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Limit theorems for the logarithm of the order of a random A-mapping

  • Arsen L. Yakymiv EMAIL logo
Published/Copyright: October 11, 2017

Abstract

Let š”–n be a semigroup of mappings of a set X with n elements into itself, A be some fixed subset of the set N of natural numbers, and Vn(A) be a set of mappings from š”–n, with lengths of cycles belonging to A. The mappings from Vn(A) are called A-mappings. We suppose that the set A has an asymptotic density ϱ > 0, and that |k : k ≤ n, k ∈ A, m āˆ’ k ∈ A|/n → ϱ2 as n → āˆž uniformly over m ∈ [n, Cn] for each constant C > 1. A number M(α) of different elements in a set {α, α2, α3, …} is called an order of mapping α ∈ š”–n. Consider a random mapping σ = σn(A) having uniform distribution on Vn(A). In the present paper it is shown that random variable ln M(σn(A)) is asymptotically normal with mean l(n)=āˆ‘k∈A(n)ln⁔(k)/k and variance ϱln3(n)/24, where A(t) = {k : k ∈ A, k ≤ t}, t > 0. For the case A = N this result was proved by B. Harris in 1973.


Originally published in Diskretnaya Matematika (2017) 29, ā„–1, 136–155 (in Russian).


Award Identifier / Grant number: 14-01-00318

Funding statement: This study was supported by the Russian Foundation for Basic Research, grant 14-01-00318.

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Received: 2016-7-28
Revised: 2016-11-21
Published Online: 2017-10-11
Published in Print: 2017-10-26

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