# Quenched large deviation principle for words in a letter sequence

Matthias Birkner, Andreas Greven, Frank den Hollander

Arxiv ID: 0807.2611•Last updated: 10/5/2022

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
\emph{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 \emph{quenched} LDP, i.e., we condition on a typical letter sequence. We
focus on the case where the renewal process has an \emph{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.
The previous version (arXiv:0807.2611v2) appeared in Probab. Theory Relat.
Fields 148, no. 3/4 (2010), 403--456. Meanwhile, it has turned out that the
original proof of the representation of the rate function is flawed when the
mean word length is infinite. We add an erratum in which we fix the flaw in the
proof. Along the way we derive new representations of the rate function that
are interesting in their own right. A key ingredient in the proof is the
observation that if the rate function in the annealed large deviation principle
is finite at a stationary word process, then the letters in the tail of the
long words in this process are typical.

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