Local limit theorems for weighted sums of independent random variables

E. M. Shoukry

Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 1, pp. 135-142

Résumé

In this paper, we study the behaviour of $\displaystyle S_n=\sum_{k=-\infty}^{\infty}a_{kn}\xi_k$ as $n$ tends to infinity, where $\xi_k$ are independent identically distributed random variables and their common distribution function belongs to the domain of attraction of a certain stable law $G$ with index $\alpha$. Let the following two conditions on the matrix of coefficients ($a_{kn}$) be satisfied: 1) $\displaystyle\sum_{k=-\infty}^{\infty}|a_{kn}|^{\alpha}\widetilde h(a_{kn})=b_n\to 1\qquad(n\to\infty),\\$ where $\widetilde h(x)$ is the slowly varying function from the representation for the characteristic function of $G$; 2) $\displaystyle\gamma_n=\sup_k|a_{kn}|\to 0\qquad(n\to\infty).\\$ Then it is shown that the distribution function of $S_n$ converges to a stable distribution function, and, if $\displaystyle \int_{-\infty}^{\infty}|f(t)|^p\,dt\infty$, $p>0$, where $f(t)$ is the characteristic function of $\xi_k$ then the density function of $S_n$ exists and converges to the density function of the limit distribution.