A general strong law

Convex Komlós sets in Banach function spaces

“…Finally, we will briefly discuss a connection between our work and an extension of Komlós' Theorem to L p (1 p < 2) due to Chatterji [2].…”

“…Then, Komlós [5] proved the following theorem. (See [5,2,6] for more information on the history and generalizations of Komlós' Theorem.) We will say that a sequence { f n } n in L 1 is Komlós if there exists a subsequence {g n } n of { f n } n and a function g ∈ L 1 such that for any further subsequence {h n } n of {g n } n , 1 n n i=1 h n −→ n g μ-a.e.…”

Convergence of series of scalar- and vector-valued random variables and a subsequence principle in 𝐿₂

“…Property (i) is a well-known theorem of Revesz (see [18]) and follows easily from the martingale convergence theorem (see [3]). Indeed, 2:::=1 angn converges almost surely and in L2 (0) whenever 2:::=1 a~ < 00, and so the same is true of 2::;;0=1 ak(fnk -f).…”

On the Uniform Theory of Lacunary Series

“…(1.6) and Chatterji [22] showed that under sup n E|X n | p < ∞, 0 < p < 2 the conclusion of the previous theorem can be changed to…”