BibTex RIS Kaynak Göster
Yıl 2015, Cilt: 5 Sayı: 1, 61 - 73, 01.06.2015

Öz

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.

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

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 → ∞.

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.
Toplam 24 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Research Article
Yazarlar

Z. Hanalioglu Bu kişi benim

T. Khaniyev I. Agakishiyev Bu kişi benim

I. Agakıshıyev Bu kişi benim

Yayımlanma Tarihi 1 Haziran 2015
Yayımlandığı Sayı Yıl 2015 Cilt: 5 Sayı: 1

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