The Annals of Mathematical Statistics

Sums of Small Powers of Independent Random Variables

J. M. Shapiro

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Let $(x_{nk}), k = 1, 2, \cdots, k_n; n = 1, 2, \cdots$ 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 $S_n$. In a previous paper [3] the system of infinitesimal, rowwise independent random variables $(|x_{nk}|^r)$ was studied for $r \geqq 1$. Specifically, let $S^r_n = |x_{n1}|^r + \cdots + |x_{nk_n}|^r - B_n(r),$ where the $B_n(r)$ 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 $F^r(x)$ and for $F^r(x)$ to converge $(r \rightarrow \infty)$ to a distribution function $H(x)$ were given, together with the form that $H(x)$ must take. In Section 2 of this paper we consider the system $(|x_{nk}|^r)$ for $0 < r < 1$. Results similar to the above are found, replacing $(r \rightarrow \infty)$ by $(r \rightarrow 0^+)$. However different assumptions must be made at certain points. Various remarks are made in this paper to show where the results here differ from [3]. In particular it is shown that, if $F^r(x)$ converges $(r \rightarrow 0^+)$ to a distribution function $H(x)$, then $H(x)$ will be the distribution function of the sum of two independent random variables, one Poisson and the other Gaussian. Furthermore, while the Gaussian summand may or may not be degenerate, the Poisson summand will be nondegenerate in all but one special case.

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Ann. Math. Statist., Volume 31, Number 1 (1960), 222-224.

First available in Project Euclid: 27 April 2007

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Shapiro, J. M. Sums of Small Powers of Independent Random Variables. Ann. Math. Statist. 31 (1960), no. 1, 222--224. doi:10.1214/aoms/1177705999.

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