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Stochastic scrabble: large deviations for sequences with scores

Published online by Cambridge University Press:  14 July 2016

Richard Arratia
Affiliation:
University of Southern California
Pricilla Morris
Affiliation:
University of Southern California
Michael S. Waterman
Affiliation:
University of Southern California

Abstract

A derivation of a law of large numbers for the highest-scoring matching subsequence is given. Let Xk, Yk be i.i.d. q=(q(i))iS letters from a finite alphabet S and v=(v(i))iS be a sequence of non-negative real numbers assigned to the letters of S. Using a scoring system similar to that of the game Scrabble, the score of a word w=i1 · ·· im is defined to be V(w)=v(i1) + · ·· + v(im). Let Vn denote the value of the highest-scoring matching contiguous subsequence between X1X2 · ·· Xn and Y1Y2· ·· Yn. In this paper, we show that Vn/K log(n) → 1 a.s. where KK(q,v). The method employed here involves ‘stuttering’ the letters to construct a Markov chain and applying previous results for the length of the longest matching subsequence. An explicit form for β ∊Pr(S), where β (i) denotes the proportion of letter i found in the highest-scoring word, is given. A similar treatment for Markov chains is also included.

Implicit in these results is a large-deviation result for the additive functional, H ≡ Σn < τv(Xn), for a Markov chain stopped at the hitting time τ of some state. We give this large deviation result explicitly, for Markov chains in discrete time and in continuous time.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

Research supported by the System Development Foundation,

the National Science Foundation,

§

the Institute of Mathematics and its Applications and

††

the National Institutes of Health.

References

[1] Arratia, R. and Waterman, M. S. (1985) Critical phenomena in sequence matching. Ann. Prob. 13, 12361249.CrossRefGoogle Scholar
[2] Arratia, R. and Waterman, M. S. (1985) An Erdös–Rényi law with shifts. Adv. Math. 55, 1323.CrossRefGoogle Scholar
[3] Arratia, R., Gordon, L., and Waterman, M. S. (1986) An extreme value theory for sequence matching. Ann . Statist. 14, 971993.CrossRefGoogle Scholar
[4] Erdös, P. and Rényi, A. (1970) On a new law of large numbers. J. Anal. Math. 22, 103111.CrossRefGoogle Scholar
[5] Karlin, S. and Taylor, H. M. (1970) A First Course in Stochastic Processes , 2nd edn. Academic Press, New York.Google Scholar
[6] Rényi, A. (1970) Probability Theory , Akademia Kiado, Budapest.Google Scholar
[7] Waterman, M. S. (1984) General methods of sequence comparison. Bull. Math. Biol. 46, 473500. Reference added in proof CrossRefGoogle Scholar
[8] Arratia, R., Goldstein, L. and Gordon, L. (1988) Two moments suffice for Poisson approximations: the Chen-Stein method. Ann. Prob. To appear.Google Scholar
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