Quenched large deviation principle for words in a letter sequence

Greven A, Birkner M, den Hollander F (2010)

Publication Status: Published

Publication Type: Journal article, Original article

Publication year: 2010

Journal

Probability Theory and Related Fields Springer Verlag (Germany)

Publisher: Springer Verlag (Germany)

Book Volume: 148

Pages Range: 403-456

Journal Issue: 3-4

URI: https://link.springer.com/article/10.1007/s00440-009-0235-5

DOI: 10.1007/s00440-009-0235-5

Abstract

When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. 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 the present paper we consider the quenched LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an algebraic tail. The rate function turns 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, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Earlier work by Birkner considered the case where the renewal process has an exponential tail, in which case the rate function turns out to be the first term on the set where the second term vanishes and to be infinite elsewhere. In a companion paper the annealed and the quenched LDP are applied to the collision local time of transient random walks, and the existence of an intermediate phase for a class of interacting stochastic systems is established. © 2009 Springer-Verlag.

Authors with CRIS profile

Andreas Greven Lehrstuhl für Mathematische Stochastik

Involved external institutions

Johannes Gutenberg-Universität Mainz (JGU) DE Germany (DE) Leiden University NL Netherlands (NL)

How to cite

APA:

Greven, A., Birkner, M., & den Hollander, F. (2010). Quenched large deviation principle for words in a letter sequence. Probability Theory and Related Fields, 148(3-4), 403-456. https://doi.org/10.1007/s00440-009-0235-5

MLA:

Greven, Andreas, Matthias Birkner, and Frank den Hollander. "Quenched large deviation principle for words in a letter sequence." Probability Theory and Related Fields 148.3-4 (2010): 403-456.

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