LLONGEST k -MONOTONE CHAINS GERGELY AMBRUS
Alfr´ed R´enyi Institute of Mathematics
Abstract.
We study higher order convexity properties of random point sets in the unitsquare. Given n uniform i.i.d random points, we derive asymptotic estimates for themaximal number of them which are in k -monotone position, subject to mild boundaryconditions. Besides determining the order of magnitude of the expectation, we also provestrong concentration estimates. We provide a general framework that includes the pre-viously studied cases of k = 1 (longest increasing sequences) and k = 2 (longest convexchains). Higher order convexity
Let X n be a set of n uniform, independent random points in the unit square [0 , . It isa classical and well studied problem to determine the maximal number of points forminga monotone increasing chain in X n , i.e. a set of points p , . . . , p m in X n so that boththe x -coordinates and the y -coordinates of ( p i ) m form an increasing sequence. This is thegeometric analogue of the famous question of longest increasing subsequences in randompermutations, first mentioned in 1961 by Ulam [21], which has been studied extensivelyever since (see e.g. [1, 6, 18]). Let L n denote the maximum number of points of X n forming a monotone increasing chain. The order of magnitude of the expectation of L n wasdetermined by Hammersley about half a century ago, with the exact value of the constant inthe asymptotics determined five years later by Vershik and Kerov [23], and independently,by Logan and Shepp[16]: Theorem 1.1 ([13], [16], [23]) . As n → ∞ , E L n ∼ n / . This result serves as the starting point for the current research. We are going to studypoint sets which satisfy a more general monotonicity criteria. We start off with a basicconcept.
Definition 1.2.
A set of points p , . . . , p m is a chain if their x -coordinates form a monotoneincreasing sequence. The length of the chain is the cardinality of the point set, that is, m . Next, one may study points of the random sample X n forming a convex chain. Themotivation is two-fold. On the one hand, convex polygons with vertices among a randomsample have been studied extensively in the last 50 years (see e.g. the excellent survey ofB´ar´any [7] or the monograph of Schneider and Weil [19]). On the other hand, given how E-mail address : [email protected] . Date : September 30, 2020.Research of the author was supported by NKFIH grants PD125502 and K116451 and by the BolyaiResearch Scholarship of the Hungarian Academy of Sciences. a r X i v : . [ m a t h . M G ] S e p LONGEST K -MONOTONE CHAINS fruitful and far-reaching the research of the monotone increasing subsequences has been, itis a natural attempt to transfer the results to the convex analogue.The first steps in that direction were took in our joint paper with I. B´ar´any [5], where westudied the order of magnitude of the maximal number of points of X n forming a convexchain together with (0 ,
0) and (1 , L n denote the maximal number of points of X n in a convex chain lying under thediagonal y = x . Theorem 1.3 ([5]) . There exists a positive constant α , so that as n → ∞ , E L n ∼ α n / . We also proved a limit shape result for the longest convex chains and established upperand lower estimates for α . Alternative proofs to some the results are given in [3] and [4].The goal of the present paper is to study the analogous questions for higher order con-vexity, and to describe a unified framework to the above results. Note that the propertiesstudied above are equivalent to non-negativity of the first (monotone increasing property)and second (convexity property) “ discrete derivatives” of the chains. Therefore, it is natu-ral to define higher order convexity along this scheme. Eli´as and Matouˇsek introduced thefollowing concept in order to establish Erd˝os-Szekeres type results: Definition 1.4 (Eli´as and Matouˇsek, [10]) . The ( k +1) -tuple ( p , . . . , p k +1 ) of distinct pointsin the plane is called positive , if it lies on the graph of a function whose k -th derivativeexists everywhere, and is nowhere negative. The points ( p , . . . , p m ) in the plane form a k -monotone chain if their x -coordinates are monotone increasing, and every ( k + 1) -tupleof them is positive. Note that in the present paper, “monotone” will always refer to monotone increasing .It would be an alternative to use the term “ k -convex”. However, there are already variousother concepts existing by that name, thus we stick to “ k -monotone”.A second, important remark points out the difference between cases of k = 1 ,
2, and largervalues of k . In the above definition, positivity of different ( k + 1)-tuples may be demon-strated by different functions. For k = 1 ,
