Large deviation asymptotics and control variates for simulating large functions

Annals of Applied Probability

Volumn 16, Issue 1, 2006, Pages 310-339

  Meyn, Sean P ^a,b  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

^b University of Illinois at Urbana Champaign   (United States)

Author keywords

Computational methods in Markov chains; Ergodic theorems; General methods of simulation; Large deviations; Markov operators; Monte Carlo methods; Spectral theory

Indexed keywords

EID: 33645689714     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/105051605000000737     Document Type: Article

References (49)

1. ASMUSSEN, S. (1987). Applied Probability and Queues. Wiley, New York. MR889893
2. BAHADUR, R. R. and RANGA RAO, R. (1960). On deviations of the sample mean. Ann. Math. Statist. 31 1015-1027. MR117775
3. BALAJI, S. and MEYN, S. P. (2000). Multiplicative ergodicity and large deviations for an irreducible Markov chain. Stoch. Process. Appl. 90 123-144. MR1787128
4. CHAGANTY, N. R. and SETHURAMAN, J. (1993). Strong large deviation and local limit theorems. Ann. Probab. 21 1671-1690. MR1235434
5. CHEN, M., PANDIT, C. and MEYN, S. P. (2003). In search of sensitivity in network optimization. Queueing Systems Theory Appl. 44 313-363. MR2001506
6. DAI, J. G. and MEYN, S. P. (1995). Stability and convergence of moments for multiclass queueing networks via fluid limit models. IEEE Trans. Automat. Control 40 1889-1904. MR1358006
7. DAI, J. G. and WEISS, G. (2002). A fluid heuristic for minimizing makespan in job shops. Oper. Res. 50 692-707. MR1918551
8. DE ACOSTA, A. (1990). Large deviations for empirical measures of Markov chains. J. Theoret. Probab. 3 395-431. MR1057523
9. DE ACOSTA, A. and NEY, P. (1998). Large deviation lower bounds for arbitrary additive functionals of a Markov chain. Ann. Probab. 26 1660-1682. MR1675055
10. DELLNITZ, M. and JUNGE, O. (1999). On the approximation of complicated dynamical behavior. SIAM J. Numer. Anal. 36 491-515. MR1668207
11. DEMBO, A. and ZEITOUNI, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York. MR1619036
12. DEUSCHEL, J. D. and STROOCK, D. W. (1989). Large Deviations. Academic Press, Boston. MR997938
13. DONSKER, M. D. and VARADHAN, S. R. S. (1975). Asymptotic evaluation of certain Markov process expectations for large time. I. II. Comm. Pure Appl. Math. 28 1-47; 279-301. MR386024
14. DONSKER, M. D. and VARADHAN, S. R. S. (1976). Asymptotic evaluation of certain Markov process expectations for large time. III. Comm. Pure Appl. Math. 29 389-461. MR428471
15. DONSKER, M. D. and VARADHAN, S. R. S. (1983). Asymptotic evaluation of certain Markov process expectations for large time. IV. Comm. Pure Appl. Math. 36 183-212. MR690656
16. EICHELSBACHER, P. and SCHMOCK, U. (2001). Large deviations for products of empirical measures of dependent sequences. Markov Process. Related Fields 7 435-468. MR1856491
17. GLASSERMAN, P. (2004). Monte Carlo Methods in Financial Engineering. Springer, New York. MR1999614
18. GLYNN, P. and SZECHTMAN, R. (2002). Some new perspectives on the method of control variates. In Monte Carlo and Quasi-Monte Carlo Methods (K. T. Fang, F. J. Hickernell and H. Niederreiter, eds.) 27-49. Springer, Berlin. MR1958845
19. GLYNN, P. W. (1994). Some topics in regenerative steady-state simulation. Acta Appl. Math. 34 225-236. MR1273856
20. GLYNN, P. W. and MEYN, S. P. (1996). A Liapounov bound for solutions of the Poisson equation. Ann. Probab. 24 916-931. MR1404536
21. GLYNN, P. W. and ORMONEIT, D. (2002). Hoeffding's inequality for uniformly ergodic Markov chains. Statist. Probab. Lett. 56 143-146. MR1881167
22. GLYNN, P. W. and WHITT, W. (2002). Necessary conditions for limit theorems in cumulative processes. Stoch. Process. Appl. 98 199-209. MR1887533
