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

Abstract

Abstract In this study, a random walk process (𝑋(𝑡)) with two barriers at 0 and 𝛽>0 levels and triangular distributed interference of chance, arising in buffer stock problem, is investigated. Under certain conditions, the process 𝑋(𝑡) is to be ergodic, and the exact formula is obtained for the characteristic function of the ergodic distribution of the process 𝑋(𝑡). Then, characteristic function of the ergodic distribution of this process is expressed by the characteristics of the boundary functionals 𝑁(𝑧) and 𝑆𝑁(𝑧). Here, 𝑁(𝑧) is the first exit time of random walk {𝑆𝑛 },𝑛≥1, from the interval (−𝑧,𝛽−𝑧),𝑧∈(0,𝛽) and 𝑆𝑁(𝑧)=Σ𝑁(𝑧)𝑖=1 𝜂𝑖. Moreover, using the obtained results, the limit form of the characteristic function of the ergodic distribution of the standardized process 𝑊𝛽(𝑡)≡(𝑋(𝑡)−𝛽2)⁄𝛽2 is found , when 𝛽→∞. Afterwards, asymptotic results are obtained for all moments of the ergodic distributions of the processes 𝑊𝛽(𝑡) and 𝑋(𝑡). Finally, the asymptotic expansions for expected value, variance, standard derivation, skewness and kurtosis coefficients of the processes 𝑊𝛽(𝑡) and 𝑋(𝑡) are obtained, when 𝛽→∞.