On mixing and entropy

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.

Operations Research Letters, 2008

We show that when w 1 | • • • |w n , optimization over S can be performed in time O(n m+1), and in time O(n log n) when w 1 = • • • = w n = 1.

We show that for infinite measure-preserving transformations, power weak mixing does not imply multiple recurrence. We also show that the infinite measure-preserving "Chacon transformation" known to have infinite ergodic index is not power weakly mixing, and is 3-recurrent but not multiply recurrent. We also construct some doubly ergodic infinite measurepreserving transformations that are not of positive type but have conservative Cartesian square. Finally, we study the power double ergodicity property.

Probability Surveys, 2005

This is an update of, and a supplement to, a 1986 survey paper by the author on basic properties of strong mixing conditions.

1973

Discrete and Continuous Dynamical Systems, 2013

For a mixing and uniformly expanding interval map f : I → I we pose the following questions. For which permutation transformations σ : I → I is the composition σ • f again mixing? When σ • f is mixing, how does the mixing rate of σ • f typically compare with that of f ? As a case study, we focus on the family of maps f (x) = mx mod 1 for 2 ≤ m ∈ N. We split [0, 1) into N equal subintervals, and take σ to be a permutation of these. We analyse those σ ∈ S N for which σ • f is mixing, and show that, for large N , typical permutations will preserve the mixing property. In contrast to the situation for continuous time diffusive systems, we find that composition with a permutation cannot improve the mixing rate, but may make it worse. We obtain a precise result on the worst mixing rate which can occur as σ varies, with m, N fixed and gcd(m, N) = 1.

Nonlinearity, 2004

For a stationary source with finite alphabet, let R n be the number of nonoverlapping n-blocks of symbols, occurring before the initial n-block reappears. When the source is ψ-mixing, we prove that the difference between the expectation of log R n and the entropy of n-blocks converges to the constant of Euler divided by − ln(2). This can be considered the correct version of a conjecture presented in Maurer (1992 J. Cryptol. 5 89-105). Our theorem generalizes recent results presented in

Mathematical Systems Theory, 1973

1. Introduction. The following discussion is an application of Ornstein's recent results concerning Bernoulli automorphisms to the study of induced transformations. Let (Y, ~), tz) be a non-atomic Lebesgue space and let T be an automorphism of X. If A is a measurable subset of X and / /°° T~A wi=o = X, then T induces on the Lebesgue space A with the conditional measure t~a the induced automorphism Ta defined by the formula TA(x)= Tk(x) (x c A), where k = k(x) is the smallest positive integer j such that Ti(x) ~ A. It will be shown that for a certain class of subsets of a Bernoulli shift weak mixing is a sufficient condition to ensure that the induced transformation be Bernoulli. As an application, one deduces that any two Bernoulli shifts are equivalent in the sense of Kakutani; that is, they can be imbedded as induced transformations in a single ergodic system of finite measure. Conversely, given a weak mixing system (Y, S), we ask what knowledge can be derived about S knowing the existence of a subset A of Y for which SA is Bernoulli. More precisely, it is shown that if co is the partition of A into return times of T, B a Bernoulli partition for (A, Ta), and H~,A(oJ+/B +) < oo (where + denotes the "future" of a partition under Ta), then S is weakly mixing if and only if S is a Bernoulli shift. 2. Preliminaries. We will adhere to the definitions and notation of entropy theory established in [11]. If ~ is a measurable partition of (X, ~), then o(~) will denote the sub<r-algebra consisting of those M-measurable sets which are unions of elements of ~. The infinite partitions V~T i~ and V-~T i~ will be denoted by ~+ and ~-respectively. If G is a finite generator of the or-algebra of X such that {TiG:-oo < i < ~ } are jointly independent, then G will be called a Bernoulli partition for T. Given finite partitions P and Q of X and E > 0 we will write P_[_'~Q if and only if Zi[~(Pi/QJ)-u(Pi)] < ~ for all but a set of atoms of Q whose union * These results are contained in the author's doctoral dissertation written at the University of California at Berkeley under the guidance of Professor Jacob Feldman to whom the author is very grateful for both enc,~uragement and advice. In addition, the author wishes to thank Professor William Parr,' for several helpful suggestions.

Israel Journal of Mathematics, 1979

Proceedings of the American Mathematical Society, 1976

Given a linear contraction T T on a Banach space X X and x ∈ X x \in X , the convergence \[ ∀ x ∗ ∈ X ∗ N − 1 ∑ i = 1 N | ⟨ x ∗ , T i x ⟩ | → N → ∞ 0 \forall {x^ \ast } \in {X^ \ast }{N^{ - 1}}\sum \limits _{i = 1}^N {|\langle {x^ \ast },{T^i}x\rangle |} \xrightarrow [{N \to \infty }]{}0 \] is shown to be equivalent to the convergence \[ sup | | x ∗ | | ⩽ 1 N − 1 ∑ j = 1 N | ⟨ x ∗ , T k j x ⟩ | → 0 \sup \limits _{||{x^ \ast }|| \leqslant 1} {N^{ - 1}}\sum \limits _{j = 1}^N {|\langle {x^ \ast },{T^{{k_j}}}x\rangle | \to 0} \] for every subsequence with k j / j {k_j}/j bounded. A sufficient condition is that, for some { n i } , T n i x → 0 \{ {n_i}\} ,{T^{{n_i}}}x \to 0 weakly.