aa r X i v : . [ m a t h . P R ] D ec Asymptotics of the convex hull of spherical samples
Enkelejd Hashorva
Department of Actuarial Science, Faculty of Business and EconomicsHEC Lausanne, University of Lausanne
Abstract:
In this paper we consider the convex hull of a spherically symmetric sample in R d . Our main contributionsare some new asymptotic results for the expectation of the number of vertices, number of facets, area and the volumeof the convex hull assuming that the marginal distributions are in the Gumbel max-domain of attraction. Further, webriefly discuss two other models assuming that the marginal distributions are regularly varying or O -regularly varying. Key words and phrases:
Convex hull; max-domain of attractions; asymptotic results; Carnal distributions; extremevalue distributions.
AMS 2000 subject classification:
Primary 52A22; Secondary 60D05, 60F05,60G70.
Let X , . . . , X n , n ≥ R d , d ≥ CH [ X , . . . , X n ] their convexhull. Distributional and asymptotical properties of the random polytope CH [ X , . . . , X n ] are discussed by manyauthors, see e.g., R´eny and Sulanake (1963), Efron (1965), Raynaud (1970), Carnal (1970), Eddy and Gale (1981),Groeneboom (1988), Aldous et al. (1991), Carnal and H¨usler (1991), Dwyer (1991), Hueter (1992, 1999, 2004, 2005),Reitzner (2002, 2004), Buchta (2005), B´ar´any and Vu (2007), Mayer and Molchanov (2007) and the references therein.In this paper we deal with spherically symmetric random vectors assuming the stochastic representation X i d = R U , i = 1 , . . . , n, (1.1)with R > U which is uniformly distributed on the unit hypersphere of R d (here d = and below ⊤ stand for the equality of the distribution function, and the transpose sign, respectively).Next, if X = ( X , . . . , X d ) ⊤ is a spherically symmetric random vector with stochastic representation (1.1), then inview of Cambanis et al. (1981) X i d = R B / , ( d − / , i = 1 , . . . , d, (1.2)with B / , ( d − / a Beta distributed random variable with parameters 1 / , ( d − / R .Since X k , k ≤ d are symmetric about 0 by (1.2) X k , k ≤ d have the same distribution function denoted by Q d .Our main interest lies in the asymptotic properties of CH [ X , . . . , X n ]; specifically we focus on the asymptoticbehaviour of the expectation of the number of the vertices, facets, the surface area and the volume of the convexhull. Interesting asymptotic results for these quantities are derived in the seminal paper Carnal (1970) under explicitassumptions on the tail asymptotics of the distribution function F of R (bivariate setup d = 2).In fact, from the extreme value point of view, Carnal assumed that F is in the max-domain of attraction (MDA)of a univariate extreme value distribution G . It is well-known that G is either the Gumbel distribution Λ( x ) =exp( − exp( − x )) , x ∈ R , the Fr´echet distribution Φ γ ( x ) = exp( − x − γ ) , x > , γ >
0, or the Weibull distributionΨ γ ( x ) = exp( −| x | γ ) , x <
0. Naturally, we raise the question whether Carnal’s results can be derived under asymptoticrestrictions on Q d ? The answer is positive when Q d is in the MDA of some univariate distribution function, see Section3.Dwyer (1991) extends Carnal’s finding to the multidimensional setup assuming again that F is in the MDA of G . Inthe latter paper it is demonstrated that the investigation of the expectation of the number of vertices and facets isof interest for determine the running time of algorithms for constructing a representation of the facial lattice of theconvex hull of a given point set.In the Gumbel case ( G = Λ) the results of Carnal (1970) and Dwyer (1991) are valid for special distribution functions F with light exponential tails and infinite upper endpoint (referred below as Carnal distributions). Asymptotic resultsfor the expectation of the number of the vertices of the convex hull are to date not available when F is in the GumbelMDA and has a finite upper endpoint.Without going to mathematical details, we briefly mention the main contributions of this paper:a) Making use of extreme value theory, we extend the known results for the Carnal distributions F to the largerclass of univariate distribution functions in the Gumbel MDA. Furthermore, we obtain asymptotic estimates for theexpectation and the variance of the number of vertices of the convex hull as well as a CLT extending a fundamentaltheorem of Hueter (1999).b) We show that several existing results can be derived with similar assumptions on the marginal distribution function Q d giving a positive answer to the above question.c) A new result derived in this paper is the boundedness of the sequence of the expectation of the number of thevertices of the convex hull if either F , or Q d are O -regularly