Geodesic
On ergodic properties of homogeneous Markov chains
Vladikavkazskij matematičeskij žurnal, Tome 14 (2012) no. 1, pp. 37-46 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper we continue our investigations initiated in [1]. Namely, we study the spectrum of Kolmogorov matrices with at least one column separated from zero. It is shown that $\lambda=0$ is an eigenvalue with multiplicity 1, while the rest of the spectrum is separated from zero. Therefore, a Markov process generated by such a matrix converges to its uniquely defined final distribution exponentially fast. We give an explicit estimate for the rate of this convergence. 
@article{VMJ_2012_14_1_a3,
     author = {E. V. Golovneva},
     title = {On ergodic properties of homogeneous {Markov} chains},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {37--46},
     year = {2012},
     volume = {14},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2012_14_1_a3/}
}
TY  - JOUR
AU  - E. V. Golovneva
TI  - On ergodic properties of homogeneous Markov chains
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2012
SP  - 37
EP  - 46
VL  - 14
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VMJ_2012_14_1_a3/
LA  - ru
ID  - VMJ_2012_14_1_a3
ER  - 
%0 Journal Article
%A E. V. Golovneva
%T On ergodic properties of homogeneous Markov chains
%J Vladikavkazskij matematičeskij žurnal
%D 2012
%P 37-46
%V 14
%N 1
%U http://geodesic.mathdoc.fr/item/VMJ_2012_14_1_a3/
%G ru
%F VMJ_2012_14_1_a3
E. V. Golovneva. On ergodic properties of homogeneous Markov chains. Vladikavkazskij matematičeskij žurnal, Tome 14 (2012) no. 1, pp. 37-46. http://geodesic.mathdoc.fr/item/VMJ_2012_14_1_a3/

[1] Vorobeva E. V., “O nekotorykh ergodicheskikh svoistvakh odnorodnoi markovskoi tsepi s nepreryvnym parametrom”, Vestn. Syktyvkarskogo un-ta. Cer. 1, 2006, no. 6, 39–44 | MR

[2] Perron O., “Zur Theorie der Matrizen”, Math. Ann., 64 (1907), 248–263 | DOI | MR | Zbl

[3] Frobenius G., “Uber Matrizen aus nichtnegativen Elementen”, Akad. Wiss. Berlin, 1912, 456–477 | Zbl

[4] Bulinskii A. V., Shiryaev A. N., Teoriya sluchainykh protsessov, Fizmatlit, M., 2003, 399 pp.

[5] Khille E., Fillips R. C., Funktsionalnyi analiz i polugruppy, IL, M., 1962, 830 pp.

[6] Klement F., Kheimans Kh. i dr., Odnoparametricheskie polugruppy, Mir, M., 1992, 352 pp. | MR

[7] Goldschtein J. A., Semigroups of linear operators and applications, Oxford Math. Monogr., Oxford University Press, New York, 1985, 256 pp. | MR

[8] Lankaster P., Teoriya matrits, Nauka, M., 1982, 269 pp. | MR

[9] Gerschgorin S. A., “Uber die Abgrenzung der Eigenwerte einer Matrix”, Izv. Akademii nauk SSSR. Ser. 7. Otdelenie matematicheskikh i estestvennykh nauk, 1931, no. 6, 749–754 (German) | Zbl

[10] Kolotilina L. Yu., “Mnozhestva, soderzhaschie singulyarnyi spektr kvadratnoi matritsy”, Chislennye metody i voprosy organizatsii vychislenii. XXI, Zap. nauchn. sem. POMI, 359, SPb., 2008, 52–77 | MR | Zbl

[11] Kolotina L. Yu., “Problema vyrozhdennosti/nevyrozhdennosti dlya matrits, udovletvoryayuschikh usloviyam diagonalnogo preobladaniya, formuliruemym v terminakh orientirovannykh grafov”, Chislennye metody i voprosy organizatsii vychislenii. XVII, Zap. nauchn. sem. POMI, 309, SPb., 2004, 40–83 | MR | Zbl

[12] Kantorovich L. V., Vulikh B. 3., Pinsker A. G., Funktsionalnyi analiz v poluuporyadochennykh prostranstvakh, Gostekhizdat, M.–L., 1950, 550 pp. | Zbl