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Short Communications

A local limit theorem for unequally distributed random variables

V. M. Kruglov

Moscow

Abstract: Let $\xi_1,\dots,\xi_n$ be a sequence of independent random variables. Form another sequence
$$ \eta_n=\frac{\xi_1+\dots+\xi_n}{B_n}-A_n.\eqno(1) $$
Suppose that for any $n$ $\xi_n$ has one of $\tau$ absolutely continuous distributions
$$ F_1(x),F_2(x),\dots,F_\tau(x) $$
The following assertion is proved.
For the sequence of the densities $p_n(x)$ of the sums (1) to converge uniformly to the density of a limit law for some $B_n>0$, $A_n$ it is necessary and sufficient that
1. $\mathbf P\{\eta_n<x\}\to G(x)$ weakly ($G$ is the limit law).
2. There exists such an $N$ that $p_N(x)$ is bounded.

Received: 20.10.1966

 English version: Theory of Probability and its Applications, 1968, 13:2, 332–334

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