Abstract. Let $(X,{\cal F},\mu )$ be a probality space and let $L^2(X,0)$ be the collection of all $f \in L^2(X)$ with zero integrals. A collection ${\cal A}$ 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 ${\cal A} $ and there exists a constant $C < \infty $ such that the following condition holds: Whenever $T_1,\ldots,T_k$ are finitely many operators in ${\cal A}$, 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 $({\rm Re} T_ig, {\rm Re} T_jg)$ are approximately equal to the corresponding inner products $({\rm Re} T_if, {\rm Re} T_jf)$, $i,j=1,\ldots,k$, and such that the joint distribution of the functions ${\rm Re} T_1g,\ldots,{\rm Re} T_kg$ is approximately Gaussian. It has been proved recently ([3]) that if $(S_n)^\infty_1$ is a sequence of uniformly bounded linear operators on $L^2 (X)$ with $S_n {\bf1} = {\bf1}$, $n=1,2,\ldots $, and such that $(S_n)^\infty_1$ satisfies the Bourgain's infinite entropy condition and the G.D.P., then there exists an $h\in L^2(X)$ such that $\lim_{n\to\infty } S_n h$ fails to exist $\mu $-a.e. as a finite limit on $X$. The purpose of this paper is to show that various classes of operators commonly encountered in ergodic theory do actually satisfy the G.D.P.. In particular, we will show that continuous ergodic automorphism of a compact abelian group, translations and endomorphisms of the $n$-tori, multiple Riemann-sum operators, and operators arising from the conjectures of Bellow and Khintchine all satisfy the G.D.P.. Applications to convergence problems are also given.

AMS Subject Classification (1991): 28D99, 60F99

Received May 26, 1997, and in final form November 12, 1999. (Registered under 2720/2009.)