On Inequalities for the Ladder Height Distribution
under Small Drift

Lotov, Vladimir (Novosibirsk, Russia)
lotov@math.nsc.ru

Let $\{\xi_n\},\,n\ge 1$ be a sequence of i.i.d. random variables such that ${\mathsf E}\,\xi_1=-m<0$. Put $F(t)={\mathsf P}\,(\xi_1<t)$, $S_n=\xi_1+\ldots+\xi_n,\ n\ge 1$,$\eta_-=\min\{n\ge 1:\ S_n\le 0\}$, and $\chi_-=S_{\eta_-}$. As a rule, explicit formulas for $U(t)={\mathsf P}\,(\chi_-<t)$ are not available. The following inequality is well known: $$F(t)\le U(t)\le rF(t),\ \ t\le 0,$$where $r=(1-q)^{-1}, \ q={\mathsf P}\,(\sup(0, S_1,\ldots)\gt)$. It is clear that the constant r can take large values for small values of m. We give an estimate for U(t) which is uniform in $m\gt$. Some other inequalities are discussed.