Exponential bounds for the tail probability of the supremum of an inhomogeneous random walk

Abstract

Let $\{{\xi }_1,{\xi }_2,\dots \}$ be a sequence of independent but not necessarily identically distributed random variables. In this paper, the sufficient conditions are found under which the tail probability $\mathbb{P}({\sup }_{n⩾0}{\sum }_{i=1}^n{\xi }_i>x)$ can be bounded above by ${\varrho }_1\exp \{-{\varrho }_{2}x\}$ with some positive constants ${\varrho }_1$ and ${\varrho }_2$. A way to calculate these two constants is presented. The application of the derived bound is discussed and a Lundberg-type inequality is obtained for the ultimate ruin probability in the inhomogeneous renewal risk model satisfying the net profit condition on average.

Published at https://doi.org/10.15559/18-VMSTA99 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)