Strengthening classical results on convergence rates in strong limit theorems

Let X₁, X₂, . . . be i.i.d. random variables, and set S_n=X₁+ . . . +X_n. Several authors proved convergence of series of the type f(ɛ)=∑_nc_nP(|S_n|>ɛa_n),ɛ>α, under necessary and sufficient conditions. We show that under the same conditions, in fact i.e. the finiteness of ∑_nc_nP(|S_n|>ɛa_n),ɛ>α, is equivalent to the convergence of the double sum ∑_k∑_nc_nP(|S_n|>ka_n). Two exceptional series required deriving necessary and sufficient conditions for E[sup_n|S_n|(logn)^η/n]<∞,0≤η≤1.     

The Partitioned Version of the Erdős—Szekeres Theorem

   Abstract. Let k≥ 4 . A finite planar point set X is called a convex k -clustering if it is a disjoint union of k sets X ₁ , . . . ,X _k of equal sizes such that x ₁ x ₂ . . . x _k is a convex k -gon for each choice of x ₁ ∈ X ₁ , . . . ,x _k ∈ X _k . Answering a question of Gil Kalai, we show that for every k≥ 4 there are two constants c=c(k) , c'=c'(k) such that the following holds. If X is a finite set of points in general position in the plane, then it has a subset X' of size at most c' such that X \ X' can be partitioned into at most c convex k -clusterings. The special case k=4 was proved earlier by Pór. Our result strengthens the so-called positive fraction Erdos—Szekeres theorem proved by Barany and Valtr. The proof gives reasonable estimates on c and c' , and it works also in higher dimensions. We also improve the previous constants for the positive fraction Erdos—Szekeres theorem obtained by Pach and Solymosi.     

Edge proximity and matching extension in projective planar graphs

A graph $G$ with at least $2m+2$ vertices is said to be distance $d$ $m$-extendable if, for any matching $M$ of $G$ with $m$ edges in which the edges lie at distance at least $d$ pairwise, there exists a perfect matching of $G$ containing $M$. In this paper we prove that every 5-connected triangulation on the projective plane of even order is distance 3 7-extendable and distance 4 $m$-extendable for any $m$.     

Positive dependence properties of elliptically symmetric distributions

Let X₁, …, X_p have p.d.f. g(x₁² + … + x_p²). It is shown that (a) X₁, …, X_p are positively lower orthant dependent or positively upper orthant dependent if, and only if, X₁,…, X_p are i.i.d. N(0, σ²); and (b) the p.d.f. of |X₁|,…, |X_p| is TP₂ in pairs if, and only if, In g(u) is convex. Let X₁, X₂ have p.d.f. $\text{f(x}{}_1\text{, x}{}_2\text{) =}\text{|Σ|}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{g((x}{}_1\text{, x}{}_2\text{)}\text{Σ}{}^{\mathrm{?1}}\text{(x}{}_1\text{, x}{}_2\text{)′)}$. Necessary and sufficient conditions are given for f(x₁, x₂) to be TP₂ for fixed correlation ?. It is shown that if f is TP₂ for all ? >0. then (X₁, X₂)′ ～ N(0, Σ). Related positive dependence results and applications are also considered.