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Teor. Veroyatnost. i Primenen., 1960, Volume 5, Issue 1, Pages 114–124 (Mi tvp4817)  

Short Communications

On Approximation of a Multinomial Distribution by Infinitely Divisible Laws

L. D. Meshalkin

Moscow

Abstract: Let $F_p^n(x)$ be an $(n,p)$ binomial distribution function, $\mathfrak{G}$ a set of all infinitely divisible laws and
$$\rho(F_p^n,\mathfrak G)=\inf\limits_{G\in\mathfrak G}\sup\limits_x|F_p^n(x)-G(x)|.$$
Then,
a) $\sup\limits_{0\leq p\leq1}\rho_1(F_p^n,\mathfrak G)<C_0 n^{-2/3}$,
b) $\rho_1(F^n_{n^{-2/3}},\mathfrak G_1^M(n^{1/3}))>C(M)n^{-2/3}(\lg n)^{-1/4}$, where $C_0$ is an absolute constant $C(M)>0$ depends on $M$ only, and
$$\mathfrak G_1^M(a)=\{G:G\in\mathfrak G;\int_{-\infty}^\infty e^{itx} dG(x)=\exp[i\gamma t+\sum_{|k|<M}(e^{itk}-1)q_k]
\int_{-\infty}^\infty x dG(x)=a,\quad q_k\geq0,k=0,\pm1….\}.$$
The result a) is generalized for the case of a multinomial distribution.

Full text: PDF file (837 kB)

English version:
Theory of Probability and its Applications, 1960, 5:1, 106–114

Received: 30.10.1959

Citation: L. D. Meshalkin, “On Approximation of a Multinomial Distribution by Infinitely Divisible Laws”, Teor. Veroyatnost. i Primenen., 5:1 (1960), 114–124; Theory Probab. Appl., 5:1 (1960), 106–114

Citation in format AMSBIB
\Bibitem{Mes60}
\by L.~D.~Meshalkin
\paper On Approximation of a Multinomial Distribution by Infinitely Divisible Laws
\jour Teor. Veroyatnost. i Primenen.
\yr 1960
\vol 5
\issue 1
\pages 114--124
\mathnet{http://mi.mathnet.ru/tvp4817}
\transl
\jour Theory Probab. Appl.
\yr 1960
\vol 5
\issue 1
\pages 106--114
\crossref{https://doi.org/10.1137/1105009}


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