Sums of Independent Truncated Random Variables

Abstract

Let $(x_{nk}), (k = 1, 2, \cdots, k_n; n = 1, 2, \cdots)$ be a double sequence of infinitesimal (i.e. $\lim_{n \rightarrow \infty} \max_{1 \leqq k \leqq k_n} P\{|x_{nk}| > \epsilon\} = 0$ for every $\epsilon > 0$) random variables such that for each $n, x_{n1}, \cdots, x_{nk_n}$ are independent. Let $S_n = x_{n1} + \cdots + x_{nk_n}$ and let $F_n(x)$ be the distribution function of $S_n$. For any $a > 0$ let the random variables $x^a_{nk}$ be defined by \begin{equation*}x^a_{nk} = \begin{cases}x_{nk}, & \text{if} -a < x_{nk} \leqq a, \\ 0, & \text{\otherwise},\end{cases}\end{equation*} and let $F^a_n(x)$ be the distribution function of $S^a_n = x^a_{n1} + \cdots + x^a_{nk_n}$. In the next section certain necessary and sufficient conditions are given for $F^a_n(x)$ to converge $(n \rightarrow \infty)$ to a limiting distribution and in particular it is shown that if $F^a_n(x)$ converges to $F(x)$, then $F(x)$ has finite moments of all orders. In Sec. 3 it is shown that if $F^a_n(x)$ converges to $F(x)$ then for each positive integer $k$ the $k$th moment of $F^a_n(x)$ approaches the $k$th moment of $F(x)$ as $n \rightarrow \infty$. We shall call the random variables $(x_{nk})$ a truncated system if there exists a $b > 0$ independent of $k$ and $n$ such that $P\{|x_{nk}| > b\} = 0$. We note that if we start with a truncated system we can choose $a > 0$ such that $x^a_{nk} = x_{nk}$.