Rare numbers

Suppose thatX ₁,X ₂,... is a sequence of iid random variables taking values inZ ⁺. Consider the random sequenceA(X)(X ₁,X ₂,...). LetY _n be the number of integers which appear exactly once in the firstn terms ofA(X). We investigate the limit behavior ofn ^–(1–) Y _n for [0, 1].     

NA随机变量三角阵列的钟开莱形式的大数定理（英文）

设{Xnm;n≥1,m≥1}每行都是一个NA随机变量列.给出了在各种矩条件随机变量的钟开莱—太勒形式的强大数定理.     

On the laws of large numbers for double arrays of independent random elements in Banach spaces

For a double array of independent random elements {Vmn,m ≥ 1,n ≥ 1} in a real separable Banach space,conditions are provided under which the weak and strong laws of large numbers for the double sums mi=1 nj=1Vij,m ≥ 1,n ≥ 1 are equivalent.Both the identically distributed and the nonidentically distributed cases are treated.In the main theorems,no assumptions are made concerning the geometry of the underlying Banach space.These theorems are applied to obtain Kolmogorov,Brunk–Chung,and Marcinkiewicz–Zygmund type strong laws of large numbers for double sums in Rademacher type p(1 ≤ p ≤ 2) Banach spaces.     

On the Convergence of Sampling-Based Decomposition Algorithms for Multistage Stochastic Programs

The paper presents a convergence proof for a broad class of sampling algorithms for multistage stochastic linear programs in which the uncertain parameters occur only in the constraint right-hand sides. This class includes SDDP, AND, ReSa, and CUPPS. We show that, under some independence assumptions on the sampling procedure, the algorithms converge with probability 1.The first author acknowledges support by the Swiss National Science Foundation. The second author acknowledges support by NZPGST Grant UOAX0203. The authors are grateful to the anonymous referees for comments improving the exposition of this paper.     

Some strong laws of large numbers for blockwise martingale difference sequences in martingale type p Banach spaces

For a blockwise martingale difference sequence of random elements {Vn , n ≥ 1} taking values in a real separable martingale type p (1 ≤ p ≤ 2) Banach space, conditions are provided for strong laws of large numbers of the form limn→∞∑ n i=1 Vi /gn = 0 almost surely to hold where the constants gn ↑∞. A result of Hall and Heyde [Martingale Limit Theory and Its Application, Academic Press, New York, 1980, p. 36] which was obtained for sequences of random variables is extended to a martingale type p (1 p ≤ 2) Banach space setting and to hold with a Marcinkiewicz-Zygmund type normalization. Illustrative examples and counterexamples are provided.     

Expected number of steps of a random optimization method

Some explanatory remarks are presented, relevant to the objections by Pang (Ref. 1) to Dorea's paper (Ref. 2).     

A strong law for weighted sums of i.i.d. random variables

A strong law is proved for weighted sumsS _n=a _in X _i whereX _i are i.i.d. and {a _in} is an array of constants. When sup(n ^–1|a _in | ^q )^1/q <, 1<q andX _i are mean zero, we showE|X| ^p <,p ^l+q ^–1=1 impliesS _n /n 0. Whenq= this reduces to a result of Choi and Sung who showed that when the {a _in} are uniformly bounded,EX=0 andE|X|< impliesS _n /n 0. The result is also true whenq=1 under the additional assumption that lim sup |a _in |n ^–1 logn=0. Extensions to more general normalizing sequences are also given. In particular we show that when the {a _in} are uniformly bounded,E|X|^1/< impliesS _n /n 0 for >1, but this is not true in general for 1/2<<1, even when theX _i are symmetric. In that case the additional assumption that (x ^1/ log^1/–1 x)P(|X|x)0 asx provides necessary and sufficient conditions for this to hold for all (fixed) uniformly bounded arrays {a _in}.     

Counting the number of twin Niven numbers

Given an integer q≥2, we say that a positive integer is a q-Niven number if it is divisible by the sum of its digits in base q. Given an arbitrary integer r∈[2,2q], we say that (n,n+1,…,n+r−1) is a q-Niven r -tuple if each number n+i, for i=0,1,…,r−1, is a q-Niven number. We show that there exists a positive constant c=c(q,r) such that the number of q-Niven r-tuples whose leading component is <x is asymptotic to cx/(log x) ^r as x→∞. Research of J.M. De Koninck supported in part by a grant from NSERC. Research of I. Kátai supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA.     