23. HÄGGSTRÖM, O. (2005). On the central limit theorem for geometrically ergodic Markov chains. Probab. Theory Related Fields 132 74-82. MR2136867
24. HENDERSON, S. G. (1997). Variance reduction via an approximating Markov process. Ph.D. thesis, Stanford Univ.
25. HENDERSON, S. G. and GLYNN, P. W. (2002). Approximating martingales for variance reduction in Markov process simulation. Math. Oper. Res. 27 253-271. MR1908526
26. HENDERSON, S. G., MEYN, S. P. and TADIĆ, V. B. (2003). Performance evaluation and policy selection in multiclass networks. Discrete Event Dynamic Systems Theory Appl. 13 149-189. MR1972053
27. HENDERSON, S. G. and MEYN, S. P. (2005). Variance reduction for simulation in multiclass queueing networks. IIE Trans. Oper. Engineering. To appear.
28. HORDIJK, A. and SPIEKSMA, F. (1990). On ergodicity and recurrence properties of a Markov chain with an application to an open Jackson network. Adv. in Appl. Probab. 24 343-376.
29. JAIN, N. C. (1990). Large deviation lower bounds for additive functional of Markov processes. Ann. Probab. 18 1071-1098. MR1062059
30. 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.
31. 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.
32. KONTOYIANNIS, I., LASTRAS-MONTAÑO, L. A. and MEYN, S. P. (2005). Relative entropy and exponential deviation bounds for general Markov chains. In Proceedings of the IEEE International Symposium on Information Theory, Adelaide Convention Centre, Adelaide, Australia, 4-9 September.
33. KONTOYIANNIS, I. and MEYN, S. P. (2003). Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab. 13 304-362. MR1952001
34. KONTOYIANNIS, I. and MEYN, S. P. (2005). Large deviations asymptotics and the spectral theory of multiplicatively regular Markov processes. Electron. J. Probab. 10 61-123 (electronic). MR2120240
35. KUMAR, P. R. and MEYN, S. P. (1996). Duality and linear programs for stability and performance analysis queueing networks and scheduling policies. IEEE Trans. Automat. Control 41 4-17. MR1312612
36. LAW, A. M. and KELTON, W. D. (2000). Simulation Modeling and Analysis, 3rd ed. McGraw-Hill, New York. MR630193
37. MEYN, S. P. (2005). Workload models for stochastic networks: Value functions and performance evaluation. IEEE Trans. Automat. Control 50 1106-1122. MR2156040
38. MEYN, S. P. and TWEEDIE, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London. Available at http://black.csl.uiuc.edu/~meyn/pages/ book.html. MR1287609
39. MEYN, S. P. and TWEEDIE, R. L. (1994). Computable bounds for convergence rates of Markov chains. Ann. Appl. Probab. 4 981-1011. MR1304770
40. NELSON, B. L. (1990). Control-variate remedies. Oper. Res. 38 974-992. MR1095954
41. NEY, P. and NUMMELIN, E. (1987). Markov additive processes. I. Eigenvalue properties and limit theorems. Ann. Probab. 15 561-592. MR885131
42. NEY, P. and NUMMELIN, E. (1987). Markov additive processes II. Large deviations. Ann. Probab. 15 593-609. MR885132
43. NUMMELIN, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge Univ. Press. MR776608
44. PANDIT, C. and MEYN, S. P. (2005). Worst-case large-deviations with application to queueing and information theory. Stoch. Process. Appl. To appear.
45. REY-BELLET, L. and THOMAS, L. E. (2002). Fluctuations of the entropy production in anharmonic chains. Ann. H. Poincaré 3 483-502. MR1915300
46. VARADHAN, S. R. S. (1984). Large Deviations and Applications. SIAM, Philadelphia. MR758258
47. VARADHAN, S. R. S. (1985). Large deviations and applications. Exp. Math. 3 251-272. MR861018
48. WU, L. (1991). Grandes déviations pour les processus de Markov essentiellement irréductibles. I. Temps discret. C. R. Acad. Sci. Paris Sér. I Math. 312 609-614. MR1101043
49. WU, L. (2004). Essential spectral radius for Markov semigroups. I. Discrete time case. Probab. Theory Related Fields 128 255-321. MR2031227