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Large deviation principles for words drawn from correlated letter sequences | | Frank den Hollander
; Julien Poisat
; | | Date: |
21 Mar 2013 | | Abstract: | When an i.i.d. 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 emph{annealed}
LDP (large deviation principle) 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 cite{BGdH10}
the emph{quenched} LDP (= conditional on a typical letter sequence) was
derived for the case where the renewal times have an emph{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 i.i.d. It is shown that both LDP’s carry over when the letter
sequence satisfies a mixing condition called emph{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. | | Source: | arXiv, 1303.5383 | | Services: | Forum | Review | PDF | Favorites | |
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