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Teoriya Veroyatnostei i ee Primeneniya, 1981, Volume 26, Issue 4, Pages 827–832
(Mi tvp3513)
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This article is cited in 8 scientific papers (total in 8 papers)
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 $\delta>0$
$$
|\mathbf P\{\xi_t\in[x,x+h)\}-\mathbf P\{\xi_{t+\delta}\in[x,x+h)\}|<
C_1\gamma_{ht}(1+|ln\gamma_{ht}|)^4\ln\biggl(1+\frac{\delta}{t}\biggr),
$$
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.
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
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