On the Approximation of Polynomial Distributions by Infinitely Divisible Laws
| Autor: | L. D. Meshalkin |
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| Rok vydání: | 1960 |
| Předmět: | |
| Zdroj: | Theory of Probability & Its Applications. 5:106-114 |
| ISSN: | 1095-7219 0040-585X |
| DOI: | 10.1137/1105009 |
| Popis: | Let $F_p^n (x)$ be an $(n,p)$ binomial distribution function, $\mathfrak{G}$ a set of all infinitely divisible laws and \[ \rho \left( {F_p^n ,\mathfrak{G}} \right) = \mathop {\inf }\limits_{G \in \mathfrak{G}} \mathop {\sup }\limits_x \left| {F_p^n (x) - G(x)} \right| . \] Then, a) $\mathop {\sup }\limits_{0 \leqq p \leqq 1} \rho _1 \left( {F_p^n ,\mathfrak{G}} \right) C(M)n^{ - {2/ 3}} \left( {\log n} \right)^{ - 1/ 4} $, whrer $C_0 $ is an absolute constant $C(M) > 0$ depends on M only, and \[ \begin{gathered} \mathfrak{G}_1^M (a) = \left\{ {G:G \in \mathfrak{G};\int_{ - \infty }^\infty {e^{itx} dG(x) = \exp \left[ {i\gamma t + \mathop \Sigma \limits_{|k| < M} \left( {e^{itk} - 1 } \right)q_k } \right]} ,} \right. \hfill \\ \phantom{ \mathfrak{G}_1^M (a) = \ } \left. {\int_{ - \infty }^\infty {xdG(x) = a,\quad q_k \geqq 0,k = 0, \pm 1 \cdots } } \right\}. \hfill \\ \end{gathered} \]The result ... |
| Databáze: | OpenAIRE |
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