Sums of Powers of Independent Random Variables
Let $(x_{nk}), k = 1, \cdots, k_n; n = 1, 2, \cdots$ be a double sequence of infinitesimal random variables which are rowwise independent (i.e., $\lim_{n\rightarrow\infty} \max_{1\leqq k \leqq k_n} P (| x_{nk} | > \epsilon) = 0$ for every $\epsilon > 0,$ and for each $n x_{n1}, \cdots, x_{nk_n}$ are independent). Let $S_n = x_{n1} + \cdots + x_{nk_n} - A_n$ where the $A_n$ are constants and let $F_n(x)$ be the distribution function of $S_n.$ Necessary and sufficient conditions for $F_n(x)$ to converge to a distribution function $F(x)$ are known, and in particular we know that $F(x)$ is infinitely divisible. In this paper we shall investigate the system of infinitesimal, rowwise independent random variables $(| x_{nk} | ^r), r \geqq 1.$ In particular we shall be interested in large values of $r$. Specifically, let $S^r_n = | x_{n1} | ^r + \cdots + | x_{n1} | ^r - B_n(r),$ where $B_n(r)$ are suitably chosen constants. Let $F_n^r(x)$ be the distribution function of $S^r_n.$ Necessary and sufficient conditions for $F_n^r(x)$ to converge $(n \rightarrow \infty)$ to a distribution function $F^r(x)$ are given, and also necessary and sufficient conditions for $F^r(x)$ to converge $(r \rightarrow \infty)$ to a distribution function $H(x)$ are given. The form that $H(x)$ must take is obtained and under rather general conditions it is shown that $H(x)$ is a Poisson distribution. In any case it is shown that $H(x)$ is the sum of two independent random variables, one Gaussian and the other Poisson (including their degenerate cases).