Given a sequence $$(X_n)$$ ( 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 $$\sum _{n=1}^\infty X_n$$ ∑ n = 1 ∞ X n is almost surely convergent. For independent random variables, it is well known that if $$\sum _{n=1}^\infty \mathbb {E}(\Vert X_n\Vert ^2) <\infty $$ ∑ n = 1 ∞ E ( ‖ X n ‖ 2 ) < ∞ , then $$\sum _{n=1}^\infty X_n$$ ∑ n = 1 ∞ 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 $$X_n$$ X n is given as a linear combination $$a_{n,1}Z_1+ \cdots +a_{n,n}Z_n$$ a n , 1 Z 1 + ⋯ + a n , n Z n where $$(Z_n)$$ ( Z n ) is a sequence of independent symmetrical random variables of unit variance and $$(a_{n,k})$$ ( 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 $$\sum _{n=1}^\infty X_n$$ ∑ n = 1 ∞ X n which is also sufficient for the almost sure convergence of $$\sum _{n=1}^\infty \pm X_n$$ ∑ n = 1 ∞ ± X n for all (non-random) changes of sign. The result is based on an important bound of the mean of the random variable $$\sup (\Vert X_1 + \cdots +X_k\Vert : 1\le k \le n)$$ sup ( ‖ X 1 + ⋯ + X k ‖ : 1 ≤ k ≤ n ) which extends the classical Lévy’s inequality and has some independent interest.

中文翻译：

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关于相关随机变量序列的收敛性

给定一个序列 $$(X_n)$$ ( X n ) 在希尔伯特空间中取值的对称随机变量，一个有趣的开放问题是确定序列 $$\sum _{n=1}^\ infty X_n$$ ∑ n = 1 ∞ X n 几乎肯定收敛。对于独立的随机变量，众所周知，如果 $$\sum _{n=1}^\infty \mathbb {E}(\Vert X_n\Vert ^2) <\infty $$ ∑ n = 1 ∞ E ( ‖ X n ‖ 2 ) < ∞ ，那么 $$\sum _{n=1}^\infty X_n$$ ∑ n = 1 ∞ X n 几乎肯定收敛。这已扩展到因变量（即负相关随机变量）的某些情况，但在因变量的一般设置中，问题仍然存在。本文考虑每个变量 $$X_n$$ X n 被给定为线性组合 $$a_{n,1}Z_1+ \cdots +a_{n,n}Z_n$$ an , 1 Z 1 + ⋯ +一个 ，n Z n 其中 $$(Z_n)$$ ( Z n ) 是单位方差的独立对称随机变量序列，$$(a_{n,k})$$ ( an , k ) 是常数。对于高斯随机变量，这是一般设置。我们得到了 $$\sum _{n=1}^\infty X_n$$ ∑ n = 1 ∞ X n 几乎肯定收敛的充分条件，这对于 $$\sum _{ 几乎肯定收敛也是充分的n=1}^\infty \pm X_n$$ ∑ n = 1 ∞ ± X n 对于所有（非随机）符号变化。结果基于随机变量 $$\sup (\Vert X_1 + \cdots +X_k\Vert : 1\le k \le n)$$ sup ( ‖ X 1 + ⋯ + X k ‖ : 1 ≤ k ≤ n ) 它扩展了经典的 Lévy 不等式并具有一些独立的兴趣。这是一般设置。我们得到了 $$\sum _{n=1}^\infty X_n$$ ∑ n = 1 ∞ X n 几乎肯定收敛的充分条件，这对于 $$\sum _{ 几乎肯定收敛也是充分的n=1}^\infty \pm X_n$$ ∑ n = 1 ∞ ± X n 对于所有（非随机）符号变化。结果基于随机变量 $$\sup (\Vert X_1 + \cdots +X_k\Vert : 1\le k \le n)$$ sup ( ‖ X 1 + ⋯ + X k ‖ : 1 ≤ k ≤ n ) 它扩展了经典的 Lévy 不等式并具有一些独立的兴趣。这是一般设置。我们得到了 $$\sum _{n=1}^\infty X_n$$ ∑ n = 1 ∞ X n 几乎肯定收敛的充分条件，这对于 $$\sum _{ 几乎肯定收敛也是充分的n=1}^\infty \pm X_n$$ ∑ n = 1 ∞ ± X n 对于所有（非随机）符号变化。结果基于随机变量 $$\sup (\Vert X_1 + \cdots +X_k\Vert : 1\le k \le n)$$ sup ( ‖ X 1 + ⋯ + X k ‖ : 1 ≤ k ≤ n ) 它扩展了经典的 Lévy 不等式并具有一些独立的兴趣。