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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
$$\rho(F_p^n,\mathfrak G)=\inf\limits_{G\in\mathfrak G}\sup\limits_x\left|F_p^n(x)-G(x)\right|.$$
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
$$\mathfrak G_1^M(a)=\biggl\{G:G\in\mathfrak G;\int_{-\infty}^\infty e^{itx}\,dG(x)=\exp\biggl[i\gamma t+\sum_{|k|<M}(e^{itk}-1)q_k\biggr]\\\int_{-\infty}^\infty x\,dG(x)=a,\quad q_k\geq0,k=0,\pm1\dots.\biggr\}.$$
The result a) is generalized for the case of a multinomial distribution.

Received: 30.10.1959

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