varying.Organisation of the paper: The main results are presented in Section 3 followed by a section dedicated to the proofs.We conclude the paper with an Appendix. We introduce first our notation, provide few results from extreme value theory, and review some known results for theconvex hull CH [ X , . . . , X n ] of a spherically symmetric sample X , . . . , X n as given in the Introduction.If H is the distribution function of a random variable Y (henceforth abbreviated as Y ∼ H ), then we write H := 1 − H for its survival function. Further we define the generalised inverse of H by H − ( s ) := inf { x : H ( x ) ≥ s } and denoteby x H := sup { x : H ( x ) < } the upper endpoint of H . We use similar notation for other distributions.Throughout in the following B α,β stands for a Beta random variable with positive parameters α, β with densityfunction x α − (1 − x ) β − Γ( α + β )Γ( α )Γ( β ) , x ∈ (0 , , where Γ( · ) is the Euler Gamma function.From extreme value theory (see e.g., Reiss (1989), Embrechts et al. (1997), Falk et al. (2004)) the univariate distributionfunction N is in the MDA of the univariate distribution function G , if for some constants a n > , b n , n ∈ N lim n →∞ sup t ∈ R (cid:12)(cid:12)(cid:12) N n ( a n t + b n ) − G ( t ) (cid:12)(cid:12)(cid:12) = 0 . (2.1)As mentioned above only three choices are possible for G , namely Λ , Φ γ or Ψ γ , with γ ∈ (0 , ∞ ). When G = Λ the upperendpoint x N of N can be finite of infinite. For both other cases, x N is either finite (Weibull) or infinite (Fr´echet).The characterisation of both Weibull and Fr´echet max-domain of attractions is closely related to the concept of theregularly varying functions. In the following a positive measurable function L is called slowly varying at infinity iflim u →∞ L ( us ) / L ( u ) = 1 for any s >
0. A regularly varying function with index γ ∈ R is the product of some L ( x )with x γ .Next, we briefly review some known results for the convex hull. Let v n , f n denote the number of the vertices and thefacets of CH [ X , . . . , X n ]. Referring to Dwyer (1991) we may write Z ∞ Q n − d ( s ) | dF ( s ) | ≤ E { v n } n ≤ (1 + o (1))2 d − Z ∞ [1 − − d +1 Q d ( s )] n − | dF ( s ) | , n → ∞ (2.2)(we abuse slightly the notation writing P k ( s ) instead of ( P ( s )) k , k ∈ R for P some arbitrary function). Furthermore E { v n } ≤ E { f n } ∼ n d d κ d Z ∞ δ ( r ) q dd ( r ) exp( − nQ d ( r )) dr, n → ∞ , (2.3)where q d is the density function of Q d (which exists, see (5.1) in Appendix) and δ ( r ) is bounded by (see Lemma 4 inDwyer (1991)) δ ( r ) ≤ (1 + o (1)) dτ d − κ d − q d ( r ) Z ∞ r ( u − r ) d − u dF ( u ) , r → ∞ , (2.4)where τ d := √ d (1 + 1 /d ) ( d +1) / Γ( d + 1) , κ d := 2 π d/ Γ( d/ . (2.5)In (2.3) and below a u ∼ b u , u ↑ ω, with ω ∈ ( −∞ , ∞ ] means that lim u ↑ ω a u /b u = 1. Further, we write a n ∼ b n insteadof lim n →∞ a n /b n = 1.In the two-dimensional setup d = 2 (see Carnal (1970))2 E { v n } ∼ n Z ∞ Q n − ( s ) | dH ( s ) | , (2.6)with H ∼ min( R , R ) q B / , / , R d = R d = R, (2.7)where R , R , B / , / are mutually independent, see A1 for the proof.Referring again to Dwyer (1991), we have for the surface area A n and the volume V n of the convex hull E { A n } ∼ n d d τ d Z ∞ δ ( r ) q dd ( r ) exp( − nQ d ( r )) dr (2.8)and E { V n } ∼ n d τ d Z ∞ rδ ( r ) q dd ( r ) exp( − nQ d ( r )) dr, (2.9)where δ can be bounded asymptotically by δ ( r ) ≤ (1 + o (1)) dτ d − κ d − q d ( r ) Z ∞ r ( u − r ) (3 d − / u dF ( u ) , r → ∞ . (2.10)In the bivariate setup 2 E { A n } ∼ n Z ∞ Q n − d ( s ) | dK ( s ) | , (2.11)with K ( s ) = 1 − K ( s ) = 1 π (cid:20)Z ∞ s p y − s dF ( y ) (cid:21) , s ≥ . We note in passing that K ( s ) , s ≥ A2 . Asymptotic results for the sequence of the expectation of the number of vertices E { v n } , n ≥ d = 2) by investigating the asymptotic behaviour ( n → ∞ ) of R R Q n ( s ) | dH ( s ) | . Our Lemma 5.1 turns out to be quiteuseful; furthermore it sheds some light explaining the role of extreme value theory in our analysis. More specifically,in view of Lemma 5.1 E { v n } , n ≥ n → ∞ ) if and only if (iff) the tails of Q and F satisfy a certain asymptotic condition (see (5.3)). In particularlim n →∞ E { v n } = c ∈ (0 , ∞ ) (3.1)iff lim u ↑ x H H ( u ) Q ( u ) = c ∈ (0 , ∞ ) . (3.2)On the other hand, H and Q are in a strong relation with the distribution function F via (1.2). Thus it is notstraightforward to check whether (3.2) holds for some given F . One simple instance is when for some positiveconstants c , c ∈ (0 , ∞ ) H ( u ) ∼ c F ( u ) , Q ( u ) ∼ c F ( u ) , u ↑ x H (3.3)implying that (3.1) is valid with c := c /c .If F is in the MDA of an extreme value distribution, then by Berman (1992) it follows that both Q and H are inthe same MDA, and further