Modern Stochastics: Theory and Applications

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 $\mathbb{P}(\sup _{n\geqslant 0}{\sum _{i=1}^{n}}\xi _{i}>x)$ 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.