On the strong limit theorems for double arrays of blockwise M-dependent random variables

For a double array of blockwise M-dependent random variables {X _mn ,m ?? 1, n ?? 1}, strong laws of large numbers are established for double sums ?? _i=1 ^m ?? _j=1 ⁿ X _ij , m ?? 1, n ?? 1. The main results are obtained for (i) random variables {X _mn ,m ?? 1, n ?? 1} being non-identically distributed but satisfy a condition on the summability condition for the moments and (ii) random variables {X _mn ,m ?? 1, n ?? 1} being stochastically dominated. The result in Case (i) generalizes the main result of Móricz et al. [J. Theoret. Probab., 21, 660?C671 (2008)] from dyadic to arbitrary blocks, whereas the result in Case (ii) extends a result of Gut [Ann. Probab., 6, 469?C482 (1978)] to the bockwise M-dependent setting. The sharpness of the results is illustrated by some examples.     

On the Glivenko–Cantelli theorem for generalized empirical processes based on strong mixing sequences

Given \s{X_i, i 1\s} as non-stationary strong mixing (n.s.s.m.) sequence of random variables (r.v.'s) let, for 1 i n and some γ ε [0, 1],

F₁(x)=γP(X_i<x)+(1-γ)P(X_ix)

and

I_i(x)=γI(X_i<x)+(1-γ)I(X_ix)

. For any real sequence \s{C_i\s} satisfying certain conditions, let

.

In this paper an exponential type of bound for P(D_n ), for any >0, and a rate for the almost sure convergence of D_n are obtained under strong mixing. These results generalize those of Singh (1975) for the independent and non-identically distributed sequence of r.v.'s to the case of strong mixing.     

A generalization of the formula for the triangular number of the sum and product of natural numbers

This note generalizes the formula for the triangular number of the sum and product of two natural numbers to similar results for the triangular number of the sum and product of r natural numbers. The formula is applied to derive formula for the sum of an odd and an even number of consecutive triangular numbers.     

关于随机序列加权和的收敛性的注记

通过对数组加权系数的列截尾,研究了B值随机一致有界序列加权和的收敛性,获得了相应的几乎处处收敛和完全收敛到0的结果.最后建立了B值随机适应序列加权和的矩收敛定理.     

任意随机变量序列级数的强收敛性

利用鞅差序列级数的收敛定理和条件三级数定理研究了任意随机变量序列级数的强收敛性,推广了某些经典的鞅差序列和独立随机变量序列及两两NQD序列的强极限定理.     

Weak law of large numbers for almost periodically correlated processes

This note contains two simple observations concerning the weak law of large numbers for almost periodically correlated processes.

     

混合序列加权和的强收敛性

本文给出混合序列加权和的强收敛性的一些充分条件，这些结论推广和改进了文［１］定理３，文［２］定理３；文［３］定理４．１５以及文［４］定理４．     

鞅差序列加权和的几乎处处收敛(英文)

本文研究了一类鞅差序列加权和的收敛性的问题.利用一些基本不等式和截尾技术,获得了加权和的几乎处处收敛性,推广了关于独立同分布的随机变量序列的相关结果.     

Some Limit Theorems for Weighted Sums of Arrays of Nod Random Variables

In this paper the authors study the complete, weak and almost sure convergence for weighted sums of NOD random variables and obtain some new limit theorems for weighted sums of NOD random variables, which extend the corresponding theorems of Stout [1], Thrum [2] and Hu et al. [3].     

一类随机变量序列部分和的强大数定律

设{Xn,n≥1}是随机变量序列.文[4]在二阶矩限制下,获得了任意随机变量序列的Hajek-Renyi型不等式,并给出了随机变量序列的强大数定律.本文利用胡舒合等获得的强大数定律,给出了随机变量序列的一些几乎必然收敛性,并给出了结果在PA,NA和两两NQD序列场合下的应用.     

Some Limit Theorems for Sequences of Pairwise NQD Random Variables

In this article, the authors study some limit properties for sequences of pairwise NQD random variables, which are not necessarily identically distributed. They obtain Baum and Katz complete convergence and the strong stability of Jamison＇s weighted sums for pairwise NQD random variables, which may have different distributions. Some wellknown results are improved and extended.     

人口数相依的两性分支过程的极限性质

本文研究了后代分布依赖于人口数的两性Galton-Watson分支过程, 在对后代分布的适当假设下,对于上临界的情况, 我们研究了有关过程的几乎处处收敛的极限性质.