Quenched large deviation principle for words in a letter sequence

• Published in: Probability theory and related fields Vol. 148; no. 3-4; pp. 403 - 456
• Main Authors: Birkner, Matthias, Greven, Andreas, den Hollander, Frank
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.11.2010; Springer; Springer Nature B.V
• Subjects: Algebra; Annealing; Deviation; Economics; Entropy; Exact sciences and technology; Finance; General topics; Insurance; Law; Limit theorems; Management; Markov processes; Mathematical analysis; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Origins; Probability; Probability and statistics; Probability Theory and Stochastic Processes; Quantitative Finance; Quenching; Quenching (cooling); Random walk; Renewals; Sciences and techniques of general use; Statistics for Business; Stochastic processes; Stochastic systems; Studies; Theoretical; Annealed vs. quenched; Specific relative entropy; Empirical process; Large deviation principle; 60G10; 60F10; Letters and words; Renewal process; Rate function; Random walk; Probability theory; Stochastic system; Local time; Large deviation; Transients; Letter; Relative entropy; Exponential tail
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-009-0235-5

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.