( δ , δ ) = Γ ( X n , r, c ) and H ( x , x n ) is as in Equation (12).Let ε ∈ (0 , ( r − /r ) and c = ( r − /r . Then P (cid:0) X (1) < ε, X ( n ) > − ε (cid:1) → n → ∞ with the rateof convergence depending on F . Moreover, for sufficiently large n , ( X (1) + r − /r > ( r − /r a.s.; in fact,( X (1) + r − /r ↓ ( r − /r as n → ∞ (in probability) and X ( n ) /r > max(( r − /r, ( X (1) + r − /r ) a.s. since r ∈ (1 , n , we have Γ ( X n , r, ( r − /r ) = [( r − /r, ( X (1) + r − /r ) a.s. and p n ( F, r, ( r − /r ) ≈ Z ε Z − ε n ( n − f ( x ) f ( x n ) h F ( x n ) − F ( x )+ F (( r − /r ) − F (( x + r − /r ) i n − dx n dx = Z ε nf ( x ) h − F ( x ) + F (( r − /r ) − F (( x + r − /r ) i n − − h − ε − F ( x ) + F (( r − /r ) − F (( x + r − /r ) i n − ! dx ≈ Z ε nf ( x ) h − F ( x ) + F (( r − /r ) − F (( x + r − /r ) i n − dx . (36)Let G ( x ) = 1 − F ( x ) + F (( r − /r ) − F (( x + r − /r ) . The integral in Equation (36) is critical at x = 0, since G (0) = 1, and for x ∈ (0 , 1) the integral converges to0 as n → ∞ . Let α i := − d i +1 G ( x ) dx i +11 (cid:12)(cid:12)(cid:12) (0 + , + ) = f ( i ) (0 + ) + r − ( i +1) f ( i ) (cid:16)(cid:0) r − r (cid:1) + (cid:17) . Then by the hypothesis of thetheorem, we have α i = 0 and f ( i ) (cid:16)(cid:0) r − r (cid:1) + (cid:17) = 0 for all i = 0 , , , . . . , ( k − f ( x ) around x = 0 + up to order k and G ( x ) around 0 + up to order ( k + 1) so that x ∈ (0 , ε ), are asfollows: f ( x ) = 1 k ! f ( k ) (0 + ) x k + O (cid:0) x k +11 (cid:1) and G ( x ) = G (0 + ) + 1( k + 1)! (cid:18) d k +1 G (0 + ) dx k +11 (cid:19) x k +11 + O (cid:0) x k +21 (cid:1) = 1 − α k ( k + 1)! x k +11 + O (cid:0) x k +21 (cid:1) . Then substituting these expansions in Equation (36), we obtain p n ( F, r, ( r − /r ) ≈ Z ε n " k ! f ( k ) (0 + ) x k + O (cid:0) x k +11 (cid:1) − α k ( k + 1)! x k +11 + O (cid:0) x k +21 (cid:1) n − dx . Now we let x = w n − / ( k +1) to obtain p n ( F, r, ( r − /r ) ≈ Z ε n / ( k +1) n " n k/ ( k +1) k ! f ( k ) (0 + ) w k + O (cid:0) n − (cid:1) − n (cid:18) α k ( k + 1)! w k +1 + O (cid:16) n − ( k +2) / ( k +1) (cid:17)(cid:19) n − (cid:18) n / ( k +1) (cid:19) dw letting n → ∞ , ≈ Z ∞ k ! f ( k ) (0 + ) w k exp (cid:20) − α k ( k + 1)! w k +1 (cid:21) dw = f ( k ) (0 + ) α k = f ( k ) (0 + ) f ( k ) (0 + ) + r − ( k +1) f ( k ) (cid:16)(cid:0) r − r (cid:1) + (cid:17) , (37)27s n → ∞ at rate O ( κ ( f ) · n − ( k +2) / ( k +1) ).For the general case of Y = { y , y } , the transformation φ ( x ) = ( x − y ) / ( y − y ) maps ( y , y ) to (0 , 1) andthe transformed random variables U = φ ( X i ) are distributed with density g ( u ) = ( y − y ) f ( y + u ( y − y ))on ( y , y ). Replacing f ( x ) by g ( x ) in Equation (37), the desired result follows. (cid:4) Proof of Theorem 4.5 Case (i) follows trivially from Theorem 2.3. The special cases for n = 1 and r = { , ∞} follow