Sep 21, 2017 term paper 2

FIND THE STATE PROBABILITIES, SINGLE LETTER, DIGRAM, AND TRIGRAM RELATIVE FREQUENCIES, AND THE…

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Find the state probabilities, single letter, digram, and trigram relative frequencies, and the zeroth-, first-, and second-order simulants of the following pfssa.

Notice that for k ≥ 2, a k-gram in the text generated by this pfssa must end in one of ab,ac,bb,bc,ca. For a non-empty word u ∈ {a,b,c}+, let u denote the word obtained by deleting the first letter of u (on the left). Thus, if lgth(u) = 1, then u = λ, the empty word. We will use this notation to talk about kth-order contexts in the text generated by D, and the kth order simulants of D. For instance, for k ≥ 3, a state uab in D(k) (lgth(u) = k −2) has arcs to the states u’ abb and u’ abc.

(a) For k ≥ 3, describe the states in D(k) receiving arcs from the states of the form uab,uac,ubb,ubc, and uca. Give the probability and letter labels on these arcs.

(b) By 7.5.6 and 7.5.7, H(k) (D) = H(k) (D(k) ) = H(∞) (D(k) ). Use this to show that the sequence (H(k) (D))k≥0 is strictly decreasing, and that, therefore, D is not equivalent to any of its simulants


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