Ergodic theorems for extended real-valued random variables

In this paper, we prove a new version of the Birkhoff ergodic theorem (BET) for random variables depending on a parameter (alias integrands). This involves variational convergences, namely epigraphical, hypographical and uniform convergence and requires a suitable definition of the conditional expectation of integrands. We also have to establish the measurability of the epigraphical lower and upper limits with respect to the σ-field of invariant subsets. From the main result, applications to uniform versions of the BET to sequences of random sets and to the strong consistency of estimators are briefly derived.

Colloquium Mathematicum

Given a σ-finite infinite measure space (Ω, µ), it is shown that any Dunford-Schwartz operator T : L 1 (Ω) → L 1 (Ω) can be uniquely extended to the space L 1 (Ω) + L ∞ (Ω). This allows to find the largest subspace Rµ of L 1 (Ω) + L ∞ (Ω) such that the ergodic averages 1 n n−1 k=0 T k (f) converge almost uniformly (in Egorov's sense) for every f ∈ Rµ and every Dunford-Schwartz operator T. Utilizing this result, almost uniform convergence of the averages 1 n n−1 k=0 β k T k (f) for every f ∈ Rµ, any Dunford-Schwartz operator T and any bounded Besicovitch sequence {β k } is established. Further, given a measure preserving transformation τ : Ω → Ω, Assani's extension of Bourgain's Return Times theorem to σ-finite measure is employed to show that for each f ∈ Rµ there exists a set Ω f ⊂ Ω such that µ(Ω \ Ω f) = 0 and the averages 1 n n−1 k=0 β k f (τ k ω) converge for all ω ∈ Ω f and any bounded Besicovitch sequence {β k }. Applications to fully symmetric subspaces E ⊂ Rµ are given.

Israel Journal of Mathematics, 2005

We prove extensions of Menchoff's inequality and the Menchoff-Rademacher theorem for sequences {f n } ⊂ L p , based on the size of the norms of sums of sub-blocks of the first n functions. The results are applied to the study of a.e. convergence of series n a n T n g n α when T is an L 2-contraction, g ∈ L 2 , and {a n } is an appropriate sequence. Given a sequence {f n } ⊂ L p (Ω, µ), 1 < p ≤ 2, of independent centered random variables, we study conditions for the existence of a set of x of µ-probability 1, such that for every contraction T on L 2 (Y, π) and g ∈ L 2 (π), the random power series n f n (x)T n g converges π-a.e. The conditions are used to show that for {f n } centered i.i.d. with f 1 ∈ L log + L, there exists a set of x of full measure such that for every contraction T on L 2 (Y, π) and g ∈ L 2 (π), the random series n fn(x)T n g n converges π-a.e.

Transactions of the American Mathematical Society, 1998

It is shown that any bounded weight sequence which is good for all probability preserving transformations (a universally good weight) is also a good weight for any L 1 L_{1} -contraction with mean ergodic (ME) modulus, and for any positive contraction of L p L_{p} with 1 &gt; p &gt; ∞ 1 &gt; p &gt;\infty . We extend the return times theorem by proving that if S S is a Dunford-Schwartz operator (not necessarily positive) on a Lebesgue space, then for any g g bounded measurable { S n g ( ω ) } \{S^{n} g(\omega )\} is a universally good weight for a.e. ω . \omega . We prove that if a bounded sequence has &quot;Fourier coefficents&quot;, then its weighted averages for any L 1 L_{1} -contraction with mean ergodic modulus converge in L 1 L_{1} -norm. In order to produce weights, good for weighted ergodic theorems for L 1 L_{1} -contractions with quasi-ME modulus (i.e., so that the modulus has a positive fixed point supported on its conservative part), we show that the modulus of the tenso...

arXiv (Cornell University), 2015

For a Dunford-Schwartz operator in the L p −space, 1 ≤ p < ∞, of an arbitrary measure space, we prove pointwise convergence of the conventional and Besicovitch weighted ergodic averages. Pointwise convergence of various types of ergodic averages in fully symmetric spaces of measurable functions with non-trivial Boyd indices is studied. In particular, it is shown that for such spaces Bourgain's Return Times theorem is valid. Definition 1.1. A measure space (Ω, A, µ) is called semifinite if every subset of Ω of non-zero measure admits a subset of finite non-zero measure. A semifinite measure space (Ω, A, µ) is said to have the direct sum property if the Boolean algebra (A/ ∼) of equivalence classes of measurable sets is complete, that is, every subset of (A/ ∼) has a least upper bound. Note that every σ−finite measure space has the direct sum property. A detailed account on measures with direct sum property is found in [5]; see also [12].