Related papers: Kac's Lemma and countable generators for actions o…

We continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving actions of countable groups introduced in Part I. In this paper we prove a non-ergodic finite generator theorem and use it to establish…

We show that the Erd\H{o}s-Kac theorem is valid in almost all intervals $\left[x,x+h\right]$ as soon as $h$ tends to infinity with $x$. We also show that for all $k$ near $\log\log x$, almost all interval $\left[x,x+\exp\left(\left(\log\log…

Iterates of quantum operations and their convergence are investigated in the context of mean ergodic theory. We discuss in detail the convergence of the iterates and show that the uniform ergodic theorem plays an essential role. Our results…

This brief pedagogical note re-proves a simple theorem on the convergence, in $L_2$ and in probability, of time averages of non-stationary time series to the mean of expectation values. The basic condition is that the sum of covariances…

In this paper, we introduce a variation of the group testing problem where each test is specified by an ordered subset of items and returns the first defective item in the specified order or returns null if there are no defectives. We refer…

The famous equivalence theorem is reexamined in order to make it applicable to the case of intrinsically quantum infinite-component effective theories. We slightly modify the formulation of this theorem and prove it basing on the notion of…

This article is concerned with measure equivalence and uniform measure equivalence of locally compact, second countable groups. We show that two unimodular, locally compact, second countable groups are measure equivalent if and only if they…

The first part of this article deals with theorems on uniqueness in law for \sigma-finite and constructive countable random sets, which in contrast to the usual assumptions may have points of accumulation. We discuss and compare two…

This is the first in a series of papers on projective positive energy representations of gauge groups. Let $\Xi \rightarrow M$ be a principal fiber bundle, and let $\Gamma_{c}(M,\mathrm{Ad}(\Xi))$ be the group of compactly supported (local)…

Let $X$ be a measure space and $T:X\to X$ a measurable transformation. For any measurable $E\subseteq X$ and $x\in E$, the possibly infinite return time is $n_E(x):=\inf\{n>0: T^n x\in E\}$. If $T$ is an ergodic tranformation of the…

An expression is proposed for the quantum mechanical state of a pre- and post-selected ensemble, which is an ensemble determined by the final as well as the initial state of the quantum systems involved. It is shown that the probabilities…

We study in this paper the validity of the mean ergodic theorem along \emph{left} F\o lner sequences in a countable amenable group $G$. Although the \emph{weak} ergodic theorem always holds along \emph{any} left F\o lner sequence in $G$, we…

We establish an Ergodic Theorem for lower probabilities, a generalization of standard probabilities widely used in applications. As a by-product, we provide a version for lower probabilities of the Strong Law of Large Numbers.

Let Gamma be a finitely generated, amenable group. Using an idea of E Ghys, we prove that if Gamma has a nontrivial, orientation-preserving action on the real line, then Gamma has an infinite, cyclic quotient. (The converse is obvious.)…

An action trace is a function naturally associated to a probability measure preserving action of a group on a standard probability space. For countable amenable groups, we characterise stability in permutations using action traces. We…

Estimating the probability distribution 'q' governing the behaviour of a certain variable by sampling its value a finite number of times most typically involves an error. Successive measurements allow the construction of a histogram, or…

In this note we show existence of bounded, transitive cocycles over a transitive action of a finitely generated group, and bounded, ergodic cocycles over an ergodic, probability preserving action of $\Bbb Z^d$.

The algebra of observables associated with a quantum field theory is invariant under the connected component of the Lorentz group and under parity reversal, but it is not invariant under time reversal. If we take general covariance…

Logical entropy gives a measure, in the sense of measure theory, of the distinctions of a given partition of a set, an idea that can be naturally generalized to classical probability distributions. Here, we analyze how fundamental concepts…

Sequential hypothesis testing asks for decision rules that update as data arrive. A natural goal is \emph{eventual correctness}: the rule may change its mind early on, but it should make only finitely many wrong decisions almost surely.…