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Central limit theorem for renewal theory for several patterns

M S Tanushev1, R Arratia

  • 1Department of Mathematics, University of Southern California, Los Angeles 90089-1113, USA.

Journal of Computational Biology : a Journal of Computational Molecular Cell Biology
|April 1, 1997
PubMed
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We establish a joint central limit theorem for nonoverlapping word occurrences, modeled as competing renewals in i.i.d. sequences. This provides a formula for limit covariance in DNA sequence analysis.

Area of Science:

  • Probability Theory
  • Stochastic Processes
  • Bioinformatics

Background:

  • The study addresses the statistical analysis of patterns in sequences, drawing motivation from restriction enzyme activity in DNA.
  • Existing work includes central limit theorems for overlapping occurrences (Lundstrom, 1990) and foundational work on overlap-matching polynomials (Guibas & Odlyzko, 1980).

Purpose of the Study:

  • To prove a joint central limit theorem for the counts of nonoverlapping occurrences of multiple specified words.
  • To develop an explicit formula for the limit covariance in this competing renewals model.

Main Methods:

  • Modeling occurrences of m given words as competing renewals within an independent and identically distributed (i.i.d.) sequence over a finite alphabet.
  • Utilizing the matrix of overlap-matching polynomials to derive the limit covariance.

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Main Results:

  • A joint central limit theorem is proven for the vector of counts of nonoverlapping word occurrences.
  • A simple, explicit formula for the limit covariance is provided, based on overlap-matching polynomials.
  • The results extend to a general case of competing renewals with interacting processes.

Conclusions:

  • The study provides a novel statistical framework for analyzing nonoverlapping patterns in sequences.
  • The derived formula for limit covariance offers a practical tool for applications, such as in bioinformatics.
  • The findings contribute to the broader theory of competing renewal processes.