2, there exists a single monotone/convex functioncontaining all the points on its graph. The same property was conjectured to hold also forlarger values of k by Eli´aˇs and Matouˇsek [10]. However, Rote found a counterexample for k = 3 [10].An alternative but equivalent definition may be given, see Corollary 2.3 of [10]: a ( k + 1)-tuple is positive iff its k th divided difference is nonnegative, where divided differences aredefined as follows. Assume p , . . . , p n are points in the plane of the form p i = ( x i , y i )(note that here, the x -coordinates do not necessarily form an increasing sequence). The j th (forward) divided difference ∆ j ( p i , . . . , p i + j +1 ) of the ( j + 1)-tuple p i , . . . , p i + j is definedrecursively by ∆ ( p i ) := y i ∆ j ( p i , . . . , p i + j ) := ∆ j − ( p i +1 , . . . , p i + j ) − ∆ j − ( p i , . . . , p i + j − ) x i + j − x i (1)for every 0 (cid:54) i (cid:54) n − j . Note that divided differences (and, hence, positivity of a ( k + 1)-tuple) are invariant under permutations.Divided differences are used in polynomial approximation; in particular, they providethe coefficients for the summands of Newton’s interpolating polynomial: ONGEST k -MONOTONE CHAINS 3 Lemma 1.5 (Newton interpolating polynomial) . Let p , . . . , p k +1 be points in the planewith distinct x -coordinates. Assume that p i = ( x i , y i ) . The unique polynomial P ( x ) ofdegree k whose graph contains all the points p i for i = 1 , . . . , k + 1 may be expressed as (2) P ( x ) = k (cid:88) j =0 ∆ j ( p , . . . , p j +1 ) j (cid:89) i =1 ( x − x i ) . Divided differences are also related to higher order derivatives by the following general-ization of the mean value theorem (see [17], Eq. 1.33):
Lemma 1.6 (Cauchy) . Assume that the points p , . . . , p k +1 have increasing x -coordinates a := x < . . . < x k +1 =: b , and they lie on the graph of a function f which is k timesdifferentiable everywhere on the interval [ a, b ] . Then there exists ξ ∈ ( a, b ) so that (3) ∆ k ( p , . . . , p k +1 ) = f ( k ) ( ξ ) k ! . Next, we extend the definition of divided differences to multisets of points. For a point p , introduce the notation p ◦ i = { p, . . . , p (cid:124) (cid:123)(cid:122) (cid:125) i } , that is, the multiset of p with multiplicity i . Assume p = ( x, f ( x )) is a point on the graphof a function f , which is k times differentiable at x . In accordance with (3), we define the i th divided difference of p ◦ ( i +1) with respect to f by(4) ∆ i ( p ◦ ( i +1) ; f ) := f ( i ) ( x ) i !for every i (cid:54) k . Note that this agrees with the limit of ∆ i (˜ p , . . . , ˜ p i +1 ) as ˜ p , . . . , ˜ p i +1 converge to p along the graph of f .By repeatedly applying (1), we may define divided differences up to order k with respectto a function f of any multiset of points lying on the graph of a k -times differentiablefunction f . Therefore, we may extend the k -monotonicity property to multisets of pointswith respect to f , provided that all points of multiplicity larger than 1 lie on the graphof f .From now on, we assume that all ( k + 1)-tuples of X n are in k -general position , that is,they do not lie on the graph of a polynomial of degree at most k −
1. This property holdswith probability 1.Under this assumption, positivity of ( k + 1)-tuples is a transitive property: Lemma 1.7 ([10], Lemma 2.5) . Assume the distinct points p , . . . , p k +2 form a chain, theyare in k -general position, and that both ( k + 1) -tuples ( p , . . . , p k +1 ) and ( p , . . . , p k +2 ) arepositive. Then any ( k + 1) -element subset of ( p , . . . , p k +2 ) is positive. In other words, the 2-coloring of ( k + 1)-tuples given by positivity/non-positivity is atransitive coloring (defined in [10] and [11]). This will prove to be crucial in the subsequentarguments. In particular, it implies that in order to check k -monotonicity of a chain, itsuffices to check positivity of all of its intervals (i.e. sets of consecutive points) of length k + 1. LONGEST K -MONOTONE CHAINS Results
Our goal is to determine the order of magnitude of the maximum number of pointsin a uniform random sample from the unit square which form a k -monotone chain. Fortechnical reasons, we also impose boundary conditions on the chain – these conditions willensure that a k -monotone chain is also l -monotone for every l (cid:54) k . In the case k = 1,the boundary condition simply requires the monotone chain to start at (0 ,
0) and finish at(1 , y = x . For general k , we introduce the curveΓ k = ( x, x k ) , x (cid:62) x (cid:62)
0, let(5) γ k ( x ) = ( x, x k ) ∈ Γ k . For the point γ k ( x ), we write∆ i ( γ k ( x ) ◦ ( i +1) ) := ∆ i ( γ k ( x ) ◦ ( i +1) ; x k ) = k ( k − . . . ( k − i + 1) x k − i , that is, we consider k -monotonicity with respect to f ( x ) = x k . Note that for x = 0 and x = 1, γ k ( x ) is the same for every k ; however, ambiguity is avoided by specifying the valueof k .The setup is the following. Fix k (cid:62)
1. For any n (cid:62)
1, as set before, let X n be a set of n i.i.d. uniform random points in the unit square [0 , . Definition 2.1.
Let M k ( X n ) be the set of all chains ( p , . . . , p m ) of X n so that (6) ( γ k (0) ◦ k , p , p , . . . , p m , γ k (1) ◦ k ) is a k -monotone chain. Furthermore, let L k ( X n ) =: L kn denote the maximal cardinality ofelements of M k ( X n ) . Thus, L kn is a random variable defined on the space of n -element i.i.d. uniform samplesfrom the square. Note that the definition of divided differences and the boundary conditionat γ k (0) implies that up to order k , the j th divided differences of the consecutive ( j + 1)-tuples of the chain form a monotone increasing sequence, starting at 0. Therefore,(7) M k ( X n ) ⊂ M j ( X n )for every j (cid:54) k .We first extend Hammersley’s result [13] by generalizing Theorem 1.1 and Theorem 1.3. Theorem 2.2.
