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Teor. Veroyatnost. i Primenen., 1968, Volume 13, Issue 2, Pages 348–351 (Mi tvp854)  

Short Communications

A local limit theorem for unequally distributed random variables

V. M. Kruglov

Moscow

Abstract: Let $\xi_1,…,\xi_n$ be a sequence of independent random variables. Form another sequence
$$ \eta_n=\frac{\xi_1+…+\xi_n}{B_n}-A_n.\eqno(1) $$
Suppose that for any $n$ $\xi_n$ has one of $\tau$ absolutely continuous distributions
$$ F_1(x),F_2(x),…,F_\tau(x) $$
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.

Full text: PDF file (225 kB)

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

Bibliographic databases:

Received: 20.10.1966

Citation: V. M. Kruglov, “A local limit theorem for unequally distributed random variables”, Teor. Veroyatnost. i Primenen., 13:2 (1968), 348–351; Theory Probab. Appl., 13:2 (1968), 332–334

Citation in format AMSBIB
\Bibitem{Kru68}
\by V.~M.~Kruglov
\paper A~local limit theorem for unequally distributed random variables
\jour Teor. Veroyatnost. i Primenen.
\yr 1968
\vol 13
\issue 2
\pages 348--351
\mathnet{http://mi.mathnet.ru/tvp854}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=234510}
\zmath{https://zbmath.org/?q=an:0167.46801|0165.20103}
\transl
\jour Theory Probab. Appl.
\yr 1968
\vol 13
\issue 2
\pages 332--334
\crossref{https://doi.org/10.1137/1113040}


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