Diskretnaya Matematika, Tome 20 (2008) no. 4, pp. 113-119
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V. G. Mikhailov. On the asymptotic behaviour of the probability of existence of equivalent tuples with nontrivial structure in a random sequence. Diskretnaya Matematika, Tome 20 (2008) no. 4, pp. 113-119. http://geodesic.mathdoc.fr/item/DM_2008_20_4_a9/
@article{DM_2008_20_4_a9,
author = {V. G. Mikhailov},
title = {On the asymptotic behaviour of the probability of existence of equivalent tuples with nontrivial structure in a~random sequence},
journal = {Diskretnaya Matematika},
pages = {113--119},
year = {2008},
volume = {20},
number = {4},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/DM_2008_20_4_a9/}
}
TY - JOUR
AU - V. G. Mikhailov
TI - On the asymptotic behaviour of the probability of existence of equivalent tuples with nontrivial structure in a random sequence
JO - Diskretnaya Matematika
PY - 2008
SP - 113
EP - 119
VL - 20
IS - 4
UR - http://geodesic.mathdoc.fr/item/DM_2008_20_4_a9/
LA - ru
ID - DM_2008_20_4_a9
ER -
%0 Journal Article
%A V. G. Mikhailov
%T On the asymptotic behaviour of the probability of existence of equivalent tuples with nontrivial structure in a random sequence
%J Diskretnaya Matematika
%D 2008
%P 113-119
%V 20
%N 4
%U http://geodesic.mathdoc.fr/item/DM_2008_20_4_a9/
%G ru
%F DM_2008_20_4_a9
In a long enough sequence of discrete random variables, as a rule, an $s$-tuple exists of nontrivial structure, that is, a tuple with at least one repeated symbol. We consider the case where the sequence consists of $n+s-1$ independent random variables taking the values $1,\dots,N$ with equal probabilities. It is shown that as $n\to\infty$, $ns^3N^{-2}\to0$ the probability of that in the sequence $s$-tuples exist with the same nontrivial structure is equal to $1-(1+n/N)^se^{-sn/N}(1+o(1))$.
[1] Mikhailov V. G., Shoitov A. M., “Strukturnaya ekvivalentnost $s$-tsepochek v sluchainykh diskretnykh posledovatelnostyakh”, Diskretnaya matematika, 15:4 (2003), 7–34 | MR | Zbl
[2] Barbour A. D., Holst L., Janson S., Poisson approximation, Oxford Univ. Press, Oxford, 1992 | MR | Zbl
