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Lower bounds for moments of global scores of pairwise Markov chains

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  • Lember, Jüri
  • Matzinger, Heinrich
  • Sova, Joonas
  • Zucca, Fabio

Abstract

Let X1,… and Y1,… be random sequences taking values in a finite set A. We consider a similarity score Ln≔L(X1,…,Xn;Y1,…,Yn) that measures the homology of words (X1,…,Xn) and (Y1,…,Yn). A typical example is the length of the longest common subsequence. We study the order of moment E|Ln−ELn|r in the case where the two-dimensional process (X1,Y1),(X2,Y2),… is a Markov chain on A×A. This general model involves independent Markov chains, hidden Markov models, Markov switching models and many more. Our main result establishes a condition that guarantees that E|Ln−ELn|r≍nr2. We also perform simulations indicating the validity of the condition.

Suggested Citation

  • Lember, Jüri & Matzinger, Heinrich & Sova, Joonas & Zucca, Fabio, 2018. "Lower bounds for moments of global scores of pairwise Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 128(5), pages 1678-1710.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:5:p:1678-1710
    DOI: 10.1016/j.spa.2017.08.009
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    References listed on IDEAS

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    1. Glynn, Peter W. & Ormoneit, Dirk, 2002. "Hoeffding's inequality for uniformly ergodic Markov chains," Statistics & Probability Letters, Elsevier, vol. 56(2), pages 143-146, January.
    2. Derrode, Stéphane & Pieczynski, Wojciech, 2013. "Unsupervised data classification using pairwise Markov chains with automatic copulas selection," Computational Statistics & Data Analysis, Elsevier, vol. 63(C), pages 81-98.
    3. Hilary S. Booth & Shevarl F. MacNamara & Ole M. Nielsen & Susan R. Wilson, 2004. "An Iterative Approach to Determining the Length of the Longest Common Subsequence of Two Strings," Methodology and Computing in Applied Probability, Springer, vol. 6(4), pages 401-421, December.
    4. Stanislaw Barder & Jüri Lember & Heinrich Matzinger & Märt Toots, 2012. "On Suboptimal LCS-alignments for Independent Bernoulli Sequences with Asymmetric Distributions," Methodology and Computing in Applied Probability, Springer, vol. 14(2), pages 357-382, June.
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    Cited by:

    1. Kristi Kuljus & Jüri Lember, 2023. "Pairwise Markov Models and Hybrid Segmentation Approach," Methodology and Computing in Applied Probability, Springer, vol. 25(2), pages 1-32, June.
    2. Lember, Jüri & Sova, Joonas, 2020. "Existence of infinite Viterbi path for pairwise Markov models," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1388-1425.

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