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Error bounds on multivariate Normal approximations for word count statistics

Part of: Markov processes Distribution theory

Published online by Cambridge University Press:  01 July 2016

Haiyan Huang
Haiyan Huang*
Affiliation:
University of Southern California
*
∗ Current address: Department of Biostatistics, Harvard School of Public Health, Boston, MA 02115, USA. Email address: hhuang@hsph.harvard.edu
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Abstract

Given a sequence S and a collection Ω of d words, it is of interest in many applications to characterize the multivariate distribution of the vector of counts U = (N(S,w1), …, N(S,wd)), where N(S,w) is the number of times a word w ∈ Ω appears in the sequence S. We obtain an explicit bound on the error made when approximating the multivariate distribution of U by the normal distribution, when the underlying sequence is i.i.d. or first-order stationary Markov over a finite alphabet. When the limiting covariance matrix of U is nonsingular, the error bounds decay at rate O((log n) / √n) in the i.i.d. case and O((log n)3 / √n) in the Markov case. In order for U to have a nondegenerate covariance matrix, it is necessary and sufficient that the counted word set Ω is not full, that is, that Ω is not the collection of all possible words of some length k over the given finite alphabet. To supply the bounds on the error, we use a version of Stein's method.

Keywords

First-order stationary Markov chainfull word setStein's methodword counts

MSC classification

Primary: 62E17: Approximations to distributions (nonasymptotic) 62E20: Asymptotic distribution theory 60J22: Computational methods in Markov chains 92D20: Protein sequences, DNA sequences
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 3 , September 2002 , pp. 559 - 586
Copyright
Copyright © Applied Probability Trust 2002 

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