Exponential bounds for the tail probability of the supremum of an inhomogeneous random walk
Volume 5, Issue 2 (2018), pp. 129–143
Pub. online: 15 March 2018
Type: Research Article
Open Access
Open Access
1
The second author was supported by grant No S-MIP-17-72 from the Research Council of Lithuania.
Received
13 October 2017
13 October 2017
Revised
10 February 2018
10 February 2018
Accepted
13 February 2018
13 February 2018
Published
15 March 2018
15 March 2018
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 $\mathbb{P}(\,{\sup _{n\geqslant 0}}\,{\sum _{i=1}^{n}}{\xi _{i}}>x)$ 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.
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