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Distribution of the number of words with a prescribed frequency and tests of randomness

Part of: Limit theorems Distribution theory Distribution theory - Probability Parametric inference

Published online by Cambridge University Press:  01 July 2016

Andrew L. Rukhin
Andrew L. Rukhin*
Affiliation:
University of Maryland, Baltimore County
*
* Postal address: Department of Mathematics and Statistics, UMBC, Baltimore, MD 21250, USA. Email address: rukhin@math.umbc.edu
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Abstract

The goal of this paper is to investigate properties of statistical procedures based on numbers of different patterns by using generating functions for the probabilities of a prescribed number of occurrences of given patterns in a random text. The asymptotic formulae are derived for the expected value of the number of words occurring a given number of times and for the covariance matrix. The form of the optimal linear test based on these statistics is established. These problems appear in testing for the randomness of a string of binary bits, DNA sequencing, source coding, synchronization, quality control protocols, etc. Indeed, the probabilities of repeated (overlapping) patterns are important in information theory (the second-order properties of relative frequencies) and molecular biology problems (finding patterns with unexpectedly low or high frequencies).

Keywords

Asymptotic normalitycorrelation polynomialdistribution of m-patternsefficacygenerating functionsoptimal linear test

MSC classification

Primary: 60E05: Distributions
Secondary: 60F99: None of the above, but in this section 62E20: Asymptotic distribution theory 62F03: Hypothesis testing
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 4 , December 2002 , pp. 775 - 797
Copyright
Copyright © Applied Probability Trust 2002 

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