The compound Poisson distribution of the number of matches of values of a discrete function of s-tuples in segments of a sequence of random variables
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A. M. Shoitov
For a sequence X = {X1, ... , Xn, ... } of independent identically distributed random variables, we construct the s-tuples Yi(s) = (Xi, ... , Xi+s-1), i = 1, 2, ... , n, and consider the random variables Fi = f(Yi(s)), i = 1, 2, ... , where f is a function defined on the set Rs and taking non-negative integer values.
We consider the sequence F = {F1, F2, ... } and study two random variables, the variable
equal to the number of matches of symbols on a segment of length n of the sequence F (here I{·} stands for the indicator of a random event), and the variable
equal to the number of matches of values of the function f of non-overlapping s-tuples of a segment of the sequence X of length n + s – 1.
With the use of the Stein method, we find sufficient conditions for the distribution of the random variables Zn(F) and Z′n(F) to converge to the compound Poisson law for the function f of a general form. As corollaries to these results we obtain both known and new limit theorems for the number of matches of values of a function of segments of sequences in a polynomial scheme for a series of particular types of the function f .
Copyright 2007, Walter de Gruyter
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- The compound Poisson distribution of the number of matches of values of a discrete function of s-tuples in segments of a sequence of random variables
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- Homomorphisms and endomorphisms of linear and alinear quasigroups
- On finite groups close to completely factorisable groups
- Non-asymptotic bounds for probabilities of the rank of a random matrix over a finite field
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Articles in the same Issue
- The compound Poisson distribution of the number of matches of values of a discrete function of s-tuples in segments of a sequence of random variables
- On stability of an efficient solution of a vector Boolean problem of maximisation of absolute values of linear functions
- Optimal logarithmic functions for lifting of a solution of an exponential congruence
- On the number of independent sets in graphs with fixed independence number
- Homomorphisms and endomorphisms of linear and alinear quasigroups
- On finite groups close to completely factorisable groups
- Non-asymptotic bounds for probabilities of the rank of a random matrix over a finite field
- On representation of k-valued logic functions by a sum of products of subfunctions
- A lower bound for the 4-satisfiability threshold
- A Markov chain of order s with r partial connections and statistical inference on its parameters
