Large deviation principles for words drawn from correlated letter sequences

Frank den Hollander (Leiden Universiteit)
Julien Poisat (Leiden Universiteit)

Abstract


When an i.i.d. sequence of letters is cut into words according to i.i.d. renewal times, an i.i.d.\ sequence of words is obtained. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In Birkner, Greven and den Hollander, the quenched LDP (= conditional on a typical letter sequence) was derived for the case where the renewal times have an algebraic tail. The rate function turned out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. The purpose of the present paper is to extend both LDP's to letter sequences that are not i.i.d. It is shown that both LDP's carry over when the letter sequence satisfies a mixing condition called summable variation. The rate functions are again given by specific relative entropies w.r.t. the reference law of words, respectively, letters. But since neither of these reference laws is i.i.d., several approximation arguments are needed to obtain the extension.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-16

Publication Date: March 1, 2014

DOI: 10.1214/ECP.v19-2681

References

  • Berbee, Henry. Chains with infinite connections: uniqueness and Markov representation. Probab. Theory Related Fields 76 (1987), no. 2, 243--253. MR0906777
  • Birkner, Matthias. Conditional large deviations for a sequence of words. Stochastic Process. Appl. 118 (2008), no. 5, 703--729. MR2411517
  • Birkner, Matthias; Greven, Andreas; den Hollander, Frank. Quenched large deviation principle for words in a letter sequence. Probab. Theory Related Fields 148 (2010), no. 3-4, 403--456. MR2678894
  • Dembo, Amir; Zeitouni, Ofer. Large deviations techniques and applications. Second edition. Applications of Mathematics (New York), 38. Springer-Verlag, New York, 1998. xvi+396 pp. ISBN: 0-387-98406-2 MR1619036
  • Deuschel, Jean-Dominique; Stroock, Daniel W. Large deviations. Pure and Applied Mathematics, 137. Academic Press, Inc., Boston, MA, 1989. xiv+307 pp. ISBN: 0-12-213150-9 MR0997938
  • Donsker, M. D.; Varadhan, S. R. S. Asymptotic evaluation of certain Markov process expectations for large time. IV. Comm. Pure Appl. Math. 36 (1983), no. 2, 183--212. MR0690656
  • Donsker, M. D.; Varadhan, S. R. S. Large deviations for stationary Gaussian processes. Comm. Math. Phys. 97 (1985), no. 1-2, 187--210. MR0782966
  • Halmos, Paul R. Lectures on ergodic theory. Chelsea Publishing Co., New York 1960 vii+101 pp. MR0111817
  • den Hollander, Frank; Steif, Jeffrey E. Mixing properties of the generalized $T,T^ {-1}$-process. J. Anal. Math. 72 (1997), 165--202. MR1482994
  • Johansson, Anders; Öberg, Anders. Square summability of variations of $g$-functions and uniqueness of $g$-measures. Math. Res. Lett. 10 (2003), no. 5-6, 587--601. MR2024717
  • Johansson, Anders; Öberg, Anders; Pollicott, Mark. Countable state shifts and uniqueness of $g$-measures. Amer. J. Math. 129 (2007), no. 6, 1501--1511. MR2369887
  • Ledrappier, F. Sur la condition de Bernoulli faible et ses applications. (French) Théorie ergodique (Actes Journées Ergodiques, Rennes, 1973/1974), pp. 152--159. Lecture Notes in Math., Vol. 532, Springer, Berlin, 1976. MR0507548
  • Orey, Steven. Large deviations in ergodic theory. Seminar on stochastic processes, 1984 (Evanston, Ill., 1984), 195--249, Progr. Probab. Statist., 9, Birkhäuser Boston, Boston, MA, 1986. MR0896730
  • Orey, Steven; Pelikan, Stephan. Large deviation principles for stationary processes. Ann. Probab. 16 (1988), no. 4, 1481--1495. MR0958198
  • K.S. Parthasarathy, Probability Measures on Metric Spaces, Vol. 352, American Mathematical Society, 1967.


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.