Uniform convergence of random Fourier transforms almost surely. (English. Russian original) Zbl 0631.60035

Klesov, O. I.

Theory Probab. Math. Stat. 33, 27-33 (1986); translation from Teor. Veroyatn. Mat. Stat. 33, 25-31 (1985).

This paper is concerned with conditions for the convergence of the random Fourier transforms \(\zeta (t)=\sum \eta_ n\exp (i\lambda_ nt)\) where \(\{\eta_ n,n\geq 1\}\) are random variables with E \(\eta\) \({}_ n=0\), E \(\eta\) \({}^ 2_ n<\infty\) for each n and \(\{\lambda_ n,n\geq 1\}\) are real numbers with \(| \lambda_ n| \to \infty\) as \(n\to \infty.\)
Sufficient conditions for convergence in terms of the covariance structure of the \(\{\eta_ n\}\) process are given together with some rate of convergence results for the particular case where the \(\eta\) ’s are uncorrelated.

Reviewer: C C.Heyde

MSC:

60F15	Strong limit theorems
42A20	Convergence and absolute convergence of Fourier and trigonometric series
41A25	Rate of convergence, degree of approximation

Keywords:

convergence of the random Fourier transforms; conditions for convergence in terms of the covariance structure