This article is cited in 9 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

 English version: Theory of Probability and its Applications, 1982, 26:4, 815–819

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