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Aurel Spătaru 《Probability Theory and Related Fields》2006,136(1):1-18
Let X1, X2, . . . be i.i.d. random variables, and set Sn=X1+ . . . +Xn. Several authors proved convergence of series of the type f(ɛ)=∑ncnP(|Sn|>ɛan),ɛ>α, under necessary and sufficient conditions. We show that under the same conditions, in fact i.e. the finiteness of ∑ncnP(|Sn|>ɛan),ɛ>α, is equivalent to the convergence of the double sum ∑k∑ncnP(|Sn|>kan). Two exceptional series required deriving necessary and sufficient conditions for E[supn|Sn|(logn)η/n]<∞,0≤η≤1. 相似文献
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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. 相似文献
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A graph with at least vertices is said to be distance -extendable if, for any matching of with edges in which the edges lie at distance at least pairwise, there exists a perfect matching of containing . 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 -extendable for any . 相似文献
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Allan R Sampson 《Journal of multivariate analysis》1983,13(2):375-381
Let X1, …, Xp have p.d.f. g(x12 + … + xp2). It is shown that (a) X1, …, Xp are positively lower orthant dependent or positively upper orthant dependent if, and only if, X1,…, Xp are i.i.d. N(0, σ2); and (b) the p.d.f. of |X1|,…, |Xp| is TP2 in pairs if, and only if, In g(u) is convex. Let X1, X2 have p.d.f. . 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. 相似文献
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Yehoram Gordon 《Israel Journal of Mathematics》1969,7(2):151-163
Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ
p(X) as inf{Σ
i
=1/m
|x*(x
i)|p
p Σ
i
=1/m
‖x
i‖p
p]1
p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x
1,x
2, …,x
m} ⊂X such that Σ
i
=1/m
‖x
i‖>0. It follows immediately from [2] thatμ
p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ
p(X) for various spaces, and obtain some asymptotic estimates ofμ
p(X) for general finite dimensional Banach spaces.
This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof.
A. Dvoretzky and Prof. J. Lindenstrauss. 相似文献