Rate of Weak Convergence of Random Walk with a Generalized Reflecting Barrier

Yıl 2025, Erken Görünüm, 1 - 1

Başak Gever , Tahir Khanıyev

https://doi.org/10.35378/gujs.1383377

Öz

In this study, a random walk process with generalized reflecting barrier is considered and an inequality for rate of weak convergence of the stationary distribution of the process of interest is propounded. Though the rate of convergence is not thoroughly examined, the literature does provide a weak convergence theorem under certain conditions for the stationary distribution of the process under consideration. Nonetheless, one of the most crucial issues in probability theory is the convergence rate in limit theorems, as it affects the precision and effectiveness of using these theorems in practice. Therefore, for the rate of convergence for the examined process, comparatively simple inequality is represented. The obtained inequality demonstrates that the rate of convergence is correlated with the tail of the distribution of ladder heights of the random walk.

Anahtar Kelimeler

Inequality for Asymptotic Rate, Weak Convergence, Random Walk, Reflecting Barrier

Etik Beyan

No conflict of interest was declared by the authors.

Teşekkür

Dedicated to the memory of Professor A. V. Skorohod.

Kaynakça

• [1] Feller, W. “An Introduction to Probability Theory and Its Applications II”, New York: John Wiley, (1971).
• [2] Aliyev, R., Khaniyev, T., and Gever, B., “Weak convergence theorem for ergodic distribution of stochastic processes with discrete interference of chance and generalized reflecting barrier”, Theory of Probability and Application, 60(3): 246–258, (2016).
• [3] Hanalioglu, Z., Khaniyev, T., “Asymptotic Results for an Inventory Model of Type (s, S) with Asymmetric Triangular Distributed Interference of Chance and Delay”, Gazi University Journal of Science, 31(1): 174-187, (2018).
• [4] Khaniev, T. A., Unver, I., and Maden, S., “On the semi-Markovian random walk with two reflecting barriers”, Stochastic Analysis and Applications, 19(5): 799–819, (2001).
• [5] Kobayashi, M., Miyazawa, M., “Tail asymptotics of the stationary distribution of a two-dimensional reflecting random walk with unbounded upward jumps”, Advances in Applied Probability, 46(2): 365–399, (2014).
• [6] Miyazawa, M., Zwart, B., “Wiener-Hopf factorizations for a multidimensional Markov additive process and their applications to reflected processes”, Stochastic Systems, 2: 67–114, (2012).
• [7] Aliyev, R. T., Khaniyev, T. A., “On the rate of convergence of the asymptotic expansion for the ergodic distribution of a semi – Markov (s,S) inventory model”, Cybernetics and System Analysis, 48(1): 117 – 121, (2012).
• [8] Anisimov, V. “Switching processes in queueing models”, New York:John Wiley & Sons, (2013).
• [9] Gihman, I. I., Skorohod, A. V. “Theory of Stochastic Processes II”. Berlin: Springer –Verlag, (1975).
• [10] Gokpinar, F., Khaniyev, T., and Mammadova, Z., “The weak convergence theorem for the distribution of the maximum of a Gaussian random walk and approximation formulas for its moments”, Methodology and Computing in Applied Probability, 15(2): 333–347, (2013).
• [11] Hanalioglu, Z., Khaniyev, T., and Agakishiyev, I., “Weak convergence theorem for the ergodic distribution of a random walk with normal distributed interference of chance”, TWMS Journal of Applied and Engineering Mathematics, 5(1): 61–73, (2015).
• [12] Kesemen, T., Aliyev, R., and Khaniyev, T., “Limit distribution for semi-Markovian random walk with Weibull distributed interference of chance”, Journal of Inequalities and Applications, 134(1): 1–8, (2013).
• [13] Khaniyev, T., Ardic Sevinc, O., “Limit Theorem for a Semi-Markovian Random Walk with General Interference of Chance”, Sains Malaysiana, 49(4): 919–928, (2020).
• [14] Hanalioglu, T., Aksop, C., “A sharp bound for the ergodic distribution of an inventory control model under the assumption that demands and inter-arrival times are dependent”, Journal of Inequalities and Applications, 2014(75): 1–10, (2014).
• [15] Lukac, E. “Characteristic Function”, London, Griffin, (1970).
• [16] Rogozin, B. A., “On the distribution of the first jump”, Theory of Probability and Applications, 9: 450–465, (1964).