Summary
Let (S n ) be a sequence ofR d-valued random variables adapted to the internal history of a stationary sequence of random elements (X n ). We formulate conditions under which the principle of large deviations holds true for the sequence (S n ).
References
Bahadur, R.R.: Some limit theorems in statistics. SIAM, Philadelphia (1971)
Chernoff, H.: A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Stat.23, 493–507 (1952)
Cramér, H.: Sur un nouveaux théorème-limite de la théorie des probabilités Colloquium on the theory of probability. Paris: Hermann 1938
Ellis, R.S.: Large deviations and other limit theorems for a class of dependent random variables with applications to statistical mechanics (preprint, Dept. of Math., Univ. of Massachusetts 1981)
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 mechanics. Berlin Heidelberg New York: Springer 1985
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 process. Adv. in Appl. Math.6, 373–412 (1985)
Martin-Löf, A.: Mixing properties, differentiability of the free energy, and the central limit theorem for a pure phase in the Ising model at low temperature. Commun. Math. Phys.32, 75–92 (1973)
Orey, S.: Large deviations in ergodic theory. In: Seminar on stochastic processes, 1984. Boston: Birkhäuser 1986
Rockafellar, R.T.: Convex analysis. New Jersey: Princeton 1970
Steinebach, J.: Convergence rates of large deviation probabilities in the multidimensional case. Ann. Probab.6, 751–759 (1978)
Varadhan, S.R.S.: Large deviations and applications. SIAM, Philadelphia (1984)
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Nummelin, E. Large deviations for functionals of stationary processes. Probab. Th. Rel. Fields 86, 387–401 (1990). https://doi.org/10.1007/BF01208257
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DOI: https://doi.org/10.1007/BF01208257
Keywords
- Stochastic Process
- Stationary Process
- Probability Theory
- Mathematical Biology
- Stationary Sequence