Summary
Letμ 1 andμ 2 be Borel probability measures on ℝd with finite moment generating functions. The main theorem in this paper proves the large deviation principle for a random walk whose transition mechanism is governed byμ 1 when the walk is in the left halfspace Λ1 = {x∈ℝd :x 1≦0} and whose transition mechanism is governed byμ 2 when the walk is in the right halfspace Λ2 = {x∈ℝd :x 1>0}. When the measuresμ 1 andμ 2 are equal, the main theorem reduces to Cramér's Theorem.
Article PDF
Similar content being viewed by others
References
Azencott, R., Ruget, G.: Mélanges d'équations differentielles et grand écart à la loi des grandes nombres. Z. Wahrscheinlichkeitstheor. Verw. Geb.38, 1–54 (1977)
Bolthausen, E.: Markov process large deviations in τ-topology. Stochastic Processes Appl.25, 95–108 (1987)
Cramér, H.: Sur un nouveau théorème-limite de la théorie des probabilités. Colloque consacré à la théorie des probabilités, vol. 3. Paris: Hermann. Actual. Sci. Ind.736, 2–23 (1938)
de Acosta, A.: Upper bounds for large deviations of dependent random vectors. Z. Wahhrscheinlichkeitstheor. Verw. Geb.69, 551–565 (1985)
de Acosta, A.: Large deviations for vector-valued functionals of a Markov chain: lower bounds. Ann. Probab.16, 925–960 (1988)
de Acosta, A.: Large deviations for empirical measures of Markov chains. J. Theoret. Probab.3, (1990) 395–431
Deuschel, J.-D., Stroock, D.W.: Large deviations. New York: Academic Press 1989
Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time, I. Commun. Pure Appl. Math.28, 1–47 (1975)
Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time, II. Commun. Pure Appl. Math.28, 279–301 (1975)
Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time, III. Commun. Pure Appl. Math.29, 389–461 (1976)
Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time, IV. Commun. Pure Appl. Math.36, 183–212 (1983)
Dupuis, P., Ellis, R.S., Weiss, A.: Large deviations for Markov processes with discontinuous statistics, I: General upper bounds. 1989 Ann. Probab.9 1280–1297 (1991)
Dupuis, P., Ishii, H., Soner, H.M.: A viscosity solution approach to the asmptotic analysis of queueing systems. Ann. Probab.18, 226–255 (1990)
Ellis, R.S.: Large deviations for a general class of random vectors. Ann. Probab.12, 1–12 (1984)
Ellis, R.S.: Entropy, large deviations and statistical mechnics. Berlin Heidelberg New York: Springer 1985
Ellis, R.S.: Large deviations for the empirical measure of a Markov chain with an application to the multivariate empirical measure. Ann. Probab.16, 1496–1508 (1988)
Ellis, R.S., Wyner, A.D.: Uniform large deviation property of the empirical process of a Markov chain. Ann. Probab.17, 1147–1151 (1989)
Freidlin, M.I., Wentzell, A.D.: Random perturbations of dynamical systems. Berlin Heidelberg New York: Springer 1984
Gärtner, J.: On large deviations from the invariant measure. Theory Probab. Appl.22, 24–39 (1977)
Iscoe, I., Ney, P., Nummelin, E.: Large deviations of uniformly recurrent Markov additive processes. Adv. Appl. Math.6, 373–412 (1985)
Jain, N.: Large deviation lower bounds for additive functionals of Markov processes. Ann. Probab.18, 1071–1098 (1990)
Korostelev, A.P., Leonov, S.L.: Action functional for a diffusion in discontinuous media. Technical report, Institute for Systems Studies, USSR Academy of Sciences, 1991
Korostelev, A.P., Leonov, S.L.: Action functional for diffusion process with discontiuous drift. Technical report, Institute for Systems Studies, USSR Academy of Sciences, 1991. Theory Probab. Appl. to appear
Kushner, H.: Introduction to stochastic control. New York: Holt, Rinehart and Winston 1970
Kushner, H.J.: Approximation and weak convergence methods for random processes with applications to stochastic system theory. Cambridge: MIT Press 1984
Kushner, H.J.: Weak convergence methods and singularly perturbed stochastic control and filtering problems. Boston: Birkhäuser 1990
Parekh, S., Walrand, J: Quick simulation method for excessive backlogs in networks of queues. In: Lions, P.-L., Fleming, W., (eds.) Proceedings of the IMA, 1986
Rockafellar, R.T.: Convex analysis. Princeton: Princeton University 1970
Stroock, D.W.: An introduction to the theory of large deviations. Berlin Heidelberg New York: Springer 1984
Wentzell, A.D.: Rough limit theorems on large deviations for Markov stochastic processes I. Theory Probab. Appl.21, 227–242 (1976)
Wentzell, A.D.: Rough limit theorems on large deviations for Markov stochastic processes, II. Theory Probab. Appl.21, 499–512 (1976)
Wentzell, A.D.: Rough limit theorems on large deviations for Markov stochastic processes, III. Theory Probab. Appl.24, 675–692 (1979)
Wentzell, A.D.: Rough limit theorems on large deviations for Markov stochastic processes, IV. Theory Probab. Appl.27, 215–234 (1982)
Author information
Authors and Affiliations
Additional information
This research was supported in part by a grant from the National Science Foundation (NSF-DMS-8902333)
This research was supported in part by a grant from the National Science Foundation (NSF-DMS-8901138) and in part by a Lady Davis Fellowship while visiting the Faculty of Industrial Engineering and Management at the Technion during the spring semester of 1989
Rights and permissions
About this article
Cite this article
Dupuis, P., Ellis, R.S. Large deviations for Markov processes with discontinuous statistics, II: random walks. Probab. Th. Rel. Fields 91, 153–194 (1992). https://doi.org/10.1007/BF01291423
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01291423