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Improvements on the distribution of maximal segmental scores in a Markovian sequence

Part of: Markov processes Special processes Limit theorems Stochastic processes

Published online by Cambridge University Press:  04 May 2020

S. Grusea  and
S. Mercier
S. Grusea*
Affiliation:
Institut de Mathématiques de Toulouse
S. Mercier*
Affiliation:
Institut de Mathématiques de Toulouse
*
*Postal address: Institut National des Sciences Appliquées, 135 avenue de Rangueil, 31400, Toulouse, France.
***Postal address: Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse 2 Jean Jaurès, 5 allées Antonio Machado, 31058, Toulouse, Cedex 09, France.
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Abstract

Let $(A_i)_{i \geq 0}$ be a finite-state irreducible aperiodic Markov chain and f a lattice score function such that the average score is negative and positive scores are possible. Define $S_0\coloneqq 0$ and $S_k\coloneqq \sum_{i=1}^k f(A_i)$ the successive partial sums, $S^+$ the maximal non-negative partial sum, $Q_1$ the maximal segmental score of the first excursion above 0, and $M_n\coloneqq \max_{0\leq k\leq\ell\leq n} (S_{\ell}-S_k)$ the local score, first defined by Karlin and Altschul (1990). We establish recursive formulae for the exact distribution of $S^+$ and derive a new approximation for the tail behaviour of $Q_1$ , together with an asymptotic equivalence for the distribution of $M_n$ . Computational methods are explicitly presented in a simple application case. The new approximations are compared with those proposed by Karlin and Dembo (1992) in order to evaluate improvements, both in the simple application case and on the real data examples considered by Karlin and Altschul (1990).

Keywords

local scoreMarkov theorylimit theoremsmaximal segmental score

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60F05: Central limit and other weak theorems 60G70: Extreme value theory; extremal processes
Secondary: 60F10: Large deviations 60K15: Markov renewal processes, semi-Markov processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 1 , March 2020 , pp. 29 - 52
Copyright
© Applied Probability Trust 2020

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