柯尔莫哥洛夫强大数定律的推广

柯尔莫哥洛夫强大数定律的推广 ON THE GENERALIZATION OF THE STRONG LAW OF LARGE NUMBERS OF KOLMOGOROV 邹新堤 Zou Xindi first-author < 正 > 柯尔莫哥洛夫曾证明了如下重要定理: 设§ 1 ,§ 2 ,…是相互独立同分布的随机变数序列,E|§ 1 | < +∞,则本文将在非同分布的条件下,来推广上述结果。定理1 设§ 1 ,§ 2 ,…是一随机变数序列,§是一随机变数, 若记 G n （x）=P（|§ n | > x）, n=1,2,… G（x）=P（|§| > x） 并满足下列条件: ⅰ存在某一正数A,对一切n有 G n （x）≤G（x） 当x≥A时, ⅱ E|§| K +∞,对某一正数K成立。 In this paper we generalized the strong law of large numbers of kolmogorov. Our results obtained are as follows Theorem Ⅰ。Let {ξ n } be a sequence of independent random variables and be a random variable which satisfy the following conditions 1. E|ξ n | < +∞, n=1, 2, …; E|ξ| < +∞. 2. There exists a Positive constant A and a integer N such that for all n≥N F n （x） ≤F（x） for x≤-A F n （x）≥F（x） for x≥A where F n （x） and F（x） are the distribution functions of the random Variables ξ n and ξ respectively. Then {ξ n }obeys the strong law of large number, i. e. Theorem 2. Let {ξ n }be a sequence of random variables and ξ be a random variable which satisfy the following condions There exists positive constants A and k such that and G n （x）≤G（x） for x≥A, n=1, 2, … where G n （x） and G（x） are the functions defined by G n （x）=P（|ξ n | > x） G（x）=P（|ξ| > x） Then E|ξ n | k < C, n=1, 2, … Where C is a constant which only depends on the distribution of ξ and A. 1983-03-01 2021-04-01 3