the asymptotics of Q ( u ) and H ( u ) as u → x F are determined by F ( u ) and some knownfunctions.In view of Hashorva and Pakes (2010) also the converse holds, i.e., F is in the MDA of an extreme value distributionfunction iff Q d , d ≥ F in the MDA of some univariate extreme value distribution can be retrieved if we impose instead the sameassumption on Q d .We deal first with the Gumbel case; when x F ∈ (0 , ∞ ) no asymptotic results for the quantities of interest are knownto date. When x F = ∞ we have both results of Carnal and Dwyer for any F being a Carnal distribution function.We conclude this section by briefly discussing both the Fr´echet and Weibull max-domains of attraction. It is well-known that condition (2.1) is valid for some univariate distribution function N with G = Λ and upperendpoint x N ∈ ( −∞ , ∞ ], iff for some positive scaling function w lim u ↑ x N N ( u + x/w ( u )) N ( u ) = exp( − x ) , ∀ x ∈ R . (3.4)Furthermore w ( u ) ∼ N ( u ) R ∞ u N ( s ) ds , u ↑ x N . (3.5)So far in the literature the Gumbel MDA assumption on F has not been explicitly assumed. An elegant simplificationof this assumption is suggested in Carnal (1970) which has been used in several following papers (Eddy and Gale(1981), Dwyer (1991), Hueter (1999, 2005, 2005)). More specifically, Carnal (1970) considers distribution functions F satisfying (for all large x ) x = L (1 /F ( x )) , (3.6)where L is a monotone increasing slowly varying function at infinity. We refer to (3.6) as the Carnal tail condition andto such F as Carnal distributions. As shown in Carnal (1970) if L ( s ) = exp( R s ε ( s ) /s ds ) , s ≥ s →∞ ε ( s ) = 0,then Carnal distributions have the representation F ( x ) = exp( − Z x / ( η ( s ) s ) ds ) , x ≥ , with η ( s ) = ε (1 /F ( s )) , s >
0. In the aforementioned paper (see also Dwyer (1991), Hueter (1999)) the function η satisfies some smoothness conditions being further positive and monotone non-decreasing.If N ∈ GM DA ( w ) we define next ξ N ( n ) := b n /a n , b n := N − (1 − /n ) , a n := 1 /w ( b n ) , n > . The constants a n , b n are such that (2.1) holds with G = Λ. Further, it is well-known (see e.g., Resnick (2008)) thatboth N − (1 − /n ) and ξ N ( n ) are slowly varying functions at infinity. As will be shown next this fact, Lemma 5.1 andProposition 5.4 (see Appendix) are the key ingredients needed to derive the tail asymptotics of the quantities of interest. Proposition 3.1.
Let
F, H, K, Q d , d ≥ be as in the previous section, and let CH [ X , . . . , X n ] be the convex hullof the random points X , . . . , X n which are independent with stochastic representation (1.1) , where R ∼ F . If Q d ∈ GM DA ( w ) or F ∈ GM DA ( w ) we have:a) As n → ∞ Q − d (1 − /n ) ∼ F − (1 − /n ) , and ξ Q d ( n ) ∼ ξ F ( n ) . (3.7) b) If d = 2 , then E { v n } ∼ q πξ Q ( n ) , (3.8) and for d ≥ and some ε ∈ (0 , ∞ )(1 − ε ) ξ Q d ( n ) ( d − / ≤ ( d − / Γ( d/ √ π E { v n } ≤ (1 + ε )4 d − ξ Q d ( n ) ( d − / . (3.9) c) If d = 2 , then E { A n } ∼ π [ Q − (1 − /n )] , (3.10) and for d ≥ E { A n } ≤ (1 + o (1))Γ( d + 1) (cid:18) d √ πd − (cid:19) d − ( Q − d (1 − /n )) d , n → ∞ . (3.11) d) For any d ≥ E { f n } ≤ (1 + o (1)) √ d (cid:18) πdd − (cid:19) ( d − / ( ξ Q d ( n )) − ( d − / , n → ∞ (3.12) and E { V n } ≤ (1 + o (1))Γ( d ) (cid:18) d √ πd − (cid:19) d − ( Q − d (1 − /n )) d , n → ∞ (3.13) are valid. Remarks: (a) In Carnal’s notation L ( n ) = F − (1 − /n ) and ε ( n ) := 1 /ξ F ( n ) , n ≥ . If the upper endpoint of F isfinite, say x F = 1 , then clearly lim n →∞ L ( n ) = 1 .(b) A misprint appears in the upper bound for E { v n } in Dwyer (1991), p.126. The upper bound therein should bemultiplied by d − , see (3.9) above.(c) For any N ∈ GM DA ( w ) with scaling function w defined by (3.5) we have (see e.g., Resnick (2008)) uw ( u ) → ∞ , and if x N < ∞ w ( u )( x N − u ) = ∞ , u ↑ x N . (3.14) Consequently, (3.9) implies lim n →∞ E { v n } = ∞ .(c) Utilising the expression (1.7) which gives an asymptotic formula for E { l n } with l n the perimeter of the convex hull(d=2), it follows that when F or Q d are in the Gumbel MDA with some scaling function w , then we have E { l n } ∼ πQ − (1 − /n ) . (3.15)Proposition 3.1 provides asymptotic upper and lower bounds for E { v n } .Hueter (1999) was able to give the exact asymptotic behaviour of the first and the second moment of v n ; moreovera key central limit theorem was derived therein by developing Groeneboom’s technique (see Groeneboom (1988)) inhigher dimensions.Next we extend Hueter’s CLT theorem which has been shown for Carnal distributions by considering a general sphericalrandom vector with marginal distribution or distribution of the associated random radius in the Gumbel max-domainof attraction. Proposition 3.2.
