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Spectral theory and limit theorems for geometrically ergodic Markov processes. (English) Zbl 1016.60066
Let \((\Phi_t)_{t\geq 0}\) be a time-homogeneous, gometrically ergodic Markov process in continuous or discrete time with a state space \(X\), and let \(F:X\to \mathbb{R}\) be a bounded measurable function. The authors investigate the long-time behavior of \(S_t:= \int^t_0 F(\Phi_s)ds\) in the continuous case and \(S_n:= \sum^n_{i=0} F(\Phi_i)\) in the discrete case. In particular, the solutions of a so-called multiplicative Poisson equation associated with the transition semigroup of \((\Phi_t)_{t\geq 0}\) are studied, and a multiplicative mean ergodic theorem is derived. Moreover, Edgeworth expansions with the leading term of order \(O(t^{-1/2})\) and large deviation results in a neighborhood of the mean (with exact asymptotic results) are given for \(S_t\).

MSC:
60J05 Discrete-time Markov processes on general state spaces
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
60F10 Large deviations
60J25 Continuous-time Markov processes on general state spaces
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[1] BAHADUR, R. R. and RANGA RAO, R. (1960). On deviations of the sample mean. Ann. Math. Statist. 31 1015-1027. · Zbl 0101.12603 · doi:10.1214/aoms/1177705674
[2] BALAJI, S. and MEy N, S. P. (2000). Multiplicative ergodicity and large deviations for an irreducible Markov chain. Stochastic Process. Appl. 90 123-144. · Zbl 1046.60065 · doi:10.1016/S0304-4149(00)00032-6
[3] BEZHAEVA, Z. I. and OSELEDETS, V. I. (1996). On the variance of sums for functions of a stationary Markov process. Teor. Veroy atnost. i Primenen. 41 633-639. · Zbl 0891.60062
[4] BOLTHAUSEN, E., DEUSCHEL, J.-D. and TAMURA, Y. (1995). Laplace approximations for large deviations of nonreversible Markov processes. The nondegenerate case. Ann. Probab. 23 236-267. · Zbl 0838.60023 · doi:10.1214/aop/1176988385
[5] BRy C, W. and DEMBO, A. (1996). Large deviations and strong mixing. Ann. Inst. H. Poincaré Probab. Statist. 32 549-569. · Zbl 0863.60028 · numdam:AIHPB_1996__32_4_549_0 · eudml:77545
[6] BUDHIRAJA, A. and DUPUIS, P. (2001). Large deviations for the empirical measure of reflecting Brownian motion and related constrained processes in R+.
[7] CHAGANTY, N. R. and SETHURAMAN, J. (1993). Strong large deviation and local limit theorems. Ann. Probab. 21 1671-1690. · Zbl 0786.60026 · doi:10.1214/aop/1176989136
[8] DATTA, S. and MCCORMICK, W. P. (1993). On the first-order Edgeworth expansion for a Markov chain. J. Multivariate Anal. 44 345-359. · Zbl 0770.60023 · doi:10.1006/jmva.1993.1020
[9] DE ACOSTA, A. (1990). Large deviations for empirical measures of Markov chains. J. Theoret. Probab. 3 395-431. · Zbl 0711.60023 · doi:10.1007/BF01061260
[10] DE ACOSTA, A. (1997). Moderate deviations for empirical measures of Markov chains: Lower bounds. Ann. Probab. 25 259-284. · Zbl 0877.60019 · doi:10.1214/aop/1024404288
[11] DE ACOSTA, A. and CHEN, X. (1998). Moderate deviations for empirical measures of Markov chains: Upper bounds. J. Theoret. Probab. 11 1075-1110. · Zbl 0924.60051 · doi:10.1023/A:1022673000778
[12] DE ACOSTA, A. and NEY, P. (1998). Large deviation lower bounds for arbitrary additive functionals of a Markov chain. Ann. Probab. 26 1660-1682. · Zbl 0936.60022 · doi:10.1214/aop/1022855877
