On the limiting behavior of the characteristic function of the ergodic distribution of the semi-Markov walk with two boundaries

On asymptotıc behavior of random walk process with two barriers arising in buffer stock problem

“…In the general case, the ergodicity of the processes with a discrete interference of chance is proved by Gikhman and Skorokhod (1975). Taking advantage of this general case, the ergodicity of the process 𝑋(𝑡) has been proven in study Aliyev and Khaniyev (2017). ∎…”

“…We introduce the notation: 𝜑 𝑆 𝑁(𝑧) (𝑢) = 𝐸(𝑒 𝑖𝑢𝑆 𝑁(𝑧) ), 𝑢 ∈ 𝑅, 𝑖 = √−1. The explicit form of the characteristic function 𝜑 𝑆 𝑁(𝑧) (𝑢) of boundary functional 𝑆 𝑁(𝑧) is taken from Aliyev and Khaniyev (2017). Let us give this result in Proposition 3.5 below.…”

“…Note that the characteristics of two-barrier random walk processes are mathematically difficult to find. Although relatively few, there are several interesting studies on two-barrier random walk processes in the literature (Aliyev and Khaniyev 2017;Feller 1971;Khaniev and Kucuk 2004;Khaniev et al 2001, and etc.). In the study of Khaniev at al.…”

“…(2001), a random walk process with two reflecting barriers which models the motion of high-energy particles was discussed and the stationary characteristics of the process were examined. The asymptotic behavior of the characteristic function of the ergodic distribution of the random walk process with two barriers and uniform interference of chance was investigated by Aliyev and Khaniyev (2017).…”

“…In this study, the two-barrier random walk process is investigated. But unlike Aliyev and Khaniyev (2017), in present study it is assumed that the random variables 𝜁 𝑛 , 𝑛 = 1,2, …, which describe a discrete interference of chance, are independent and identically distributed random variables having symmetrical triangular distribution in the interval [0, 𝛽].…”