Random series in 𝐿^{𝑝}(𝑋,Σ,𝜇) using unconditional basic sequences and 𝑙^{𝑝} stable sequences: A result on almost sure almost everywhere convergence

Medina, Juan; Cernuschi-Frías, B.

doi:10.1090/S0002-9939-07-08870-3

Random series in $L^p(X,\Sigma ,\mu )$ using unconditional basic sequences and $l^p$ stable sequences: A result on almost sure almost everywhere convergence
HTML articles powered by AMS MathViewer

by Juan M. Medina and B. Cernuschi-Frías PDF

Proc. Amer. Math. Soc. 135 (2007), 3561-3569 Request permission

Abstract:

Here we study the almost sure almost everywhere convergence of random series of the form $\sum _{i=1}^{\infty } {a_i f_i}$ in the Lebesgue spaces $L^p(X,\Sigma ,\mu )$, where the $a_i$’s are centered random variables, and the $f_i$’s constitute an unconditional basic sequence or an $l^p$ stable sequence. We show that if one of these series converges in the norm topology almost surely, then it converges almost everywhere almost surely.

References

Sylvie Guerre-Delabrière, Classical sequences in Banach spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 166, Marcel Dekker, Inc., New York, 1992. With a foreword by Haskell P. Rosenthal. MR 1197117
Kahane, J.P., Some Random Series of Functions, Cambridge, 1993.
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR 0415253
Michel Loève, Probability theory. I, 4th ed., Graduate Texts in Mathematics, Vol. 45, Springer-Verlag, New York-Heidelberg, 1977. MR 0651017
Menchoff, D., Sur les séries de fonctions orthogonales I. Fund. Math. 4, 1923, pages 82-105.
Peter Ørno, A note on unconditionally converging series in $L_{p}$, Proc. Amer. Math. Soc. 59 (1976), no. 2, 252–254. MR 458156, DOI 10.1090/S0002-9939-1976-0458156-7
Robert L. Taylor, Stochastic convergence of weighted sums of random elements in linear spaces, Lecture Notes in Mathematics, vol. 672, Springer, Berlin, 1978. MR 513422
P. Wojtaszczyk, Wavelets as unconditional bases in $L_p(\textbf {R})$, J. Fourier Anal. Appl. 5 (1999), no. 1, 73–85. MR 1682254, DOI 10.1007/BF01274190
Lihua Yang, Unconditional basic sequence in $L^p(\mu )$ and its $l^p$-stability, Proc. Amer. Math. Soc. 127 (1999), no. 2, 455–464. MR 1473673, DOI 10.1090/S0002-9939-99-04638-9