Let v n denote the number of the vertices of CH [ X , . . . , X n ] , where X i , i ≥ n are independentwith stochastic representation (1.1) . Suppose that F (0) = 0 , x F = ∞ and set a n := 1 /w ( b n ) , b n := Q − d (1 − /n ) , n > where Q d and F are related by (1.2) . If either F ∈ GM DA ( w ) or Q d ∈ GM DA ( w ) , and a n b n
6→ ∞ as n → ∞ , thenwe have the convergence in distribution v n − λ d ( b n /a n ) ( d − / p Var { v n } d → Z, n → ∞ , (3.16) with Var { v n } ∼ λ ∗ d ( b n /a n ) ( d − / , λ d , λ ∗ d ∈ (0 , ∞ ) , and Z a standard Gaussian random variable. Remarks : (a) The above proposition gives also the asymptotics of E { v n } and Var { v n } , n → ∞ . It turns out that theexpectation and the variance of the number of the vertices of the convex hull differ by a constant, and are both slowlyvarying functions.b) The condition a n b n
6→ ∞ as n → ∞ implies a certain asymptotic behaviour of the density function q of Q . Moreprecisely, in view of Proposition 3.1 q ( u ) ∼ w ( u ) Q ( u ) = (1 + o (1)) 12 π (cid:18) w ( u ) u (cid:19) / Q ( u ) , u → ∞ , hence q ( b n ) ∼ π (cid:18) a n b n (cid:19) / Q ( b n ) ∼ n √ a n b n , implying q ( b n ) /n as n → ∞ .(c) In Proposition 3.2 if F has a finite upper endpoint x F ∈ (0 , ∞ ) , then lim n →∞ b n = x F and lim n →∞ a n = 0 , hence lim n →∞ a n b n = 0 . We conjecture that Proposition 3.2, and in particular the asymptotics of E { v n } and Var { v n } , are also valid with thesame constants (not depending on F ), when F ∈ GM DA ( w ) with x F ∈ (0 , ∞ ) . In the 2-dimensional setup this istrue for the asymptotics of E { v n } , n → ∞ (see (3.9) above). We give next two illustrating examples.
Example 1.
Let X i d = R U , i = 1 , . . . , n be independent spherically symmetric random vectors in R . Assume thatthe distribution function F of the positive random variable R has upper endpoint 1 satisfying F ( u ) ∼ a exp( − b/ (1 − u )) , u ↑ , with a, b ∈ (0 , ∞ ). Set w ( u ) := b/ (1 − u ) , u ∈ (0 , s ∈ R F ( u + s/w ( u )) F ( u ) = (1 + o (1)) exp( − b [1 / (1 − u + s/w ( u )) − / (1 − u )]) → exp( − s ) , u ↑ , then F ∈ GM DA ( w ). Further, we have F − (1 − /n ) ∼ − b/ ln( an ) , w ( F − (1 − /n )) = b [ln( an )] , n > , consequently ξ F ( n ) = F − (1 − /n ) w ( F − (1 − /n )) ∼ b [ln n ] , n → ∞ . Hence in view of Proposition 3.1 for d = 2 E { v n } ∼ √ bπ ln n, n → ∞ . (3.17) Example 2.
Under the setup of the previous example, we suppose further that the marginal distribution function Q is in the Gumbel MDA with scaling function w ( x ) = rθx θ − / (1 + L ( x )) , r > , θ > , where L is a regularly varying function at infinity with index γθ, γ <
0. It follows that Q ( x ) ∼ exp( − rx θ (1 + L ( x ))) , x → ∞ , with L another regularly varying function at infinity with index γθ . Consequently, we have b n := Q − (1 − /n ) ∼ (cid:18) ln nr (cid:19) /θ , a n := 1 /w ( b n ) = b − θn rθ , n → ∞ implying ξ Q ( n ) = θ ln n, a n b n = b − θn rθ , n > . Hence, by Proposition 3.1 E { v n } ∼ √ πθ ln n (3.18)and lim n →∞ a n b n = 0 if θ >
2, whereas for θ ∈ (0 ,
2) we have lim n →∞ a n b n = ∞ . Note in passing that if Q is thestandard Gaussian distribution, then θ = 1 /r = 2 and lim n →∞ a n b n = 1. Further, we remark that if L is constant,then Q is a Carnal distribution. O -Regularly Varying Tails The survival function N is regularly varying (at infinity) with index γ ≤ u →∞ N ( ux ) N ( u ) = x γ . (3.19)In view of Proposition 5.3 (see Appendix A3 ), the survival function Q d satisfies (3.19) iff F also satisfies (3.19). Hencethe results of Carnal (1970) and Dwyer (1991) can be retrieved assuming the regular variation of Q d instead of thatof F . As shown in Berman (1992) it is possible to relate the asymptotics of F with that of Q , specifically Q ( u ) ∼ Γ(( γ + 1) / √ π Γ( γ/ F ( u ) , u → ∞ . Similarly, we find F satisfies (3.19) iff the survival function H is regularly varying with index 2 γ . Moreover as u → ∞ H ( u ) ∼ Γ( γ + 1 / √ π Γ( γ + 1) F ( u ) . Consequently (3.3) implies that if one of the survival functions
F , Q d or H is regularly varying with index γ ≤
0, thenfor the bivariate setup ( d = 2) we havelim n →∞ E { v n } = Γ( γ + 1 / γ/ [Γ(( γ + 1) / Γ( γ + 1) , (3.20)which is shown in Carnal (1970) assuming that F satisfies (3.19). Aldous et al. (1991) addresses the case that F satisfies (3.19) with γ = 0, which in view of our results is equivalent with Q d being slowly varying (satisfying (3.19)with γ = 0).It is interesting that the limit in (3.20) is finite, thus E { v n } , n ≥ E { v n } , n ≥ F being a O -regularly varying function, meaning that0 < lim inf u →∞ F ( ux ) F ( u ) ≤ lim sup u →∞ F ( ux ) F ( u ) < ∞ , ∀ x > . Proposition 3.3.
