Two inequalities for symmetric processes and symmetric distributions

E. L. Presman

Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 4, pp. 827-832

Résumé

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.