For any k (cid:62) there exists a positive constant α k so that lim n →∞ n − k +1 E L kn = α k . Furthermore, n − k +1 L k ( X n ) → α k almost surely, as n → ∞ . The exact value of the constant is not known except for the case k = 1, where α = 2holds. However, we may estimate it from below: Proposition 2.3.
For every k (cid:62) , α k (cid:62) . By utilizing the deviation estimates of Talagrand [20], we obtain the strong concentrationproperty of L kn : ONGEST k -MONOTONE CHAINS 5 Theorem 2.4.
For every k (cid:62) , and for every ε > , P (cid:18) | L kn − E L kn | > εn k +1) (cid:19) (cid:54) e − ε / α k holds for every sufficiently large n . Finally, we conjecture that similarly to the k = 2 case [5], the above stochastic concen-tration property leads to geometric concentration : it implies that the limit shape of longest k -monotone chains satisfying the boundary conditions is Γ k . Conjecture 2.5.
For any k (cid:62) , the longest k -monotone chains converge in probability to Γ k . That is, for any ε > , P ( ∃ longest k -monotone chain with distance > ε from Γ k ) → as n → ∞ . We stated the results for X n being chosen from the unit square. As we will see inSection 3, the boundary conditions imply that only a small fraction of the square plays arole here: all members of M k ( X n ) lie in C k (0 , / ( k k − ) (see (13)); therefore, switching the base domain from the square to C k (0 , α k by a factor of 2, without changing the order of magnitude of E L kn .3. Geometric properties
We start with a geometric characterization of positivity. Let ( p , . . . , p k +1 ) be a ( k + 1)-tuple. We define its sign to be the sign of ∆ k ( p , . . . , p k +1 ). Lemma 3.1 ([10], Lemma 2.4.) . Assume that P = ( p , . . . , p k +1 ) is a chain in k -generalposition. For any i ∈ [ k + 1] , let P i be the unique polynomial of degree k − containing allthe points p j , j (cid:54) = i . The ( k + 1) -tuple P has sign ( − k − i if p i lies below the graph of P i ,and has sign ( − k − i +1 if p i lies above the graph. We may naturally extend the above statement for chains containing multiple points. Inthe next lemma, we illustrate this for the special case when the chain consists of only 3points, with the two endpoints having multiplicity larger than 1. The same method canbe applied for the general case as long as the points of multiplicity lie on the graph of afunction f with the necessary differentiability properties: if p = ( x, f ( x )) has multiplicity β in the chain, then the approximating polynomial P is required to have derivatives agreeingwith those of f up to order β − x .Below, we define f (0) ( x ) := f ( x ). Lemma 3.2.
Let q = ( a, f ( a )) and ˜ q = ( b, f ( b )) , a < b be points on the graph of a function f , which is ( k − -times differentiable in the interval [ a, b ] . Assume furthermore that f ( k − does not vanish on [ a, b ] . Let (cid:54) i (cid:54) k +1 , and denote by Φ i,k ( a, b, f ) the unique polynomialof degree k − which satisfies Φ ( j ) i,k ( a, b, f )( a ) = f ( j ) ( a ) for every (cid:54) j (cid:54) i − , and Φ ( j ) i,k ( a, b, f )( b ) = f ( j ) ( b ) for every (cid:54) j (cid:54) k − i. (8) Assume that for the point p (cid:54) = q, ˜ q , the ( k + 1) -tuple P (cid:48) = ( q ◦ ( i − , p, ˜ q ◦ ( k − i +1) ) is a chain.Then P (cid:48) has sign ( − k − i if p lies below the graph of Φ ( j ) i,k ( a, b, f ) , and has sign ( − k − i +1 if p lies above the graph. LONGEST K -MONOTONE CHAINS Proof.
The statement follows directly from Lemma 3.1 by letting the points p , . . . , p i − converge to q , and p i +1 , . . . , p k +1 converge to ˜ q , along the graph of f . By Lemma 1.6and (4), all the divided differences of P converge to the corresponding divided differencesof ( q ◦ ( i − , p, ˜ q ◦ ( k − i +1) ). Moreover, the polynomial P i converges to Φ i,k ( a, b, f ), which isshown by convergence of the derivatives. Therefore, the statement follows. (cid:3) Note that the ( k − f on the interval ( a, b ) is not fully used;we only prescribe the derivatives of Φ i,k ( a, b, f ) at a and b up to order i − k − i − k +1)-tuple of the form ( q ◦ k , p ), and let Φ k ( a, f ) :=Φ k,k − ( a, b, f ) denote the above defined polynomial. If k is even, then for any other point p , the ( k + 1)-tuple ( q ◦ k , p ) is positive, if p lies above the graph of Φ k ( a, f ). If k is odd,then for any other point q = ( x, y ), the ( k + 1)-tuple ( q ◦ k , p ) is positive, if x < a and p liesbelow the graph of Φ k ( a, f ), or if x > a , and p lies above the graph of Φ k ( a, f ).Next, we apply Lemma 3.2 for the multisets of the form ( γ k ( a ) ◦ k , p, γ k ( b ) ◦ k ), where0 (cid:54) a < b , and k (cid:62)
1, see (5), (6). Some notations are in order. Let a, b (cid:62)
0. Define thepolynomials Φ k ( a )( x ) = x k − ( x − a ) k (9) Ψ k ( a, b )( x ) = x k − ( x − a ) k − ( x − b ) . (10)With a slight abuse of notation, we are going to denote the graphs of these polynomials bythe same symbols. It will always be clear from the context which meaning do we refer to. Definition 3.3.