Under the setup of Proposition 3.1, if either F , Q , or H is O -regularly varying at infinity, then E { v n } , n ≥ is a bounded sequence. It is well-known that F is in the Fr´echet MDA iff (3.19) holds for some γ <
0, see e.g., Embrechts et al. (1997). When F is in the Weibull MDA we have a similar behaviour of the survival function at the upper endpoint x F which isnecessarily finite, say x F = 1. More specificallylim u →∞ F (1 − x/u ) F (1 − /u ) = x γ , γ > G = Ψ γ .Our new results when F is in the Weibull MDA can be derived utilising Proposition 5.3 which implies:Condition (3.21) is equivalent with the fact that the marginal distribution Q d satisfies (3.21) with γ ∗ = γ + 1 / , γ ≥ Proof of Proposition F has a finite upper endpoint, say x F = 1. Since also Q d has the same upperendpoint, the proof follows by the fact thatlim n →∞ Q − d (1 − /n ) = x Q d = 1 , lim n →∞ F − (1 − /n ) = x F = 1 . We deal therefore with the case x F = ∞ . By Lemma 6.1 in Hashorva (2009) if N ∈ GM DA ( w ) and N has an infiniteupper endpoint, then lim u →∞ N ( cu )( uw ( u )) a N ( u ) = 0 (4.1)for any c > a ∈ R . Hence, by (3.14), (4.1), (5.12) and (5.10) Q − d (1 − /n ) ∼ F − (1 − /n ) , n → ∞ . In view of Proposition 5.4 both H and Q d , d ≥ w ,consequently ξ Q d ( n ) ∼ ξ F ( n ).b) By the assumptions and Proposition 5.4 we have H ( u ) Q ( u ) q ξ Q ( Q ( u )) ∼ √ π, u ↑ x F . Furthermore, H and Q d , d ≥ ρ = 2 and l = Γ(3)2 √ π to (2.6) we obtain 2 E { v n } ∼ n Z ∞ Q n − ( s ) | dH ( s ) | ∼ Γ(3)2 √ π. If d ≥
2, then (5.9) implies F ( u ) Q d ( u ) ξ Q d ( Q d ( u )) ( d − / ∼ − ( d − / √ π Γ( d/ , u ↑ x F . Consequently, by Lemma 5.1 n Z ∞ Q n − d ( s ) dF ( s ) ∼ − ( d − / √ π Γ( d/ ξ F ( n ) ( d − / . Similarly, n d − Z ∞ [1 − − ( d − Q d ( s )] n − dF ( s ) ∼ d − − ( d − / √ π Γ( d/ ξ F ( n ) ( d − / and thus (3.9) follows.c) With the same arguments as above we obtain K ( u )[ Q ( u ) Q − (1 − Q ( u ))] ∼ π, u ↑ x F , hence Lemma 5.1 implies 2 E { A n } ∼ n Z ∞ Q n − ( s ) | dK ( s ) | ∼ Γ(3) π [ Q − (1 − /n )] , d ≥
2. Applying Lemma 7.6 of Hashorva (2007) and using (2.11) we obtain Z x F r ( r − u ) (3 d − / u dF ( u ) ∼ Γ(3( d − / (cid:18) rw ( r ) (cid:19) (3 d − / rF ( r ) , r ↑ x F . Consequently, by (2.11), (5.9) and the fact that q d ( r ) ∼ w ( r ) Q d ( r ) , r ↑ x F (see Proposition 5.4) E { A n } ≤ (1 + o (1)) (cid:18) dd − (cid:19) d − d ( d − d + 1) / Z x F (cid:18) r w ( r ) (cid:19) d − ( rw ( r )) d − ( r ) exp( − Q d ( r )) dr ≤ (1 + o (1)) (cid:18) dd − (cid:19) d − d ( d − d + 1) / Z x F r d − exp( − Q d ( r )) dQ d ( r ) , r ↑ x F . Since Q − (1 − /n ) is regularly varying as n → ∞ the Abel formula for the Laplace transform yields E { A n } ≤ (1 + o (1))Γ( d + 1) (cid:18) d √ πd − (cid:19) d − ( Q − d (1 − /n )) d − , n → ∞ , thus the statement is established.d) By Proposition 5.4 and Lemma 7.6 in Hashorva (2007) as r ↑ x F we obtain1 q d ( r ) Z x F r ( r − u ) d − u dF ( u ) ∼ Γ( d − (cid:18) rw ( r ) (cid:19) d − r F ( r ) q d ( r ) ∼ Γ( d − d − r d − w ( r ) − d w ( r ) F ( r ) q d ( r ) ∼ Γ( d − d − − ( d − / r d − w ( r ) − d ( rw ( r )) ( d − / √ π/ Γ( d/ ∼ Γ( d − √ π Γ( d/