by construction.Case (ii): Suppose ( y , y ) = (0 , 1) and c ∈ (1 / , ε ∈ (0 , /r ). Then P (cid:0) X (1) < ε, X ( n ) > − ε (cid:1) → n → ∞ with the rate of convergence depending on F . Moreover, for sufficiently large n , X ( n ) /r < /r a.s.; infact, X ( n ) /r ↑ /r as n → ∞ (in probability) and ( X (1) + r − /r < min(1 /r, X ( n ) /r ) a.s. Then for sufficientlylarge n , Γ ( X n , r, c ) = ( X ( n ) /r, /r ] a.s. and p n ( F, r, /r ) ≈ Z − ε Z ε n ( n − f ( x ) f ( x n ) h F ( x n ) − F ( x ) + F ( x n /r ) − F (1 /r ) i n − dx dx n . = − Z − ε nf ( x n ) h F ( x n ) − F ( ε ) + F ( x n /r ) − F (1 /r ) i n − − h F ( x n ) + F ( x n /r ) − F (1 /r ) i n − ! dx n ≈ Z − ε nf ( x n ) h F ( x n ) + F ( x n /r ) − F (1 /r ) i n − dx n . (38)Let G ( x n ) = F ( x n ) + F ( x n /r ) − F (1 /r ) . The integral in Equation (38) is critical at x n = 1, since G (1) = 1, and for x n ∈ (0 , 1) the integral convergesto 0 as n → ∞ . So we make the change of variables z n = 1 − x n , then G ( x n ) becomes G ( z n ) = F (1 − z n ) + F ((1 − z n ) /r ) − F (1 /r ) , and Equation (38) becomes p n ( F, r, /r ) ≈ Z ε n f (1 − z n ) [ G ( z n )] n − dz n . (39)The new integral is critical at z n = 0. Let β i := ( − i +1 d i +1 G ( z n ) dz i +1 n (cid:12)(cid:12)(cid:12) + = f ( i ) (1 − ) + r − ( i +1) f ( i ) (cid:16)(cid:0) r (cid:1) − (cid:17) . Thenby the hypothesis of the theorem, we have β i = 0 and f ( i ) (cid:16)(cid:0) r (cid:1) − (cid:17) = 0 for all i = 0 , , , . . . , ( ℓ − f (1 − z n ) around z n = 0 + up to ℓ and G ( z n ) around 0 + up to order ( ℓ + 1) so that z n ∈ (0 , ε ), are as follows: f (1 − z n ) = ( − ℓ ℓ ! f ( ℓ ) (1 − ) z ℓn + O (cid:0) z ℓ +1 n (cid:1) G ( z n ) = G (0 + ) + 1( ℓ + 1)! (cid:18) d ℓ +1 G (0 + ) dz ℓ +1 n (cid:19) z ℓ +1 n + + O (cid:0) z ℓ +2 n (cid:1) = 1 + ( − ℓ +1 β ℓ ( ℓ + 1)! z ℓ +1 n + O (cid:0) z ℓ +2 n (cid:1) . Then substituting these expansions in Equation (39), we get p n ( F, r, /r ) ≈ Z ε n " ( − ℓ ℓ ! f ( ℓ ) (1 − ) z ℓn + O (cid:0) z ℓ +1 n (cid:1) − ( − ℓ β ℓ ( ℓ + 1)! z ℓ +1 n + O (cid:0) z ℓ +2 n (cid:1) n − dz n . z n = v n − / ( ℓ +1) , to obtain p n ( F, r, /r ) ≈ Z ε n / ( ℓ +1) n " ( − ℓ n ℓ/ ( ℓ +1) ℓ ! f ( ℓ ) (1 − ) v ℓ + O (cid:0) n − (cid:1) − n (cid:18) ( − ℓ β ℓ ( ℓ + 1)! v ℓ +1 + O (cid:16) n − ( ℓ +2) / ( ℓ +1) (cid:17)(cid:19) n − (cid:18) n / ( ℓ +1) (cid:19) dv letting n → ∞ , ≈ Z ∞ ( − ℓ ℓ ! f ( ℓ ) (1 − ) v ℓ exp (cid:20) − ( − ℓ β ℓ ( ℓ + 1)! v ℓ +1 (cid:21) dv = f ( ℓ ) (1 − ) β ℓ = f ( ℓ ) (1 − ) f ( ℓ ) (1 − ) + r − ( ℓ +1) f ( ℓ ) (cid:16)(cid:0) r (cid:1) − (cid:17) , (40)as n → ∞ at rate O ( κ ( f ) · n − ( ℓ +2) / ( ℓ +1) ).For the general case of Y = { y , y } , as in the proof of Theorem 4.4, using the transformation φ ( x ) =( x − y ) / ( y − y ) and replacing f ( x ) by g ( x ) in Equation (40), the desired result follows. (cid:4)(cid:4)