Asymptotics For The Maximum Of A Random Walk With Δ-Subexponential Distribution

The maximum of a random walk theory has always been an important part of the random walk, not only it has an important theoretical value of its own, but also in queueing theory,risk theory, the theory of large deviations, the branching process and other areas has an irreplaceable role and significance. The maximum of the heavy-tailed random walk occupies an important position in the random walk theory. In many literatures, the characters of maximum of the heavy-tailed random walk are introduced, it becomes the focus of research.In recent years, the related properties were investigated with the maximum of a random walk with exponential distribution, and achieved fruitful results. This article will generalize the results of previous research to Δ-subexponential distribution family, and discuss the asymptotic behavior about the maximum of random walk.The article content is as follows:The first chapter is the introduction and prepare knowledge. It is given the research status about the maximum of random walk and the innovation of this article, and introduces the random walk with Δ-subexponential distribution and the basic knowledge of random walk.The second chapter, according to the study and research about the random walk’s bound,we will mainly discuss the lower bound of maximum of the random walk with Δ-subexponential distribution.The third chapter uses methods of the literature researching renewal measure and the step height, which will research the asymptotic behavior of maximum of the random walk withΔ-subexponential distribution.The fourth chapter, according to the asymptotic behavior of the heavy-tailed random walk’s tail, some results will be generalized to Δ-subexponential distribution family. It will mainly study the asymptotic behavior of the random walk’s tail with Δ-subexponential distribution, and is given for an important equivalent condition about it.