Let $(X,\mathcal{F} ,\mu)$ be a probability space and let $L^{2}(X,0)$ be the collection of all $f \in L^{2}(X)$ with zero integrals. A collection $\mathcal{A}$ of linear operators on $L^{2}(X)$ is said to satisfy the Gaussian-distribution property (G.D.P.) if $L^{2}(X,0)$ is invariant under $\mathcal{A}$ and there exists a constant $C \lt \infty$ such that the following condition holds:

Whenever $T_{1},\ldots,T_{k}$ are finitely many operators in $\mathcal{A}$, and $f$ is a function in $L_{2}$ with zero integral, then, for any required degree of approximation, there is another $L_{2}$-function $g$ with $||g||_{2} \leq C ||f||_{2}$, such that all the inner products $(\mathrm{Re} T_{i}g, \mathrm{Re} T_{j}g)$ are approximately equal to the corresponding inner products $(\mathrm{Re} T_{i}f, \mathrm{Re} T_{j}f)$ for all $1 \leq i,j \leq k$ and such that the joint distribution of the functions $\mathrm{Re} T_{1}g,\ldots,\mathrm{Re} T_{k}g$ is approximately Gaussian.

It has been proved that if $(S_{n})^{\infty}_{1}$ is a sequence of uniformly bounded linear operators on $L^{2}(X)$ that satisfies the Bourgain's infinite entropy condition and the G.D.P., then there exists an $h \in L^{2}(X)$ such that $\liminf S_{n}\,h$ fails to exist $\mu$-a.e., as a finite limit on $X$.

The purpose of this paper is to provide sufficient conditions for a collection $\mathcal{A}$ of linear operators on $L^{2}(X)$ to satisfy the G.D.P.

In this paper we consider the block sequences introduced by Bellow and Losert, and show that in several cases, not only does divergence occur, but that we also have strong sweeping out. Related averages are also considered.

We give a brief proof of the theorem of P. Le Calvez and J.-C. Yoccoz on the non-existence of a minimal homeomorphism of the finitely punctured plane. The proof here is based on the Conley index.

We answer a question of [BJR] about pointwise convergence a.e. for convolution powers of measures with expectation zero and finite moments of all orders $s \lt 2$. We compare the conditions that appeared in the study of the strong sweeping out property for convolution powers.

We discuss universal estimates for the probability that there are many fluctuations in the ergodic averages of $L^{1}$ functions. Our methods involve an effective Vitali covering type of theorem and are valid for $\mathbb{Z}^{d}$ actions, for any $d \in \mathbb{N}$. For nonnegative functions we get an exponential decay for the probability of a large number of fluctuations.

A simple unified formulation for generating an estimator of a univariate distribution (not necessarily continuous) is proposed. It is a variant of the maximum likelihood method and also a generalization of the maximum product of spacings (MPS) method for estimating continuous univariate distributions. The general formulation is then applied to monotone hazard rate families which arise frequently in applications. It is shown that any asymptotic MPS estimator for any family of distributions with monotone hazard rate is always strongly consistent (whether the setting is parametric or not). The MPS estimator of a distribution function with a monotone hazard rate can be derived explicitly and is asymptotically minimax for the Kolmogorov-Smirnov type loss functions.

Suppose that $(\Omega,\mathcal{M},\mu)$ is a $\sigma$-finite measure space, $1 \lt p \lt \infty$, and $T: L^{p}(\mu) \rightarrow L^{p}(\mu)$ is a bounded, invertible, separation-preserving linear operator such that the two-sided ergodic means of the linear modulus of $T$ are uniformly bounded in norm. Using the spectral structure of $T$, we obtain a functional calculus for $T$ associated with the algebra of Marcinkiewicz multipliers defined on the unit circle$\ldots$

The purpose of this paper is to introduce a generalized notion of geometric stability for distributions with support in $\mathbf{Z}_{+}$ and $\mathbf{R}_{+}$. Several characterizations are obtained. A related concept of geometric attraction is also Studied. Importantly, Poisson mixtures are used to deduce results for the $\mathbf{R}_{+}$-case from their $\mathbf{Z}_{+}$-counterparts.

Let $\tau$ and $\sigma$ be two commuting ergodic measure preserving transformations of a probablity space, and $\mathrm{Cob}(\tau)$, $\mathrm{Cob}(\sigma)$ be the sets of their coboundaries. We show that the inclusion $\mathrm{Cob}(\sigma) \subseteq \mathrm{Cob}(\tau)$ holds if and only if $\sigma = \tau^{n}$ for some $n \in \mathbb{Z}$. The transformations $\tau$ and $\sigma$ have exactly the same coboundaries if and only if $\sigma = \tau^{\pm1}$. Some related results and open questions are discussed.

We give a characterization of $L^{1}$ coboundaries for a set of bounded operators. This set includes measure preserving transformations.

We investigate sequences of complex numbers $\mathbf{a} = \{a_{k}\}$ for which the modulated averages $\frac{1}{n}\sum^{n}_{k=1}{a_{k}T^{k} f}$ converge in norm for every weakly almost periodic linear operator $T$ in a Banach space. For Dunford-Schwartz operators on probability spaces, we study also the a.e. convergence in $L_{p}$. The limit is identified in some special cases, in particular when $T$ is a contraction in a Hilbert space, or when $\mathbf{a} = \{S^{k}\phi(\xi)\}$ for some positive Dunford-Schwartz operator $S$ on a Lebesgue space and $\phi \in L_{2}$. We also obtain necessary and sufficient conditions on $\mathbf{a}$ for the norm convergence of the modulated averages for every mean ergodic power bounded $T$, and identify the limit.

We study the boundedness in $L^{1}$ of the maximal operator of the class of martingales for a fixed increasing or decreasing filtration. We obtain necessary and sufficient conditions for several classes of martingales.

It is shown that if $\{\mu_{n}\}$ is a sequence of measures good a.e. or resp. in the p-mean for additive processes, then it is good a.e. or resp. in the p-mean, for the class of strongly bounded admissible superadditive processes. Using the method developed, it is shown also that weighted averages of strongly bounded admissible superadditive processes converge a.e. or in the p-mean for weights that are good a.e. or in the p-mean for additive processes.

We study the almost everywhere convergence of the ergodic Cesàro-$\alpha$ averages $R_{n,\alpha}f = \frac{1}{A^{\alpha}_{n}}\Sigma^{n}_{i=0}{A^{\alpha-1}_{n-i}T^{i}f}$ and the boundedness of the ergodic maximal operator $M_{\alpha}f = \mathrm{sup}_{n \in \mathbb{N}}|R_{n, \alpha}f|$, associated with a positive linear operator $T$ with positive inverse on some $L^{p}(\mu)$, $1 \lt p \lt \infty$, $0 \lt \alpha \leq 1$.