Asymptotic Expansions for the Moments of the Semi-Markovian Random Walk with Gamma Distributed Interference of Chance

Abstract

In this study, a semi-Markovian random walk with a discrete interference of chance (X t ) is considered and under some weak assumptions the ergodicity of this process is discussed. The exact formulas for the first four moments of the ergodic distribution of the process X t are obtained when the random variable 1 , which describes a discrete interference of chance, has a gamma distribution with parameters > 1 > 0. Based on these results, the asymptotic expansions are obtained for the first four moments of the ergodic distribution of the process X t , as → 0. Furthermore, the asymptotic expansions for the skewness and kurtosis of the ergodic distribution of the process X t are established. Finally, it is discussed that the alternative estimations for the stationary characteristics of this process can be offered by using obtained asymptotic expansions.

Weak convergence theorem for ergodic distribution of stochastic process with a discrete interference of change and generalized reflecting barrier

“…Мы называем процесс X(t) случайным процессом с дискретным вмешательством случая и обобщенным отражающим барьером. Похожие процессы изучались, например, в [1], [2], [7], [8].…”

Inventory model of type $(s,S)$ under heavy tailed demand with infinite variance

“…One of the most popular methods to obtain useful formulas for mentioned characteristics is using asymptotic expansion method. In recent years, many researchers have extensively studied the characteristics of an inventory model type (s,S) by using asymptotic approach (see Smith (1959), Feller (1971), Khaniyev and Aksop (2011), Aliyev and Khaniyev (2014), Kesemen et. al.…”

Limit Theorem for a Semi - Markovian Stochastic Model of Type (s,S)

“…A number of very interesting problems arising in the theories of inventory, stock control, queuing theory, reliability, mathematical insurance, stochastic finance etc., can be expressed by means of renewal processes, renewal-reward processes, random walk processes and their modifications. There are many important theoretical results about these subjects in literature (Borovkov (1984), Brown and Solomon (1975), Feller (1971), Gihman and Skorohod (1975), Janssen and Leuwaarden (2007), Khaniyev et al (2008Khaniyev et al ( , 2013, Lotov (1996), Rogozin (1964), etc.). The results of these studies have complex mathematical structures and they are not useful for applied problems.…”