It is pointed out that a strong law of large numbers for L-statistics established by van Zwet (1980) for i.i.d. sequences, remains valid for stationary ergodic data. When the underlying process is weakly Bernoulli, the result extends even to generalized L-statistics considered in Helmers et al. (1988).

We construct infinite-dimensional chains that are L¹ good for almost sure convergence, which settles a question raised in this journal [N]. We give some conditions for a coprime generated chain to be bad for L² or $L^{\infty }$, using the entropy method. It follows that such a chain with positive lower density is bad for $L^{\infty }$. There also exist such bad chains with zero density.

We discuss the classification up to orbit equivalence of inclusions 𝑆 ⊂ ℛ of measured ergodic discrete hyperfinite equivalence relations. In the case of type III relations, the orbit equivalence classes of such inclusions of finite index are completely classified in terms of triplets consisting of a transitive permutation group G on a finite set (whose cardinality is the index of 𝑆 ⊂ ℛ), an ergodic nonsingular ℝ-flow V and a homomorphism of G to the centralizer of V.

Let T be a finite entropy, aperiodic automorphism of a nonatomic probability space. We give an elementary proof of the existence of a finite entropy, countable generating partition for T.

We define the concept of directional entropy for arbitrary ${\mathbb{Z}}^2$-actions on a Lebesgue space, we examine its basic properties and consider its behaviour in the class of product actions and rigid actions.

We show that for any cellular automaton (CA) ℤ²-action Φ on the space of all doubly infinite sequences with values in a finite set A, determined by an automaton rule $F=F_{[l,r]}$, l,r ∈ ℤ, l ≤ r, and any Φ-invariant Borel probability measure, the directional entropy $h_{v⃗}\left(\Phi \right)$, v⃗= (x,y) ∈ ℝ², is bounded above by $max\left(\right|z_l|,|z_r\left|\right)logA$ if $z_l{z}_r\ge 0$ and by $|z_r-z_l|$ in the opposite case, where $z_l=x+ly$, $z_r=x+ry$. We also show that in the class of permutative CA-actions the bounds are attained if the measure considered is uniform Bernoulli.

Let T be a measure-preserving ergodic transformation of a measure space (X,,μ) and, for f ∈ L(X), let $f*=su{p}_{N}1/N{\sum }_{m=0}^{N-1}f\circ T^m$. In this paper we mainly investigate the question of whether (i) ${ʃ}_a^{\infty }|\mu (f*>t)-1/t{ʃ}_{(f*>t)}fd\mu |dt<\infty$ and whether (ii) ${ʃ}_a^{\infty }|\mu (f*>t)-1/t{ʃ}_{(f>t)}fd\mu |dt<\infty$ for some a > 0. It is proved that (i) holds for every f ≥ 0. (ii) holds if f ≥ 0 and f log log (f + 3) ∈ L(X) or if μ(X) = 1 and the random variables $f\circ T^m$ are independent. Related inequalities are proved. Some examples and counterexamples are constructed. Several known results are obtained as corollaries.