For (cid:54) a < b , the cell C k ( a, b ) is defined as follows: For k = 1: C ( a, b ) = [ a, b ] , the square with diagonal vertices γ ( a ) and γ ( b ) ; For k = 2: C ( a, b ) is the triangle bounded by Φ k ( a ) , Φ k ( b ) , and Ψ k ( a, b ) = Ψ k ( b, a ) ; For k (cid:62) C k ( a, b ) is the 4-vertex cell bounded by Φ k ( a ) , Φ k ( b ) , Ψ k ( a, b ) and Ψ k ( b, a ) . Figure 1.
The cell C k ( a, b ) between γ k ( a ) (left endpoint) and γ k ( b ) (rightendpoint), k odd, shaded with blue. The curve Γ k runs in the middle ofthe cell. The lower boundary consists of max { Φ k ( a ) , Ψ k ( b, a ) } , the upperboundary is defined by min { Ψ k ( a, b ) , Φ k ( b ) } .Let us elaborate on the k (cid:62) k is even, the lower boundary of C k ( a, b ) consists of two arcs: Φ k ( a ) for a (cid:54) x (cid:54) ( a + b ) /
2, and Φ k ( b ) for ( a + b ) / (cid:54) x (cid:54) b .The upper boundary again consists of two arcs: Ψ k ( a, b ) for a (cid:54) x (cid:54) ( a + b ) /
2, and Ψ k ( b, a ) ONGEST k -MONOTONE CHAINS 7 for ( a + b ) / (cid:54) x (cid:54) b (for k = 2, these coincide with each other). When k is odd, thelower and upper boundaries of C k ( a, b ) are the same as in the even case; however, for( a + b ) / (cid:54) x (cid:54) b , Φ k ( b ) is the upper boundary, while Ψ k ( b, a ) is the lower boundary. Thus,in any case, γ k ( a ) and γ k ( b ) are two opposite vertices of C k ( a, b ), and other two verticesboth have x -coordinates ( a + b ) / C k ( a, b ) is given by the following statement. Lemma 3.4.
Let k (cid:62) , and (cid:54) a < b . The set of points p in the plane for which ( γ k ( a ) ◦ k , p, γ k ( b ) ◦ k ) is a k -monotone chain is exactly C k ( a, b ) .Proof. We prove the statement first assuming a >
0. The case a = 0 may be obtained bya standard limit argument.Let Φ i,k ( a, b, x k ) be the polynomial defined by (8) for f ( x ) = x k . By comparing deriva-tives, we obtain that for every 1 (cid:54) i (cid:54) k + 1,(11) Φ i,k ( a, b, x k ) = x k − ( x − a ) i − ( x − b ) k +1 − i . By Lemma 3.2, our goal is to determine the intersection of the regions above Φ i,k ( a, b, x k )for i = k + 1 , k − , k − , . . . , k,
2) and the regions below Φ i,k ( a, b, x k ) for i = k, k − , k − , . . . , − mod( k,
2) (see Figure 2). Thus, we have to determinemax i ≡ k +1(mod 2)1 (cid:54) i (cid:54) k +1 Φ i,k ( a, b, x k )and min i ≡ k (mod 2)1 (cid:54) i (cid:54) k +1 Φ i,k ( a, b, x k )for every x ∈ [ a, b ]. By (11), we obtain that for x ∈ [ a, ( a + b ) / i = k + 1, and i = k , respectively, while for x ∈ [( a + b ) / , b ], the extrema are takenwhen i = 1 ,
2. Therefore, the boundary of C k ( a, b ) is constituted by the polynomials of theform (9) and (10). (cid:3) Figure 2.
Graphs of the polynomials x k − Φ i,k ( a, b, x k ), 1 (cid:54) i (cid:54) k + 1,plotted between a and b , in the k = 7 case.The next lemma states that not only C k ( a, b ) is the location of the k -monotone chains,but given two k -monotone chains in neighboring cells, we may concatenate them. LONGEST K -MONOTONE CHAINS Lemma 3.5.
