TY - JOUR

T1 - Quenched large deviation principle for words in a letter sequence

AU - Birkner, M.

AU - Greven, A.

AU - Hollander, den, W.Th.F.

PY - 2010

Y1 - 2010

N2 - 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.
Keywords: Letters and words - Renewal process - Empirical process - Annealed vs. quenched - Large deviation principle - Rate function - Specific relative entropy.

AB - 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.
Keywords: Letters and words - Renewal process - Empirical process - Annealed vs. quenched - Large deviation principle - Rate function - Specific relative entropy.

U2 - 10.1007/s00440-009-0235-5

DO - 10.1007/s00440-009-0235-5

M3 - Article

SN - 0178-8051

VL - 148

SP - 403

EP - 456

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 3

ER -