Erdős measures on the Euclidean space and on the group of A-adic integers
- Authors: Bezhaeva Z.I.1, Kulikov V.L.2, Olekhova E.F.2, Oseledets V.I.2,3
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Affiliations:
- National Research University “Higher School of Economics,”
- Financial University under the Government of the Russian Federation
- Faculty of Mechanics and Mathematics
- Issue: Vol 297, No 1 (2017)
- Pages: 28-34
- Section: Article
- URL: https://journals.rcsi.science/0081-5438/article/view/174563
- DOI: https://doi.org/10.1134/S0081543817040022
- ID: 174563
Cite item
Abstract
Let A ∈ Mn(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=1∞A−tξt and of the random variable ζ = ∑t=0∞Atξ−t 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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