Assume (cid:54) a < b < c , and that the points p , . . . , p l and p l +1 , . . . , p m are sothat ( γ k ( a ) ◦ k , p , . . . , p l , γ k ( b ) ◦ k ) and ( γ k ( b ) ◦ k , p l +1 , . . . , p m , γ k ( c ) ◦ k ) are k -monotone chainsin C k ( a, b ) and C b,c , respectively. Then ( γ k ( a ) ◦ k , p , . . . , p m , γ k ( c ) ◦ k ) is a k -monotone chain in C k ( a, c ) .Proof. By the remark following Lemma 1.7,( γ k ( a ) ◦ k , p , . . . , p l , γ k ( b ) ◦ k , p l +1 , . . . , p m , γ k ( c ) ◦ k )is a k -monotone chain. By the transitivity property provided by Lemma 1.7, we may deletefrom this chain any point, still maintaining k -monotonicity. Therefore,( γ k ( a ) ◦ k , p , . . . , p l , γ k ( b ) ◦ ( k − , p l +1 , . . . , p m , γ k ( c ) ◦ k )is also k -monotone. By iterating the erasure process, we finally erase all copies of γ k ( b ),yielding the statement. (cid:3) Next, we introduce a transformation mapping C k ( a, b ) to C k ( c, d ) which preserves k -monotonicity, showing the equivalence of the cells C k ( a, b ) with respect to problems regard-ing k -monotone chains.Let 0 < a < b and 0 < c < d , and define the transformation T a,b,c,d : R → R by(12) T a,b,c,d ( x, y ) = (cid:32) c + ( x − a ) d − cb − a , (cid:18) c + ( x − a ) d − cb − a (cid:19) k + ( y − x k ) (cid:18) d − cb − a (cid:19) k (cid:33) . Lemma 3.6.
For any < a < b and < c < d , the map T a,b,c,d preserves k -monotonicity,keeps Γ k fixed, and maps the uniform distribution on C k ( a, b ) onto the uniform distributionon C k ( c, d ) .Proof. Notice that T a,b,c,d ( a + t ( b − a ) , ( a + t ( b − a )) k + ˜ y ) = (cid:32) c + t ( d − c ) , ( c + t ( d − c )) k + ˜ y (cid:18) d − cb − a (cid:19) k (cid:33) . Thus, it is immediate that T a,b,c,d (Γ k ) = Γ k (as this corresponds to the case ˜ y = 0). Onthe boundary of C k ( a, b ), ˜ y is either ± ( t ( b − a )) k , ± ((1 − t )( b − a )) k , ± t k − (1 − t )( b − a ) k ,or ± t (1 − t ) k − ( b − a ) k , which are mapped to the same expressions with ( d − c ) in place of( b − a ). Therefore, T a,b,c,d maps C k ( a, b ) onto C k ( c, d ) by mapping the boundary curves tothe corresponding ones. Measure invariance is seen by writing T a,b,c,d = G − k ◦ A ◦ G k , where G k : ( x, y ) (cid:55)→ ( x, y − x k ), and A is an affine map. Thus, it only remains to checkthe invariance of k -monotonicity under T a,b,c,d . To this end, we may write T a,b,c,d as thecomposition of three maps: T a,b,c,d = T ,d − c,c,d ◦ T ,b − a, ,d − c ◦ T a,b, ,b − a . Assume f ( x ) has the form f ( x ) = x k + g ( x ), then f ( k ) ( x ) = k ! + g ( k ) ( x ). The first and thirdof the above maps do keep this derivative fixed, as g ( x ) is preserved by them. Finally, themap T ,b − a, ,d − c is a linear map scaling the x and y coordinates independently, therefore,it preserves the sign of the derivatives. Therefore, using Definition 1.4, T a,b,c,d preservespositivity of ( k + 1)-tuples, hence it also preserves positivity. (cid:3) ONGEST k -MONOTONE CHAINS 9 We conclude this section by calculating the area of the base cell C k ( a, b ). By (9), (10) andthe discussion afterwards, the distance between the upper and lower boundary of C k ( a, b )is ( x − a ) k − ( b − x ). Therefore, A ( C k ( a, b )) = (cid:90) ( a + b ) / a ( x − a ) k − ( b − a ) dt + (cid:90) b ( a + b ) / ( b − x ) k − ( b − a ) dt = ( b − a ) k +1 k k − . (13) 4. The Poisson model
In order to make the problem more approachable, in this section we switch to the
Poissonmodel , that has by now became an industry standard (see e.g. [18]).Let Π be a planar homogeneous Poisson process with intensity 1. Given any domain D of area A ( D ) in the plane, the number of points of Π in D has Poisson distribution withparameter A ( D ). That is, its probability mass function is given by(14) P ( | D ∩ P i | = k ) = A ( D ) k e − A ( D ) k ! , and its expectation is A ( D ).We are going to use the following standard tail estimate for Poisson random variables,see Proposition 1 of [12]. Assume that X has Poisson distribution with parameter λ . Then(15) P ( X (cid:62) m ) (cid:54) m + 1 m + 1 − λ P ( X = k ) = m + 1 m + 1 − λ e − λ λ m m ! . For arbitrary 0 (cid:54) a < b , let N k Π ( a, b ) be the cardinality of Π ∩ C k ( a, b ). By (13) and(14), N k Π ( a, b ) has a Poisson distribution with parameter (and mean) ( b − a ) k +1 / ( k k − ).Moreover, conditioning on the event N k Π ( a, b ) = N , the joint distribution of the points of Πfalling in C k ( a, b ) is the same as the joint distribution of N i.i.d. uniform points in C k ( a, b ).As the analogue of Definition 2.1, we introduce Definition 4.1.
