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This article is cited in 1 scientific paper (total in 1 paper)
Short Communications
Two inequalities for symmetric processes and symmetric distributions
E. L. Presman Moscow
Abstract:
It is proved that there exists a constant $C_1$ such that:
a) for any stochastic process $\xi_t$ with symmetric stationary independent increments and for any $\delta>0$
$$
|\mathbf P\{\xi_t\in[x,x+h)\}-\mathbf P\{\xi_{t+\delta}\in[x,x+h)\}|<
C_1\gamma_{ht}(1+|ln\gamma_{ht}|)^4\ln(1+\frac{\delta}{t}),
$$
where $\displaystyle\gamma_{ht}=\sup_x\mathbf P\{\xi_t\in[x,x+h)\}$,
b) for any symmetric probabilistic measure $F$ on the real line and for any $a>0$
$$
|a(F-E)e^{a(F-E)}\{[x,x+h)\}|<C_1\gamma_h(1+|\ln\gamma_h|)^4,
$$
where $\displaystyle\gamma_h=\sup_x e^{a(F-E)}\{[x,x+h)\}$, $\displaystyle e^{a(F-E)}=e^{-a}\sum_{k=0}^{\infty}(a^kF^k)/k!$, $F^n$ is $n$-fold convolution of $F$ with itself, $E$ is a probabilistic measure with a unit mass at zero.
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English version:
Theory of Probability and its Applications, 1982, 26:4, 815–819
Bibliographic databases:
Received: 04.09.1979
Citation:
E. L. Presman, “Two inequalities for symmetric processes and symmetric distributions”, Teor. Veroyatnost. i Primenen., 26:4 (1981), 827–832; Theory Probab. Appl., 26:4 (1982), 815–819
Citation in format AMSBIB
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\jour Teor. Veroyatnost. i Primenen.
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\vol 26
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\transl
\jour Theory Probab. Appl.
\yr 1982
\vol 26
\issue 4
\pages 815--819
\crossref{https://doi.org/10.1137/1126089}
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http://mi.mathnet.ru/eng/tvp3513 http://mi.mathnet.ru/eng/tvp/v26/i4/p827
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This publication is cited in the following articles:
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Cekanavicius V., “Approximation Methods in Probability Theory”, Approximation Methods in Probability Theory, Universitext, Springer International Publishing Ag, 2016, 1–274
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