Geodesic
On the Limiting Behavior of the Characteristic Function of the Ergodic Distribution of the Semi-Markov Walk with Two Boundaries
Matematičeskie zametki, Tome 102 (2017) no. 4, pp. 490-502.

Voir la notice de l'article provenant de la source Math-Net.Ru

The semi-Markov walk $(X(t))$ with two boundaries at the levels 0 and $\beta >0$ is considered. The characteristic function of the ergodic distribution of the process $X(t)$ is expressed in terms of the characteristics of the boundary functionals $N(z)$ and $S_{N(z)}$, where $N(z)$ is the first moment of exit of the random walk $\{S_{n}\}$, $n\ge 1$, from the interval $(-z,\beta-z)$, $z\in [0,\beta]$. The limiting behavior of the characteristic function of the ergodic distribution of the process $W_{\beta}(t)=2X(t)/\beta-1$ as $\beta \to \infty$ is studied for the case in which the components of the walk ($\eta_{i}$) have a two-sided exponential distribution.
Keywords: semi-Markov walk, characteristic function of the ergodic distribution of the semi-Markov walk. 
@article{MZM_2017_102_4_a1,
     author = {R. T. Aliev and T. A. Khaniev},
     title = {On the {Limiting} {Behavior} of the {Characteristic} {Function} of the {Ergodic} {Distribution} of the {Semi-Markov} {Walk} with {Two} {Boundaries}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {490--502},
     publisher = {mathdoc},
     volume = {102},
     number = {4},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2017_102_4_a1/}
}
TY  - JOUR
AU  - R. T. Aliev
AU  - T. A. Khaniev
TI  - On the Limiting Behavior of the Characteristic Function of the Ergodic Distribution of the Semi-Markov Walk with Two Boundaries
JO  - Matematičeskie zametki
PY  - 2017
SP  - 490
EP  - 502
VL  - 102
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2017_102_4_a1/
LA  - ru
ID  - MZM_2017_102_4_a1
ER  - 
%0 Journal Article
%A R. T. Aliev
%A T. A. Khaniev
%T On the Limiting Behavior of the Characteristic Function of the Ergodic Distribution of the Semi-Markov Walk with Two Boundaries
%J Matematičeskie zametki
%D 2017
%P 490-502
%V 102
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2017_102_4_a1/
%G ru
%F MZM_2017_102_4_a1
R. T. Aliev; T. A. Khaniev. On the Limiting Behavior of the Characteristic Function of the Ergodic Distribution of the Semi-Markov Walk with Two Boundaries. Matematičeskie zametki, Tome 102 (2017) no. 4, pp. 490-502. http://geodesic.mathdoc.fr/item/MZM_2017_102_4_a1/

[1] A. A. Borovkov, “O bluzhdanii v polose s zaderzhivayuschimi granitsami”, Matem. zametki, 17:4 (1975), 649–657 | MR | Zbl

[2] L. G. Afanaseva, E. V. Bulinskaya, “Nekotorye asimptoticheskie rezultaty dlya sluchainykh bluzhdanii v polose”, Teoriya veroyatn. i ee primen., 29:4 (1984), 654–668 | MR | Zbl

[3] B. A. Rogozin, “O raspredelenii velichiny pervogo pereskoka”, Teoriya veroyatn. i ee primen., 9:3 (1964), 498–515 | MR | Zbl

[4] V. I. Lotov, “On some boundary crossing problems for Gaussian random walks”, Ann. Probab., 24:4 (1996), 2154–2171 | DOI | MR | Zbl

[5] A. J. E. M. Janssen, J. S. H. van Leeuwaarden, “On Lerch's transcendent and the Gaussian random walk”, Ann. Appl. Probab., 17:2 (2007), 421–439 | DOI | MR | Zbl

[6] A. A. Borovkov, “Ob asimptotike raspredelenii vremen pervogo prokhozhdeniya, I”, Matem. zametki, 75:1 (2004), 24–39 | DOI | MR | Zbl

[7] A. A. Borovkov, “Ob asimptotike raspredelenii vremen pervogo prokhozhdeniya II”, Matem. zametki, 75:3 (2004), 350–359 | DOI | MR | Zbl

[8] T. Khaniyev, Z. Kucuk, “Asymptotic expansions for the moments of the Gaussian random walk with two barriers”, Statist. Probab. Lett., 69:1 (2004), 91–103 | DOI | MR | Zbl

[9] T. A. Khaniyev, İ. Unver, S. Maden, “On the semi-Markovian random walk with two reflecting barriers”, Stochastic Anal. Appl., 19:5 (2001), 799–819 | DOI | MR | Zbl

[10] T. Khaniyev, T. Kesemen, R. Aliyev, A. Kokangul, “Asymptotic expansions for the moments of a semi-Markovian random walk with an exponential distributed interference of chance”, Statist. Probab. Lett., 78:6 (2008), 785–793 | DOI | MR | Zbl

[11] V. Feller, Vvedenie v teoriyu veroyatnostei i ee prilozheniya, T. 2, Mir, M., 1984 | MR | Zbl

[12] I. I. Ezhov, V. M. Shurenkov, “Ergodicheskie teoremy, svyazannye s markovskim svoistvom sluchainykh protsessov”, Teoriya veroyatn. i ee primen., 21:3 (1976), 635–639 | MR | Zbl

[13] I. I. Gikhman, A. V. Skorokhod, Teoriya sluchainykh protsessov, T. 2, Nauka, M., 1973 | MR | Zbl