November 3, 2005
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October 9, 2019
In [4], we examined the use of coupling to obtain bounds on the mixing time of statistics on Markov chains. In the present paper, we consider the same general problem, but using strong stationary times rather than coupling. We discuss various types of behaviour that may occur when this is attempted, and analyse a variety of examples.
June 7, 2008
There is a mathematical error in the first version of this paper. A new corrected version will be posted when the error is fixed, possibly with a modified title.
June 22, 2014
We present new criteria, based on commutator methods, for the strong mixing property of discrete flows $\{U^N\}_{N\in\mathbb Z}$ and continuous flows $\{{\rm e}^{-itH}\}_{t\in\mathbb R}$ induced by unitary operators $U$ and self-adjoint operators $H$ in a Hilbert space $\mathcal H$. Our approach put into evidence a general definition for the topological degree of the curves $N\mapsto U^N$ and $t\mapsto{\rm e}^{-itH}$ in the unitary group of $\mathcal H$. Among other examples,...
March 6, 2014
For nonstationary, strongly mixing sequences of random variables taking their values in a finite-dimensional Euclidean space, with the partial sums being normalized via matrix multiplication, with certain standard conditions being met, the possible limit distributions are precisely the operator-selfdecomposable laws.
October 13, 2020
The ergodic theorems of Hopf, Wiener and Birkhoff were extended to the context of Riesz spaces with a weak order unit and conditional expectation operator by Kuo, Labuschagne and Watson in [Ergodic Theory and the Strong Law of Large Numbers on Riesz Spaces. Journal of Mathematical Analysis and Applications, 325,(2007), 422-437.]. However, the precise concept of what constitutes ergodicity in Riesz spaces was not considered. In this short paper we fill in this omission and giv...
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We investigate the subject of speed of mixing for operators on infinite dimensional Hilbert spaces which are strongly mixing with respect to a nondegenerate Gaussian measure. We prove that there is no way to find a uniform speed of mixing for all square-integrable functions. We give classes of regular functions for which the sequence of correlations decreases to zero with speed $n^{-\alpha}$ when the eigenvectors associated to unimodular eigenvalues of the operator are parame...
December 8, 2010
In this paper we study the asymptotic normality of the normalized partial sum of a Hilbert-space valued strictly stationary random field satisfying the interlaced $\rho'$-mixing condition.
November 17, 2023
We prove absolute regularity ($\beta$-mixing) for nonstationary and multivariate versions of two popular classes of integer-valued processes. We show how this result can be used to prove asymptotic normality of a least squares estimator of an involved model parameter.
May 2, 2005
The first part of the paper contains a survey of weakly mixing group actions of general group, and the second part discusses a special example of a weakly mixing action -- the SL(2,Z)-action on the torus T^2.
June 15, 2021
In this paper, we introduce and characterize the concept of directional weak mixing through independence, sequence entropy, the mean ergodic theorem, and other notions. Additionally, we deduce a directional version of the Koopman-von Neumann spectrum mixing theorem. Furthermore, we explore the relation between directional weak mixing and weak mixing.