2) 2 d/ − / r d − / w ( r ) (1 − d ) / . Hence (3.12) and Proposition 5.4 implies E { f n } ≤ (1 + o (1)) dτ d − κ d − Γ( d − √ π Γ( d/
2) 2 d/ − / Z x F r d − / w ( r ) (1 − d ) / q d ( r ) d − exp( − nQ d ( r )) dQ d ( r ) ≤ (1 + o (1)) dτ d − κ d − Γ( d − √ π Γ( d/
2) 2 d/ − / Z x F ( rw ( r )) d − / Q d ( r ) d − exp( − nQ d ( r )) dQ d ( r ) . In view of Theorem 4.1 in Hashorva et al. (2010) there exists a distribution function G d such that G d ( u ) ∼ dτ d − κ d − Γ( d − √ π Γ( d/
2) 2 d/ − / ( uw ( u )) d − / Q d ( u ) d − , u → ∞ . Applying now Lemma 5.1 to R ∞ G d ( r ) exp( − nQ d ( r )) dQ d ( r ) establishes (3.12).The proof of the last claim follows with similar arguments utilising further (3.13). ✷ Proof of Proposition F is O -regularly varying if Q d or H is O -regularly varying, andvice-versa. Next, assume that d = 2 and F is O -regularly varying. By (1.2) and (2.7) for any c > u > P { B / , ( d − / > /c } F ( cu ) ≤ Q d ( u ) ≤ F ( u )and P { B / , ( d − / > /c } F ( cu ) ≤ H ( u ) ≤ F ( u ) . Consequently (cid:18) F ( u ) P { B / , ( d − / > /c } F ( cu ) (cid:19) − ≤ H ( u ) Q d ( u ) ≤ (cid:18) F ( u ) P { B / , ( d − / > /c } F ( cu ) (cid:19) . O -regular variation of F implies that 0 < b ≤ H ( u ) /Q dd ( u ) ≤ b < ∞ for some constants b , b and for all u large, hence (2.6) yields that E { v n } , n ≥ d ≥ F ( u ) /Q d ( u ), and thus the result follows. ✷ Proof of Proposition ξ Q d ( n ) , Q − d (1 − /n ) are slowly regularly varying functions and (3.14). ✷ A1 . First note that if X ∼ Y B / , / , with Y > B / , / and X being symmetricabout 0, then the distribution of X is given by (see Carnal (1970))2 P { X > r } = P {| X | > r } = 2 π Z ∞ r arccos( r/s ) | dG ( s ) | , r ≥ , with G the distribution function of Y . Next, since H is defined by (see Carnal (1970)) H ( r ) = 2 π Z ∞ r arccos( r/s ) | dF ( s ) | , r ≥ A2 . Let R ∼ H be a positive random variable independent of B α,β , α, β ∈ (0 , ∞ ) , and denote by Q α,β the distributionfunction of RB α,β . Then Q α,β possesses the density function q α,β given by (see (22) in Hashorva et al. (2007)) q α,β ( x ) = Γ( α + β )Γ( α ) x α − Z ∞ x ( s − x ) β − s − α − β +1 dH ( s ) , ∀ x ∈ (0 , x H ) . (5.1)It is thus clear that Q α,β is a continuous distribution function.Next assume that N is a univariate distribution function with N (0) = 0 and upper endpoint x N ∈ (0 , ∞ ] such that µ N := R ∞ x dN ( x ) ∈ (0 , ∞ ). Define a new distribution function N ∗ by N ∗ ( s ) = 1 − Z ∞ s p x − s dN ( s ) /µ N , s ≥ . Then we have (see (18.5) in Reiss and Thomas (2007)) that N ∗ possesses a density function n ∗ given by n ∗ ( s ) = sµ N Z ∞ s ( x − s ) − / dN ( s ) , s ≥ N ∗ is a continuous distribution function. A3 . We give below two lemmas followed by two propositions, which are utilised in the proofs above. Note in passingthat the next lemma has been useful when dealing with the asymptotics of near extremes, see e.g. Pakes (2000).Furthermore, refinements under stronger asymptotic assumptions can be found in Li (2008). Lemma 5.1.