Let M k Π ( a, b ) be the set of all chains ( p , . . . , p m ) ⊂ Π ∩ C k ( a, b ) so that (16) ( γ k ( a ) ◦ k , p , p , . . . , p m , γ k ( b ) ◦ k ) is a k -monotone chain. Furthermore, let L k ( a, b ) denote the maximal cardinality of elementsof M k Π ( a, b ) . By Lemma 3.6 and the invariance property of the Poisson process, the distribution of L k ( a, b ) depends solely on ( b − a ); therefore, the results below involving L k (0 , n ) remainalso valid for the general variables L k ( a, b ).Next, we establish the link between the Poisson and the uniform models. By (13),the area of C k (0 , n ) is n k +1 / ( k k − ), therefore, N k Π (0 , n ) has a Poisson distribution withparameter n k +1 / ( k k − ). On the other hand, let us denote by N kn the number of points of X n in C k (0 , N kn has binomial distribution with parameters n and 1 / ( k k − ), andits mean is n/ ( k k − ). Standard Chernoff type concentration estimates for binomial andPoisson random variables (see e.g. Chapter 2 of [9]) yield the following quantitative bound. Proposition 4.2.
For any k (cid:62) , and for any c > , P (cid:32) | N k Π (0 , n ) − N kn k +1 | > c (cid:114) n k +1 k k − (cid:33) < e − c / holds for every sufficiently large n . K -MONOTONE CHAINS This also implies that the random variable L k (0 , n ) is a good approximation of L kn k +1 .Applying Proposition 4.2 with c = εn ( k +1) / allows us to transfer the statement of Theo-rem 2.2 to the Poisson model. We are going to prove the following theorem, which readilyimplies Theorem 2.2. Theorem 4.3.
For any k (cid:62) , there exists a positive constant α k so that (17) lim n →∞ n − E L k (0 , n ) = α k . Furthermore, n − L k (0 , n ) → α k almost surely, as n → ∞ . First, we need an upper bound on E L k (0 , n ). In the uniform model for k = 2, this isfairly easy to establish. The probability that a random chain of length n is convex may becalculated exactly [8], based on a beautiful argument of Valtr [22]. The probability that n uniform independent random points in the unit square form a convex chain is exactly1 n !( n + 1)! . The calculation is based on rearranging convex chains while keeping the underlying proba-bility space invariant. Unfortunately, this approach brakes down for larger values of k , andthus, such a sharp result does not hold in the more general setting. However, we may stillprove that the probability of the existence of very long k -monotone chains is minuscule. Lemma 4.4.
For every k (cid:62) there exists a constant c k so that (18) E L k (0 , n ) < c k n holds for every n (cid:62) .Proof. As the statement is known for the cases k = 1 and k = 2, we may assume that k (cid:62) n . Let C be a constantwhose value we are going to specify later. Set N = n k +1 . As we noted before, N k Π (0 , n ) hasPoisson distribution with parameter n k +1 / ( k k − ). Therefore, using (15), E L k (0 , n ) = (cid:90) ∞ P ( L k (0 , n ) (cid:62) x ) dx (cid:54) Cn + (cid:90) NCn P ( L k (0 , n ) (cid:62) x ) dx + (cid:90) ∞ N P ( L k (0 , n ) (cid:62) x ) dx (cid:54) Cn + N P ( L k (0 , n ) (cid:62) Cn ) + (cid:90) ∞ N P ( N k Π (0 , n ) (cid:62) x ) dx (cid:54) Cn + N P ( L k (0 , n ) (cid:62) Cn ) + 2 ∞ (cid:88) i = N P ( N k Π (0 , n ) = i ) (cid:54) Cn + N P ( L k (0 , n ) (cid:62) Cn ) + 14 N . (19)Thus, it suffices to show that for suitably large C (depending on k only), P ( L k (0 , n ) (cid:62) Cn ) = o ( n − k ) . Call a k -monotone chain in Π ∩ C k (0 , n ) long if its cardinality is at least Cn . To every suchlong k -monotone chain C we assign its skeleton as follows. Assume that C = { p , . . . , p m } ONGEST k -MONOTONE CHAINS 11 (with the points ordered according to their x -coordinates), where m (cid:62) Cn . Thenskel ( C ) = { γ k (0) ◦ k , p (cid:98) mn (cid:99) , p (cid:98) mn (cid:99) , . . . , p n − (cid:98) mn (cid:99) , γ k ( n ) ◦ k } =: { γ k (0) ◦ k , s , . . . , s n − , γ k ( n ) ◦ k } , (20)that is, s i = p i (cid:98) m/n (cid:99) for every i = 1 , . . . , n −
1. Also, set s i = γ k (0) for i (cid:54) s j = γ k ( n )for j (cid:62) n . Any long chain is cut into n intervals of length at least C by its skeleton.The free part of the skeleton is a chain of length n − C k (0 , n ). Thedistribution of the long chains in Π ∩ C k (0 , n ) induces a probability distribution µ on thespace of skeletons. By the law of total probability, P ( L k (0 , n ) (cid:62) Cn ) = (cid:90) P ( ∃ a long k -monotone chain C | skel ( C ) = S ) dµ ( S ) , where the integral is taken over the space of possible skeleta. Thus, (19), implies (18) aslong as P ( ∃ a long k -monotone chain C | skel ( C ) = S ) < o ( n − k )holds true for every possible skeleton S , with the constants of the asymptotic estimate beingindependent of S . This is what we are going to prove.Let us now fix S of the form (20) and assume that C is a long k -monotone chain withskel ( C ) = S . Let p ∈ C \ skel ( C ). For any point u ∈ R , let x ( u ) denote its x -coordinate.There exists a unique index i so that x ( p ) ∈ [ x ( s i ) , x ( s i +1 )]. Then, by Definition 1.4 of k -monotone chains, the ( k + 1)-tuples( s i − k +1 , s i − k +2 , . . . , s i , p )and ( s i − k +2 , s i − k +3 , . . . , s i , p, s i +1 )are positive. Let P be the unique polynomial of degree k − s i − k +1 , s i − k +2 , . . . , s i , and similarly, let P be the unique polynomial of degree k − s i − k +2 , s i − k +3 , . . . , s i , s i +1 , possibly using the extendeddefinition for multisets discussed in Section 1. That is, if γ k (0) appears with multiplicity β among the nodes for P i for i = 1 or 2, than the derivatives up to order β − P i at 0 arerequired to agree with those of x k at 0.Lemma 3.1 and its generalization to multisets implies that the point p lies in the region R i bounded by the graphs of the polynomials P and P over the interval [ x ( s i ) , x ( s i +1 )].Lemma 1.5 and formula (2) shows that P ( x ) − P ( x ) = (cid:16) ∆ k − ( s i − k +1 , s i − k +2 , . . . , s i ) − ∆ k − ( s i − k +2 , s i − k +3 , . . . , s i , s i +1 ) (cid:17) k − (cid:89) j =1 ( x − x ( s i +1 − j )) . Therefore, A ( R i ) = (cid:90) x ( s i +1 ) x ( s i ) | P ( x ) − P ( x ) | (cid:54) (cid:0) x ( s i +1 ) − x ( s i ) (cid:1)(cid:0) x ( s i +1 ) − x ( s i − k +2 ) (cid:1) k − D i (cid:54) (cid:0) x ( s i +1 ) − x ( s i − k +2 ) (cid:1) k D i (21)with D i = ∆ k − ( s i − k +2 , s i − k +3 , . . . , s i , s i +1 ) − ∆ k − ( s i − k +1 , s i − k +2 , . . . , s i ) . K -MONOTONE CHAINS Since n − (cid:88) i =0 x ( s i +1 ) − x ( s i − k +2 ) = k − (cid:88) j =0 x ( s n − j ) − x ( s − j ) < kn, there are at least (2 n ) / i in the interval [0 , n −
1] so that(22) x ( s i +1 ) − x ( s i − k +2 ) (cid:54) k . On the other hand, the k -monotonicity of C implies that (∆ k − ( s i − k +1 , s i − k +2 , . . . , s i )) n + k − i =0 is a monotone increasing sequence, which, by (4), satisfies∆ k − ( s − k +1 , s − k +2 , . . . , s ) = 0and ∆ k − ( s n , s n +1 , . . . , s n + k − ) = kn . Thus, there exist at least (2 n ) / j in the interval [0 , n −
1] so that(23) ∆ k − ( s j − k +1 , s j − k +2 , . . . , s j ) (cid:54) k. Combining (22) and (23) with (21), we obtain that there at least n/ i ∈ [0 , n − A ( R i ) (cid:54) (3 k ) k +1 . In order for C to be long, each of these regions must contain at least C points of Π. Picksuch a region R . By (15) and Stirling’s approximation, P (cid:0) | R ∩ Π | (cid:62) C (cid:1) (cid:54) (cid:32) e (3 k ) k +1 C (cid:33) C holds for any sufficiently large C . Therefore, for any given ε >
0, there exists a correspond-ing C so that the above probability is bounded above by ε . For that choice of C , P ( ∃ a long k -monotone chain C | skel ( C ) = S ) (cid:54) n − (cid:89) i =0 P (cid:0) | R i ∩ Π | (cid:62) C (cid:1) (cid:54) ε n/ , where the independence property of the Poisson process is used in the first inequality. Theproof is finished by noting that he above expression is of order o ( n − k ) for sufficiently smallvalues of ε , and all the above estimates depend on k only. (cid:3) Expectation and concentration estimates
In this section, we show that the order of magnitude of the length of the longest k -monotone chains among n random points is n / ( k +1) . We are going to prove this in thePoisson model, Theorem 4.3, which implies Theorem 2.2 of the uniform model. The proofbuilds on Kingman’s subbaditive ergodic theorem. Below, we present a version of it alongwith an important extension by Liggett. Theorem 5.1 (Kingman’s subadditive ergodic theorem with Liggett’s extension [14, 15]) . Assume X n,m , n, m ∈ N , is a family of random variables satisfying the following conditions: S1) X l,n (cid:54) X l,m + X m,n whenever (cid:54) l < m < n ; S2)
For every s (cid:62) integer, the joint distributions of the process { X m + s,n + s } are thesame as those of { X m,n } ; S3)
For each n , E | X ,n | < ∞ and E X ,n > − cn for some constant c . ONGEST k -MONOTONE CHAINS 13 Then γ = lim n →∞ E X ,n n exists, X = lim n →∞ X ,n n exists almost surely, and E X = γ. Furthermore, if the stationary sequences ( X in, ( i +1) m ) ∞ i =1 are ergodic for any m (cid:62) , then X = γ almost surely.Proof of Theorem 4.3. We show that Conditions S1), S2) and S3) of Theorem 5.1 hold forthe family of random variables X m,n := − L k ( n, m ), n, m ∈ N , see Definition 4.1. Lemma 3.5shows that L k ( a, c ) (cid:62) L k ( a, b ) + L k ( b, c )for every 0 (cid:54) a < b < c , showing the validity of S1). The invariance property S2) followsfrom Lemma 3.6. Finally, S3) is implied by Lemma 4.4.Therefore, we may apply Theorem 5.1 to obtain that E L k (0 , n ) ≈ α k n with some pos-itive constant β k . Moreover, ( L k ( in, ( i + 1) n ) ∞ i =1 is a sequence of independent, identicallydistributed random variables, hence it is ergodic. Therefore, n − L k (0 , n ) converges to α k almost surely. (cid:3) Theorem 4.3 and Proposition 4.2 implies that L kn n / ( k +1) → α k almost surely, proving Theorem 2.2.Next, we derive a lower bound on the constant α k of (17). Proof of Proposition 2.3.
