On Local Limit Theorems for Sums of Independent Random Variables
On Local Limit Theorems for Sums of Independent Random Variables
复制标题
关于独立随机变量之和的局部极限定理
DOI：
10.1137/1109044
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发表时间：
1964
影响因子：
0.6
通讯作者：
V. V. Petrov
中科院分区：
文献类型：
--
作者：
V. V. Petrov
设X_1，X_2，\cdots $是独立同分布的随机变量序列，X_1 = m$，X_1 = \sigma ^2 > 0$，|X_1| ^k < \infty $ for some integer $k \geqq 3$.证明了如下定理：假设变量Z_n =（{1 / {\sigma \sqrt n }}）（\sum\nolimits_{j = 1}^n {X_j - nm}）$有一个绝对连续的分布，其密度函数p_n（x）$对某个整数n = n_0 $有界。则存在一个函数$\vareps（n）$使得lim $\vareps（n）= 0$且满足关系式（1），当$X_1 $具有格分布时证明了一个类似的定理.这些定理的一些后果有关收敛到正常的法律平均进行了讨论。
Let $X_1 ,X_2 , \cdots $ be a sequence of independent identically distributed random variables, ${\bf E}X_1 = m$, ${\bf D}X_1 = \sigma ^2 > 0$, and ${\bf E}|X_1 |^k < \infty $ for some integer $k \geqq 3$. The following theorem is proved:Suppose that the variable $Z_n = ({1 / {\sigma \sqrt n }})(\sum\nolimits_{j = 1}^n {X_j - nm} )$ has an absolutely continuous distribution with bounded density function $p_n (x)$ for some integer $n = n_0 $. Then there exists a function $\varepsilon (n)$ such that lim $\varepsilon (n) = 0$ and relation (1) is fulfilled.A similar theorem is proved for the case when $X_1 $ has a lattice distribution. Some consequences of these theorems concerning convergence to the normal law in the mean are discussed.