×

zbMATH — the first resource for mathematics

Renewal theory for functionals of a Markov chain with compact state space. (English) Zbl 1048.60065
The main result of the paper is the key renewal theorem for Markov chains \((x_n)_{n\geq 0}\) with compact state space \(S\): Under some conditions, such as existence of stationary distribution \(\pi(dx),\) \(\lim_{n\to+\infty}(v_n/n)=\beta,\) where \((v_n)_{n\geq 0}\) is the corresponding time changed process, the following limit exists: \[ \lim_{t\to+\infty}\sum_{k=0}^{\infty} g(x_k,t-v_k)= \frac{1}{\beta}\int_S\pi(dx) \int_{-\infty}^{+\infty}g(x,t) \,dt,\quad \forall x\in S, \] where \(g\) is continuous bounded function. This is a modification of H. Kesten’s result [Ann. Probab. 2, 355–386 (1974; Zbl 0303.60090)].

MSC:
60K05 Renewal theory
60J05 Discrete-time Markov processes on general state spaces
60K15 Markov renewal processes, semi-Markov processes
60H25 Random operators and equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] Alsmeyer, G. (1994). On the Markov renewal theorem. Stochastic Process. Appl. 50 37–56. · Zbl 0789.60066
[2] Athreya, K. B., McDonald, D. and Ney, P. (1978). Limit theorems for semi-Markov processes and renewal theory for Markov chains. Ann. Probab. 6 788–797. JSTOR: · Zbl 0397.60052
[3] Borkovec, M. and Klüppelberg, C. (2001). The tail of the stationary distribution of an autoregressive process with ARCH(1) errors. Ann. Appl. Probab. 11 1220–1241. · Zbl 1010.62083
[4] Brandt, A., Franken, P. and Lisek, B. (1990). Stationary Stochastic Models. Wiley, Chichester. · Zbl 0723.60112
[5] de Haan, L., Resnick, S. I., Rootzén, H. and de Vries, C. G. (1989). Extremal behavior of solutions to a stochastic difference equation, with applications to ARCH processes. Stochastic Process. Appl. 32 213–224. · Zbl 0679.60029
[6] Doob, J. L. (1953). Stochastic Processes. Wiley, New York. · Zbl 0053.26802
[7] Engle, R. F. (1995). ARCH. Selected Readings. Oxford Univ. Press.
[8] Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2 . Wiley, New York. · Zbl 0219.60003
[9] Fun, C. D. and Lai, T. L. (2001). Asymptotic expansions in multidimensional Markov renewal theory and first passage times for Markov random walks. Adv. in Appl. Probab. 33 652–673. · Zbl 0995.60081
[10] Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 126–166. JSTOR: · Zbl 0724.60076
[11] Goldie, C. M. and Maller, R. (2000). Stability of perpetuities. Ann. Probab. 28 1195–1218. · Zbl 1023.60037
[12] Jacod, J. (1974). Corrections et compléments à l’article “Théorème de renouvellement et classification pour les chaînes semi-markoviennes.” Ann. Inst. H. Poincaré 10 201–209. · Zbl 0306.60058
[13] Kesten, H. (1973). Random difference equations and renewal theory for products of random matrixes. Acta. Math. 131 207–248. · Zbl 0291.60029
[14] Kesten, H. (1974). Renewal theory for functionals of a Markov chain with general state space. Ann. Probab. 2 355–386. · Zbl 0303.60090
[15] Klüppelberg, C. and Pergamenchtchikov, S. (2001). The tail of the stationary distribution of a random coefficient AR\((q)\) process with applications to an ARCH\((q)\) process. Ann. Appl. Probab. · Zbl 1094.62114
[16] LePage, E. (1983). Théorèmes de renouvellement pour les produits de matrixs aléatoires. Equations aux différences aléatoires. Publ. Sém. Math., Univ. Rennes.
[17] Mikosch, T. and \Starica, C. (2000). Limit theory for the sample autocorrelations and extremes of a GARCH\((1,1)\) process. Ann. Statist. 28 1427–1451. · Zbl 1105.62374
[18] Rudin, W. (1964). Principles of Mathematical Analysis , 2nd ed. McGraw-Hill, New York. · Zbl 0148.02903
[19] Shurenkov, V. M. (1984). On the theory of Markov renewal. Theory Probab. Appl. 29 247–265. · Zbl 0557.60078
[20] Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. Appl. Probab. 11 750–783. · Zbl 0417.60073
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.