Let G , G be two continuous distribution functions with upper endpoint ω ∈ ( −∞ , ∞ ] . Further let λ ∈ [0 , , l, ρ ∈ [0 , ∞ ) be given constants and L be a positive slowly varying function at 0. Then the following state-ments are equivalent:a) For any z ∈ R Z R [ λ + (1 − λ ) G ( s )] n − z dG ( s ) ∼ l ((1 − λ ) n ) ρ L (1 /n ) , n → ∞ . (5.2)2 b) G ( u ) G ρ ( u ) L ( G ( u )) ∼ l Γ( ρ + 1) , u ↑ ω. (5.3) c) Z R G ( s ) exp( − ( n − z ) G ( s )) dG ( s ) ∼ ln ρ − L (1 /n ) , n → ∞ . (5.4) Proof of Lemma b ) and c ) can be established with similar arguments. ✷ Lemma 5.2.
Let X d = RY with R ∼ F being independent of Y ∈ [0 , almost surely. Suppose that F (0) = 0 , x F = ∞ and P { Y > s } ∈ (0 , for any s ∈ (0 , and denote by G the distribution function of X . Then F is O -regularlyvarying iff G is O -regularly varying. Proof of Lemma α > , x > F ( x ) ≥ G ( x ) ≥ Z ∞ αx P { Y > x/r } dF ( r ) ≥ P { Y > /α } F ( αx ) > c > F ( cx ) P { Y > /α } F ( αx ) ≥ G ( cx ) G ( x ) ≥ P { Y > /α } F ( αcx ) F ( x ) > . Choosing α ∈ (1 , c ) the assumption F is O -regularly varying yields that also G is O -regularly varying. The proof ofthe converse follows with similar arguments, therefore it is omitted here. ✷ Proposition 5.3.
Let Y ∼ H be independent of B a,b , a, b ∈ (0 , ∞ ) with H such that x H ∈ (0 , ∞ ] , and H (0) = 0 . Let X d = Y [1 − B a,b ] /τ , τ ∈ (0 , ∞ ) with distribution function F . Then we have:i) H ∈ GM DA ( w ) is equivalent with F ∈ GM DA ( w ) , and F ( u ) ∼ Γ( a + b )Γ( b ) (cid:18) τuw ( u ) (cid:19) a H ( u ) , u ↑ x F . (5.5) Furthermore, F possesses a density function f such that f ( u ) ∼ w ( u ) F ( u ) , u ↑ x F .ii) The distribution function H satisfies (3.19) with some γ ≤ iff F satisfies (3.19) with the same γ , and moreover F ( u ) ∼ Γ( a + b )Γ( b ) Γ( b + γ/τ )Γ( a + b + γ/τ ) H ( u ) , u → ∞ . (5.6) iii) The distribution F with x F = 1 satisfies (3.21) with some γ ≥ , iff H satisfies (3.21) with γ ∗ := γ + a, γ ≥ ,and moreover F ( u ) ∼ Γ( a + b )Γ( b ) Γ( γ + 1)Γ( γ + a + 1) ( τ (1 − u )) a H ( u ) , u ↑ . (5.7) Proof of Proposition ✷ Proposition 5.4.
Let
F, H, K, q d , Q d , d ≥ be as in Proposition 3.1. Then F ∈ GM DA ( w ) iff one of the followingrelations hold:a) For any d ≥ we have Q d ∈ GM DA ( w ) . Furthermore q d ( u ) ∼ w ( u ) Q d ( u ) , u ↑ x F (5.8)3 and Q d ( u ) ∼ ( d − / Γ( d/ √ π ( uw ( u )) − ( d − / F ( u ) , u ↑ x F . (5.9) b) H ∈ GM DA (2 w ) and moreover H ( u ) ∼ p πuw ( u ) F ( u ) ∼ p πuw ( u )( Q ( u )) , u ↑ x F . (5.10) c) We have µ F := R ∞ y dF ( y ) is finite and K ∗ ( s ) = 1 − R ∞ s p y − s dF ( y ) /µ F , s ≥ is a continuous distributionfunction with K ∗ ∈ GM DA ( w ) . Furthermore, we have K ( u ) ∼ F ( u ) u/w ( u ) ∼ πu Q ( u ) , u ↑ x F . (5.11) Proof of Proposition a ) and b ) follow immediately by applying statement i ) ofProposition 5.3 in Appendix A3 (recall that both (1.2) and (2.7) hold).Next, if F ∈ GM DA ( w ), then µ F = R ∞ x dF ( x ) is finite, therefore K ∗ ( s ) = 1 − R ∞ s p y − s dF ( y ) /µ F , s ≥ x F . Applying Lemma 7.6 in Hashorva (2007) we obtain K ∗ ( u ) = 1 µ F Z ∞ u p x − u dF ( x ) ∼ µ F Γ(3 / u/w ( u )) / F ( u ) , u ↑ x F . Since locally uniformly in R w ( u + s/w ( u )) w ( u ) → , u ↑ x F (5.12)for any s ∈ R , it follows that K ∗ ∈ GM DA ( w ).In order to finish the proof we need to show the converse. Assume therefore K ∗ ∈ GM DA ( w ) and µ F ∈ (0 , ∞ ). Bythe Abel integral equation (see Heinrich (2007)) F ( x ) = 2 µ F π Z ∞ x ( y − x ) − / dK ∗ ( x ) , x ≥ . Applying again Lemma 7.6 in Hashorva (2007) we obtain F ( u ) ∼ µ F π Γ(1 / u/w ( u )) − / K ∗ ( u ) , u ↑ x F , hence the result follows. ✷ Acknowledgement : I would like to thank the referee for a very kind and deep review and several suggestions whichimproved the presentation substantially. I am thankful to Ilya Molchanov for many helpful discussions and comments,and to J¨urg H¨usler and Michael Mayer for providing some key references.
