Erdős measures on the Euclidean space and on the group of A-adic integers


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Abstract

Let AMn(ℤ) be a matrix with eigenvalues greater than 1 in absolute value. The ℤn-valued random variables ξt, t ∈ ℤ, are i.i.d., and P(ξt = j) = pj, j ∈ ℤn, 0 < p0 < 1, ∑jpj = 1. We study the properties of the distributions of the ℝn-valued random variable ζ1 = ∑t=1Atξt and of the random variable ζ = ∑t=0Atξt taking integer A-adic values. We obtain a necessary and sufficient condition for the absolute continuity of these distributions. We define an invariant Erdős measure on the compact abelian group of A-adic integers. We also define an A-invariant Erdős measure on the n-dimensional torus. We show the connection between these invariant measures and functions of countable stationary Markov chains. In the case when |{j: pj ≠ 0}| < ∞, we establish the relation between these invariant measures and finite stationary Markov chains.

About the authors

Z. I. Bezhaeva

National Research University “Higher School of Economics,”

Author for correspondence.
Email: zbejaeva@hse.ru
Russian Federation, ul. Myasnitskaya 20, Moscow, 101000

V. L. Kulikov

Financial University under the Government of the Russian Federation

Email: zbejaeva@hse.ru
Russian Federation, Leningradskii pr. 49, Moscow, 125993

E. F. Olekhova

Financial University under the Government of the Russian Federation

Email: zbejaeva@hse.ru
Russian Federation, Leningradskii pr. 49, Moscow, 125993

V. I. Oseledets

Financial University under the Government of the Russian Federation; Faculty of Mechanics and Mathematics

Email: zbejaeva@hse.ru
Russian Federation, Leningradskii pr. 49, Moscow, 125993; Moscow, 119991

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