论文标题
关于一系列依赖随机变量的收敛性
On the convergence of series of dependent random variables
论文作者
论文摘要
给定对对称的随机变量的序列$（x_n）$在希尔伯特空间中取值，一个有趣的开放问题是确定$ \ sum_ {n = 1}^\ infty x_n $几乎肯定是收敛的条件。对于独立的随机变量，众所周知，如果$ \ sum_ {n = 1}^\ infty \ mathbb {e}（\ | x_n \ |^2）<\ infty $，则$ \ sum_ {n = 1}^\ infty iftty x_n $几乎肯定地转换。这已扩展到某些因变量的情况（即相关的随机变量），但是在相关变量的一般设置中，问题仍然开放。本文考虑了每个变量$ x_n $作为线性组合$ a_ {n，1} z_1 + \ ldots + a__ {n，n，n，n} z_n $中给出的情况，其中$（z_n）$是一系列独立的对称对称的随机变量，单位变异和$（a_____ {a_ {a_ {n，k n，k}）$ sansants $ startants $ starts。对于高斯随机变量，这是一般设置。我们获得了$ \ sum_ {n = 1}^\ infty x_n $几乎确定收敛的足够条件，这对于所有（非狂热）符号的变化也几乎足以确保$ \ sum_ {n = 1}^\ infty \ pm x_n $的几乎确定收敛。结果基于随机变量$ \ sup（\ | x_1 + \ ldots + x_k \ |：1 \ leq k \ leq n）$的均值的重要界限，该范围扩大了古典lévy的不平等，并且具有独立的利益。
Given a sequence $(X_n)$ of symmetrical random variables taking values in a Hilbert space, an interesting open problem is to determine the conditions under which the series $\sum_{n=1}^\infty X_n$ is almost surely convergent. For independent random variables, it is well-known that if $\sum_{n=1}^\infty \mathbb{E}(\|X_n\|^2) <\infty$, then $\sum_{n=1}^\infty X_n$ converges almost surely. This has been extended to some cases of dependent variables (namely negatively associated random variables) but in the general setting of dependent variables, the problem remains open. This paper considers the case where each variable $X_n$ is given as a linear combination $a_{n,1}Z_1+ \ldots +a_{n,n}Z_n$ where $(Z_n)$ is a sequence of independent symmetrical random variables of unit variance and $(a_{n,k})$ are constants. For Gaussian random variables, this is the general setting. We obtain a sufficient condition for the almost sure convergence of $\sum_{n=1}^\infty X_n$ which is also sufficient for the almost sure convergence of $\sum_{n=1}^\infty \pm X_n$ for all (non-random) changes of sign. The result is based on an important bound of the mean of the random variable $\sup(\|X_1 + \ldots +X_k\|: 1\leq k \leq n)$ which extends the classical Lévy's inequality and has some independent interest.