References [1] Aldous, D.J., Fristedt, J., Griffin, P.S., and Pruitt, W.E. (1991) The number of extreme points in the convex hullof a random sample.
J. Appl. Prob. , 287–304.[2] B´ar´any, I., and Vu, V. (2007) Central limit theorems for Gaussian polytopes. Ann. Probab. , , 1593–1621.[3] Berman, M.S. (1992) Sojourns and Extremes of Stochastic Processes . Wadsworth & Brooks/ Cole, Boston.[4] Buchta, C. (2005) An identity relating moments of functionals of convex hulls.
Discrete Comput. Geom. , ,125–142.4[5] Cambanis, S., Huang, S., and Simons, G. (1981) On the theory of elliptically contoured distributions. J. Multi-variate Analysis. ,3, 368–385.[6] Carnal, H. (1970) Die konvexe H¨ulle von n rotations-symmetrisch verteilten Punkten. Z. Wahrscheinlichkeitsthe-orie Verw. Geb. , 168–176.[7] Carnal, H., and H¨usler, J. (1991) On the convex hull of n random points on a circle. J. Appl. Probab. , 231–237.[8] Dwyer, R. A. (1991) Convex hulls of samples from spherically symmetric distributions. Discrete Appl. Math. ,113–132.[9] Falk, M., H¨usler, J., and Reiss R.-D. (2004) Laws of Small Numbers: Extremes and Rare Events.
DMV Seminar , Second Edition, Birkh¨auser, Basel.[10] Eddy, W.F., and Gale, J.D. (1981) The convex hull of a spherically symmetric sample. Adv. Appl. Prob. , ,751–763.[11] Efron, B. (1965) The convex hull of a random set of points. Biometrika , 331–343.[12] Embrechts, P., Kl¨uppelberg, C., and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance.
Springer-Verlag, Berlin.[13] Groeneboom, P. (1988) Limit theorems for convex hulls.
Probab. Theory Related Fields , , 327–368.[14] Hashorva, E. (2006) Gaussian approximation of conditional elliptical random vectors. Stoch. Models. ,3, 441–457.[15] Hashorva, E. (2007) Asymptotics properties of Type I elliptical random vectors. Extremes , ,4, 175–206.[16] Hashorva, E. (2009) Conditional limits of W p scale mixture distributions. J. Stat. Plan. Inf . ,10, 3501–3511.[17] Hashorva, E, Kotz, S., and Kume, A. (2007) L p -norm generalised symmetrised Dirichlet distributions. AlbanianJ. Math. ,1, 31–56.[18] Hashorva, E., and Pakes, A.G. (2010) Distribution and asymptotics under beta random scaling. J. Math. Anal.Appl. , ,2, 496–514.[19] Hashorva, E., Pakes, A.G., and Tang, Q. (2010) Asymptotics of random contractions. Insurance: Mathematicsand Economics , bf 47, 3, 405–414.[20] Hueter, I. (1999) Limit theorems for the convex hull of random points in higher dimensions.
Trans. Amer. Math.Soc. , 4337–4363.[21] Hueter, I. (2004) Random convex hulls and the Stein method. Preprint.[22] Hueter, I. (2005) Limit theorems for convex hull peels. Preprint.[23] Hueter, I. (1992) The Convex Hull of n Random Points and its Vertex Process. Ph.D. thesis, University of Bern.[24] Li, D. (2008) On the probability of being maximal.
Australian and New Zealand Journal of Statistics , 381–394.[25] Heinrich, L. (2007) Limit distributions of some stereological estimators in Wicksell’s corpuscle problem. Imag.Anal. Stereol. , , 63–71.[26] Mayer, M., and Molchanov, I. (2007) Limit theorems for the diameter of a random sample in the unit ball. Extremes , , 129–150.5[27] Pakes, A. (2000) The number and sum of near-maxima for thin-tailed populations. Adv. in Appl. Probab. ,110017-1116.[28] Raynaud, H. (1970) Sur l’enveloppe convexe des nuages de points al´eatoires dans R n . J. Appl. Prob. , 35–48.[29] Reitzner, M. (2002) Random points on the boundary of smooth convex bodies. Trans. Amer. Math. Soc. , ,224317-2278.[30] Reitzner, M. (2004) Stochastic approximation of smooth convex bodies. Mathematika , , 11-1729.[31] R´enyi, A., and Sulanke, R. (1963) Uber die konvexe Hulle von n zuf¨allig gew¨ahlten Punkten. Z. Wahrschein-lichkeitstheorie verw. Geb. , 75–84.[32] Reiss, R-D. (1989) Approximate Distributions of Order Statistics: With Applications to Nonparametric Statistics .Springer, New York.[33] Reiss, R-D., and Thomas, M. (2007)
Statistical analysis of extreme values. From insurance, finance, hydrologyand other fields . Third Edition, Birkh¨auser, Basel.[34] Resnick, S.I. (2008)