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 ζ₁, which describes a discrete interference of chance, has a gamma distribution with parameters (α, λ), α > 1, λ > 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 λ → 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.

Keywords:

• Asymptotic expansions
• Boundary functional
• Discrete interference of chance
• Ergodic distribution
• Gamma distribution
• Ladder variables
• Moments
• Semi-Markovian random walk

Mathematics Subject Classification:

• Primary 60G50
• Secondary 60K15
• 60F99

Acknowledgments

We would like to express our regards to Professor A.V. Skorohod, Michigan State University, for his support and encouragement, which led us to do the further investigations on the processes with a discrete interference of chance and their applications. Moreover, Associate Professor Rovshan Aliyev would like to express his thanks to TUBITAK for inviting him to Karadeniz Technical University, Faculty of Art and Sciences, Department of Statistics and Computer Sciences (Turkey) and awarding him a “Fellowship for visiting scientist” scholarship. Additionally, we would like to thank the referee, Editor, and Associate Editor for their careful reading, valuable comments, and patience.