NDLI: A Poisson-Type Limit Theorem for the Number of Pairs of Matching Sequences

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Mikhailov, V. G.
Copyright Year	2009
Abstract	Two sequences $X_1,\ldots,X_m$ and $Y_1,\ldots, Y_n$ are considered constituted by independent identically distributed random variables within each of the sequences taking on values in the set $\{1,2,\ldots\}$. We study the distribution of the number $N_d$ of such pairs of s-patterns $(\overline X_i,\, \overline Y_j)$, where $\overline X_i=(X_i,\ldots,X_{i+s-1})$, $\overline Y_j=(Y_j,\ldots,Y_{j+s-1})$, in which the s-patterns $\overline X_i$ and $\overline Y_j$ differ by a relatively small number of elements d. It is shown that if ${m,n,s\to\infty}$, $d=o(s/\log s),$ and the distributions of the elements of the sequences vary in such a way that the probability ${\bf P}\{X_i=Y_j\}$ and ${\bf E} N_d$ converge to some limiting values, then the distribution of $N_d$ converges to a compound Poisson distribution. The value of the parameter d plays a role only to provide, passing to the limit, the needed rate of the parameters involved and has no influence on the form of the limit distribution. This limit distribution has the same form as that for the number of pairs $(\overline X_i, \overline Y_j)$, in which $\overline X_i= \overline Y_j$.
Starting Page	106
Ending Page	116
Page Count	11
File Format	PDF
ISSN	0040585X
DOI	10.1137/S0040585X97983390
e-ISSN	10957219
Journal	Theory of Probability & Its Applications (TPRBAU)
Issue Number	1
Volume Number	53
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2009-02-27
Access Restriction	Subscribed
Subject Keyword	pattern matching s-patterns ChenStein method compound Poisson distribution Poisson limit theorem mismatches of patterns
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistics, Probability and Uncertainty

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