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

TL;DR

This paper establishes exponential bounds for the tail probability of the supremum of an inhomogeneous random walk, providing practical methods to compute these bounds and applying them to risk models.

Contribution

It introduces sufficient conditions for exponential tail bounds of inhomogeneous random walks and offers a method to compute the constants involved, with applications to risk theory.

Findings

Derived exponential tail bounds for supremum probabilities.

Presented a method to calculate the bounding constants.

Applied bounds to ruin probability in risk models.

Abstract

Let $\{\xi_1,\xi_2,\ldots\}$ 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 $\mathbb{P}(\sup_{n\geqslant0}\sum_{i=1}^n\xi_i>x)$ can be bounded above by $\varrho_1\exp\{-\varrho_2x\}$ 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.