RANDOM SERIES IN L(X,Σ, μ) USING UNCONDITIONAL BASIC SEQUENCES AND l STABLE SEQUENCES: A RESULT ON ALMOST SURE ALMOST EVERYWHERE CONVERGENCE
نویسندگان
چکیده
Here we study the almost sure almost everywhere convergence of random series of the form ∑∞ i=1 aifi in the Lebesgue spaces L p(X,Σ, μ), where the ai’s are centered random variables, and the fi’s constitute an unconditional basic sequence or an lp stable sequence. We show that if one of these series converges in the norm topology almost surely, then it converges almost everywhere almost surely.
منابع مشابه
The Cesáro Core of Double Sequences
and Applied Analysis 3 where σ s σ σj−1 s . In this case, we write σ − limx . By V 2 σ , we denote the set of all σ-convergent and bounded double sequences. One can see that in contrast to the case for single sequences, a convergent double sequence need not be σ-convergent. But every bounded convergent double sequence is σ-convergent. So, c∞ 2 ⊂ V 2 σ ⊂ ∞ 2 . In the case σ i i 1, σ-convergence ...
متن کاملErgodic Theorems over Sparse Random Subsequences
We prove an L subsequence ergodic theorem for sequences chosen by independent random selector variables, thereby showing the existence of universally L-good sequences nearly as sparse as the set of squares. We extend this theorem to a more general setting of measure-preserving group actions. In addition, we use the same technique to prove an L almost everywhere convergence result for a modulate...
متن کاملMatrix Transformations between the Spaces of Cesہro Sequences and Invariant Means
Let ω be the space of all sequences, real or complex, and let l∞ and c, respectively, be the Banach spaces of bounded and convergent sequences x = (xn) with norm ‖x‖ = supk≥0 |xk|. Let σ be a mapping of the set of positive integers into itself. A continuous linear functional φ on l∞ is said to be an invariant mean or a σmean if and only if (i) φ(x) ≥ 0, when the sequence x = (xn) has xn ≥ 0 for...
متن کاملMatrix transformations between the spaces of Cesàro sequences and invariant means
Let ω be the space of all sequences, real or complex, and let l∞ and c, respectively, be the Banach spaces of bounded and convergent sequences x = (xn) with norm ‖x‖ = supk≥0 |xk|. Let σ be a mapping of the set of positive integers into itself. A continuous linear functional φ on l∞ is said to be an invariant mean or a σmean if and only if (i) φ(x) ≥ 0, when the sequence x = (xn) has xn ≥ 0 for...
متن کاملSome Sequence Spaces and Almost Convergence
In this paper we investigate some new sequence spaces which naturally emerge from the concept of almost convergence. Just as ordinary, absolute and strong summability, it is expected that almost convergence must give rise to almost, absolutely almost and strongly almost summability. Almost and absolutely almost summable sequences have been discussed by several authors. The object of this paper ...
متن کامل