Weak Convergence Theorem for the Ergodic Distribution of a Random Walk with Normal Distributed Interference of Chance

Yıl 2015, Cilt: 5 Sayı: 1, 61 - 73, 01.06.2015

Z. Hanalioglu T. Khaniyev I. Agakishiyev I. Agakıshıyev

Öz

In this study, a semi-Markovian random walk process X t with a discrete interference of chance is investigated. Here, it is assumed that the ζn, n = 1, 2, 3, ..., which describe the discrete interference of chance are independent and identically distributed random variables having restricted normal distribution with parameters a, σ2 . Under this assumption, the ergodicity of the process X t is proved. Moreover, the exact forms of the ergodic distribution and characteristic function are obtained. Then, weak convergence theorem for the ergodic distribution of the process Wa t ≡ X t /a is proved under additional condition that σ/a → 0 when a → ∞.

Anahtar Kelimeler

Random walk, discrete interference of chance, normal distribution, ergodicdistribution, weak convergence

Kaynakça

• Abramowitz M. and Stegun I.A., (1972), Handbook of Mathematical Functions, National Bureau of Standards, New York.
• Afanasyeva L.G. and Bulinskaya E.V., (1984), Certain results for random walks in strip, Theory of Probability and Its Applications, 29(4), pp. 677-693.
• Aliyev R.T., Khaniyev T.A. and Kesemen T., (2010), Asymptotic expansions for the moments of a semi-Markovian random walk with gamma distributed interference of chance, Communications in Statistics-Theory and Methods, 39, pp. 130-143.
• Aliyev R., Kucuk Z. and Khaniyev T., (2010), Three-term asymptotic expansions for the moments of the random walk with triangular distributed interference of chance, Applied Mathematical Modeling, 34(11), pp. 3599-3607.
• Alsmeyer G., (1991), Some relations between harmonic renewal measure and certain Şrst passage times, Statistics and Probability Letters, 12(1), pp. 19-27.
• Anisimov V.V. and Artalejo J.R., (2001), Analysis of Markov multiserver retrial queues with negative arrivals, Queuing Systems: Theory and Applications, 39(2/3), pp. 157-182.
• Borovkov A. A., (1976), Stochastic Processes in Queuing Theory, Springer - Verlag, Berlin.
• Brown M. and Solomon H.A., (1976), Second-order approximation for the variance of a renewal-reward process, Stochastic Processes and Applications, 3, pp. 301-314.
• Chang J.T., (1992), On moments of the Şrst ladder height of random walks with small drift, Annals of Applied Probability, 2(3), pp. 714-738.
• Chang J.T. and Peres Y., (1997), Ladder heights, Gaussian random walks and the Riemann zeta function, Annals of Probability, 25, pp. 787-802.
• Feller W., (1971), Introduction to Probability Theory and Its Applications II, John Wiley, New York. [12] Gihman I.I. and Skorohod A.V., (1975), Theory of Stochastic Processes II, Springer - Verlag, Berlin.
• Janssen A. J. E. M. and van Leeuwaarden J. S. H., (2007), On Lerch’s transcendent and the Gaussian random walk, Annals of Applied Probability, 17(2), pp. 421-439.
• Janssen A. J. E. M. and van Leeuwaarden J. S. H., (2007), Cumulants of the maximum of the Gaussian random walk, Stochastic Processes and Applications, 117(12), pp. 1928-1959.
• Khaniyev T.A. and Mammadova Z.I., (2006), On the stationary characteristics of the extended model of type (s,S) with Gaussian distribution of summands, Journal of Statistical Computation and Simu- lation, 76(10), pp. 861-874.
• Khaniyev T.A., Aksop C., (2011), Asymptotic results for an inventory model of type (s,S) with a generalized beta interference of chance, TWMSC Journal of Applied and Engineering Mathematics, 1(2), pp. 223-236.
• Khorsunov D., (1997), On distribution tail of the maximum of a random walk, Stochastic Processes and Applications, 72, pp. 97-103.
• Korolyuk V.S. and Borovskikh Yu. V., (1981), Analytical Problems of the Asymptotic Behavior of Probabilistic Distributions, Naukova Dumka, Kiev.
• Lotov V.I., (1996), On some boundary crossing problems for Gaussian random walks, Annals of Probability, 24(4), pp. 2154-2171.
• Nasirova T.I., (1984), Processes of Semi-Markovian Random Walk, Elm, Baku.
• Rogozin B.A., (1964), On the distribution of the Şrst jump, Theory Probability and Its Applications, 9(3), pp. 498-545.
• Siegmund D., (1986), Boundary crossing probabilities and statistical applications, Annals of Statistics, 14, pp. 361-404.
• Siegmund D., (1979), Corrected diffusion approximations in certain random walk problems, Advances in Applied Probability, 11, pp. 701-719.
• Skorohod A.V. and Slobodenyuk N.P., (1970), Limit Theorems for the Random Walks, Naukova Dumka, Kiev.
• Spitzer F., (1964), Principles of Random Walks, Van Nostrand, New York.