论文标题
依赖随机变量的大数量弱法律
A Weak Law of Large Numbers for Dependent Random Variables
论文作者
论文摘要
每个序列$ f_1，f_2，\ cdots \，随机变量的$，带有$ \，\ lim_ {m \ to \ infty} \ big（m \ sup_ {k \ in \ mathbb {n}}}}} \ Mathbb {n} f_ {k_1}，f_ {k_2}，\ cdots \，$，$及其所有子序列，是大数字的薄弱法则：$ \，\ lim_ {n \ to \ infty} \ big（（1/N）这里$ \，d_n \，$是一个“校正”随机变量，每个$ n \ in \ mathbb {n} $中的每个$ n \ in $ [ - n，n] $中的值;如果此外，这些校正器都等于零，此外，$ \，\ liminf_ {k \ to \ infty} \ mathbb {e} \ big（f_k^2 \，\ MathBf {1} _ {\ f_k | \ infty）\，。$
Every sequence $f_1, f_2, \cdots \, $ of random variables with $ \, \lim_{M \to \infty} \big( M \sup_{k \in \mathbb{N}} \mathbb{P} ( |f_k| > M ) \big)=0\,$ contains a subsequence $ f_{k_1}, f_{k_2} , \cdots \,$ that satisfies, along with all its subsequences, the weak law of large numbers: $ \, \lim_{N \to \infty} \big( (1/N) \sum_{n=1}^N f_{k_n} - D_N \big) =0\,,$ in probability. Here $\, D_N\, $ is a "corrector" random variable with values in $[-N,N]$, for each $N \in \mathbb{N} $; these correctors are all equal to zero if, in addition, $\, \liminf_{k \to \infty} \mathbb{E} \big( f_k^2 \, \mathbf{ 1}_{ \{ |f_k| \le M \} } \big) =0\,$ holds for every $M \in (0, \infty)\,.$
