Almost sure convergence of some random series

作者：Grivaux Sophie^*

来源：Comptes Rendus Mathematique, 2010, 348(3-4): 155-159.

DOI：10.1016/j.crma.2010.01.020

Let (cn)(n >= 1) be a square-summable sequence of complex numbers, d >= 2 an integer, and (e(n.d))(n >= 1) the orthonormal basis of the space L(2)([0, 1], r(d-1) dr) consisting of the radial eigenfunctions of the Laplace operator acting on the space L(2)(D(d)) of square-summable functions on the unit ball D(d) = (x is an element of R(d); r = vertical bar x vertical bar < 1] of R(d). We generalize a result of Ayache and Tzvetkov and compute in the general case the critical exponent of the sequence (c(n))(n >= 1), i.e. the infimum of the p's, p >= 2, such that the random series Sigma epsilon(n)(omega)c(n)e(n,d) converges almost surely in L(p)([0, 1], r(d-1)c dr), where (epsilon(n)) denotes a sequence of independent random choices of signs on a probability space (Omega, F, P).

• 出版日期2010-2