[13] DEMBO, A. and ZEITOUNI, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York. · Zbl 0896.60013
[14] DEUSCHEL, J. D. and STROOCK, D. W. (1989). Large Deviations. Academic, Boston. · Zbl 0675.60086 · doi:10.1214/aop/1176991495
[15] DOWN, D., MEy N, S. P. and TWEEDIE, R. L. (1995). Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23 1671-1691. · Zbl 0852.60075 · doi:10.1214/aop/1176987798
[16] ELLIS, R. S. (1988). Large deviations for the empirical measure of a Markov chain with an application to the multivariate empirical measure. Ann. Probab. 16 1496-1508. · Zbl 0661.60043 · doi:10.1214/aop/1176991580
[17] FELLER, W. (1971). An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York. · Zbl 0219.60003
[18] FENG, J. (1999). Martingale problems for large deviations of Markov processes. Stochastic Process. Appl. 81 165-212. · Zbl 0962.60008 · doi:10.1016/S0304-4149(98)00104-5
[19] FENG, J. and KURTZ, T. G. (2000). Large deviations for stochastic processes. · Zbl 1113.60002
[20] FLEMING, W. H. (1978). Exit probabilities and optimal stochastic control. Appl. Math. Optim. 4 329-346. · Zbl 0398.93068 · doi:10.1007/BF01442148
[21] FLEMING, W. H. (1997). Some results and problems in risk sensitive stochastic control. Mat. Appl. Comput. 16 99-115. · Zbl 0891.93086 · pub2.lncc.br
[22] FLEMING, W. H. and SHEU, S.-J. (1997). Asy mptotics for the principal eigenvalue and eigenfunction of a nearly first-order operator with large potential. Ann. Probab. 25 1953- 1994. · Zbl 0901.60033 · doi:10.1214/aop/1023481117
[23] GLy NN, P. W. and MEy N, S. P. (1996). A Liapunov bound for solutions of the Poisson equation. Ann. Probab. 24 916-931.
[24] HALL, P. (1982). Rates of Convergence in the Central Limit Theorem. Pitman, Boston. · Zbl 0497.60001
[25] HORDIJK, A. and SPIEKSMA, F. (1992). On ergodicity and recurrence properties of a Markov chain with an application to an open Jackson network. Adv. in Appl. Probab. 24 343-376. JSTOR: · Zbl 0766.60085 · doi:10.2307/1427696 · links.jstor.org
[26] HUANG, J., KONTOy IANNIS, I. and MEy N, S. P. (2002). The ODE method and spectral theory of Markov operators. Stochastic Theory and Control. Lecture Notes in Control and Inform. Sci. 205-221. Springer, New York.
[27] HUISINGA, W., MEy N, S. P. and SCHUETTE, C. (2001). Phase transitions and metastability in Markovian and molecular sy stems.
[28] ISCOE, I., NEY, P. and NUMMELIN, E. (1985). Large deviations of uniformly recurrent Markov additive processes. Adv. in Appl. Math. 6 373-412. · Zbl 0602.60034 · doi:10.1016/0196-8858(85)90017-X
[29] JENSEN, J. L. (1987). A note on asy mptotic expansions for Markov chains using operator theory. Adv. in Appl. Math. 8 377-392. · Zbl 0635.60024 · doi:10.1016/0196-8858(87)90016-9
[30] JENSEN, J. L. (1991). Saddlepoint expansions for sums of Markov dependent variables on a continuous state space. Probab. Theory Related Fields 89 181-199. · Zbl 0723.60019 · doi:10.1007/BF01366905
[31] KARTASHOV, N. V. (1985). Criteria for uniform ergodicity and strong stability of Markov chains with a common phase space. Theory Probab. Appl. 30 71-89. · Zbl 0586.60058
[32] KARTASHOV, N. V. (1985). Inequalities in theorems of ergodicity and stability for Markov chains with a common phase space. Theory Probab. Appl. 30 247-259. · Zbl 0657.60088 · doi:10.1137/1130034
[33] KONTOy IANNIS, I. and MEy N, S. P. (2002). Large deviation asy mptotics and the spectral theory of multiplicatively regular Markov processes.