We are going to prove the statement in the Poisson model byshowing that for sufficiently large n , E L k (0 , n ) (cid:62) n a i = 3 i for every i ∈ [0 , (cid:98) n/ (cid:99) ]. By (13), the area of C k ( a i , a i +1 ) (see Definition 3.3)is A ( C k ( a i , a i +1 )) = 3 k k k − > . Since the number of points of Π in C k ( a i , a i +1 ) has Poisson distribution with parameter A ( C k ( a i , a i +1 )), P ( | Π ∩ C k ( a i , a i +1 ) | = 0) = e − A ( C k ( a i ,a i +1 )) < e . Let Y be the number of cells of the form C k ( a i , a i +1 ) in which Π has at least one point.Then Y ∼ B ( (cid:98) n (cid:99) , p ) with p > − /e > /
2. Let λ := E Y , then λ > n (1 − /e ) /
3. By astandard Chernoff-type bound for binomial random variables (see Theorem A.1.12 of [2]), P (cid:16) Y (cid:54) λ − c (cid:112) λ log λ (cid:17) < λ − c / . Therefore, for sufficiently large n ,(24) P ( Y > n/ ≈ . K -MONOTONE CHAINS Let us now take a point p of Π in each of the non-empty cells, and let S = { s , . . . , s Y } be the collection of these points ordered with respect to their x -coordinates. By the con-struction, (cid:16) γ k (0) ◦ k , s , γ k ( a ) ◦ k , s , . . . , γ k ( a (cid:98) n/ (cid:99) ) ◦ k , γ k ( n ) ◦ k (cid:17) is a k -monotone chain, where each s i is placed in its corresponding interval so that weobtain a chain. By repeatedly applying Lemma 3.5, we deduce that (cid:16) γ k (0) ◦ k , s , s , . . . , s Y , γ k ( n ) ◦ k (cid:17) is also a k -monotone chain. Therefore, L k (0 , n ) (cid:62) Y . The proof is finished by referringto (24), which shows that E L k (0 , n ) (cid:62) n/ (cid:3) We finish this section by establishing the exponential concentration estimate for L kn .Theorem 2.4 is straightforward consequence of Talagrand’s strong concentration inequality. Theorem 5.2 (Talagrand [20]) . Suppose Y is a real-valued random variable on a productprobability space Ω ⊗ n , and that Y is 1-Lipschitz with respect to the Hamming distance,meaning that | Y ( x ) − Y ( y ) | (cid:54) whenever x and y differ in one coordinates. Moreover assume that Y is f -certifiable . Thismeans that there exists a function f : N → N with the following property: for every x and b with Y ( x ) (cid:62) b there exists an index set I of at most f ( b ) elements, such that Y ( y ) (cid:62) b holds for every y agreeing with x on I . Let m denote the median of Y . Then for every s > we have P ( Y (cid:54) m − s ) (cid:54) (cid:32) − s f ( m ) (cid:33) and P ( Y (cid:62) m + s ) (cid:54) (cid:32) − s f ( m + s ) (cid:33) . The conditions of Theorem 5.2 are clearly satisfied by the random variable L kn with thecertificate function f ( b ) = b , by fixing the points of the longest k -monotone chain in X n .Since L kn (cid:54) n , exponential concentration ensures that the mean and the median are withina distance of O ( n / k +1) ) of each other. Thus, in the above estimates, m ≈ α k n / ( k +1) ,and setting s = εn / k +1) , we obtain Theorem 2.4.The same proof yields the analogous concentration estimate for L k (0 , n ): Theorem 5.3.
For every k (cid:62) , and for every ε > , P (cid:16) | L k (0 , n ) − E L k (0 , n ) | > ε √ n (cid:17) (cid:54) e − ε / α k . holds for every sufficiently large n . Summarizing the results proved in this section, we showed that L kn is a random variableexponentially concentrated in a neighbourhood of radius O ( n / k +1) ) around its mean,which converges to α k n / ( k +1) . ONGEST k -MONOTONE CHAINS 15 References [1] D. Aldous and P. Diaconis,
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