Abstract
When an 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 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.
Highlights
NotationWrite EZ and EZ for the sets of two-sided sequences of letters and words, endowed with the product topology, and let θ and θ denote the left-shifts acting on these sets, respectively
Introduction and main resultsLet E be a finite set of letters and E = ∪ ∈NE the set of finite words drawn from E
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
Summary
Write EZ and EZ for the sets of two-sided sequences of letters and words, endowed with the product topology, and let θ and θ denote the left-shifts acting on these sets, respectively. Let X = (Xk)k∈Z be a two-sided random sequence of letters sampled according to a shift-invariant probability distribution ν on EZ. Let τ = (τi)i∈Z be a two-sided i.i.d. LDP for words drawn from correlated letter sequences sequence of renewal times drawn from a common probability law on N, independent of X. LDP for words drawn from correlated letter sequences sequence of renewal times drawn from a common probability law on N, independent of X The latter form a renewal process T = (Ti)i∈Z given by. Let Y = (Yi)i∈Z be the two-sided random sequence of words cut out from X according to τ , i.e., Yi = X(Ti−1,Ti] =
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