[34] KUMAR, P. R. and MEy N, S. P. (1996). Duality and linear programs for stability and performance analysis queueing networks and scheduling policies. IEEE Trans. Automat. Control 41 4-17. · Zbl 0845.90053 · doi:10.1109/9.481604
[35] MEy N, S. P. and TWEEDIE, R. L. (1993). Stability of Markovian processes III: Foster- Ly apunov criteria for continuous time processes. Adv. in Appl. Probab. 25 518-548. JSTOR: · Zbl 0781.60053 · doi:10.2307/1427522 · links.jstor.org
[36] MEy N, S. P. and TWEEDIE, R. L. (1993). Generalized resolvents and Harris recurrence of Markov processes. In Doeblin and Modern Probability 227-250. Amer. Math. Soc., Providence, RI. · Zbl 0784.60066
[37] MEy N, S. P. and TWEEDIE, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London. · Zbl 0925.60001
[38] MEy N, S. P. and TWEEDIE, R. L. (1994). Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Probab. 4 981-1011. · Zbl 0812.60059 · doi:10.1214/aoap/1177004900
[39] MILLER, H. D. (1961). A convexivity property in the theory of random variables defined on a finite Markov chain. Ann. Math. Statist. 32 1260-1270. · Zbl 0108.15101 · doi:10.1214/aoms/1177704865
[40] NAGAEV, S. V. (1957). Some limit theorems for stationary Markov chains. Theory Probab. Appl. 2 378-406.
[41] NAGAEV, S. V. (1961). More exact limit theorems for homogeneous Markov chains. Theory Probab. Appl. 6 62-81. · Zbl 0116.10602 · doi:10.1137/1106005
[42] NEY, P. and NUMMELIN, E. (1987). Markov additive processes I. Eigenvalue properties and limit theorems. Ann. Probab. 15 561-592. · Zbl 0625.60027 · doi:10.1214/aop/1176992159
[43] NEY, P. and NUMMELIN, E. (1987). Markov additive processes II. Large deviations. Ann. Probab. 15 593-609. · Zbl 0625.60028 · doi:10.1214/aop/1176992160
[44] NIEMI, S. and NUMMELIN, E. (1986). On nonsingular renewal kernels with an application to a semigroup of transition kernels. Stochastic Process. Appl. 22 177-202. · Zbl 0606.60080 · doi:10.1016/0304-4149(86)90001-3
[45] NUMMELIN, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge Univ. Press. · Zbl 0551.60066
[46] PETROV, V. V. (1995). Limit Theorems of Probability Theory. Clarendon, New York. · Zbl 0826.60001
[47] PINSKY, R. G. (1995). Positive Harmonic Functions and Diffusion. Cambridge Univ. Press. · Zbl 0858.31001
[48] RIESZ, F. and SZ.-NAGY, B. (1955). Functional Analy sis. Ungar, New York.
[49] SCHWERER, E. (1997). A linear programming approach to the steady-state analysis of Markov processes. Ph.D. thesis, Stanford Univ.
[50] SHURENKOV, V. M. (1984). On Markov renewal theory. Teor. Veroy atnost. i Primenen. 29 248-263. · Zbl 0544.60082
[51] STROOCK, D. W. (1984). An Introduction to the Theory of Large Deviations. Springer, New York. · Zbl 0552.60022
[52] VARADHAN, S. R. S. (1984). Large Deviations and Applications. SIAM, Philadelphia. · Zbl 0549.60023
[53] VARADHAN, S. R. S. (1985). Large deviations and applications. Exposition. Math. 3 251-272. · Zbl 0567.60030
[54] WEIS, L. (1984). Approximation by weakly compact operators in L1. Math. Nachr. 118 321- 326. · Zbl 0613.41020 · doi:10.1002/mana.19841190128
[55] WILLIAMS, R. J. (1985). Reflected Brownian motion in a wedge: Semimartingale property. Z. Wahrsch. Verw. Gebiete 69 161-176. · Zbl 0535.60042 · doi:10.1007/BF02450279
[56] WU, L. (2000). Some notes on large deviations of Markov processes. Acta Math. Sinica 16 369-394. · Zbl 0965.60038 · doi:10.1007/PL00011549
[57] BOX F, 182 GEORGE STREET PROVIDENCE, RHODE ISLAND 02912 E-MAIL: yiannis@dam.brown.edu www.dam.brown.edu/people/yiannis/ DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING AND THE COORDINATED SCIENCES LABORATORY UNIVERSITY OF ILLINOIS URL: · www.dam.brown.edu
[58] URBANA, ILLINOIS 61801 E-MAIL: s-mey n@uiuc.edu www.black.csl.uiuc.edu/ mey n/ URL: · www.black.csl.uiuc.edu
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