The Annals of Mathematical Statistics

Sums of Powers of Independent Random Variables

J. M. Shapiro

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Abstract

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).

Article information

Source
Ann. Math. Statist., Volume 29, Number 2 (1958), 515-522.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177706626

Digital Object Identifier
doi:10.1214/aoms/1177706626

Mathematical Reviews number (MathSciNet)
MR97110

Zentralblatt MATH identifier
0087.13602

JSTOR
links.jstor.org

Citation

Shapiro, J. M. Sums of Powers of Independent Random Variables. Ann. Math. Statist. 29 (1958), no. 2, 515--522. doi:10.1214/aoms/1177706626. https://projecteuclid.org/euclid.aoms/1177706626


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