Random Fourier series with Dependent Random Variables
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Lithuanian Mathematical Journal
Random Fourier series with dependent random variables Safari Mukeru Department of Decision Sciences, University of South Africa, P.O. Box 392, Pretoria, 0003, South Africa (e-mail: [email protected]) Received January 24, 2020; revised June 18, 2020
Abstract. Given a sequence of independent standard Gaussian variables (Zn ), the classical Pisier algebra P is the class of all continuous functions f on the unit circle T such that for each t ∈ T, the random Fourier series n∈Z Zn fˆ(n) × exp(2πint) converges in L2 and the corresponding sums constitute a Gaussian process that admits a continuous version. It was constructed by Pisier in 1979 to answer a long-standing question raised by Katznelson. In this paper, we consider the general random Fourier series n∈Z ξn fˆ(n) exp(2πint) where ξ = (ξn ) is a discrete Gaussian process of standard Gaussian random variables but with the restriction of independence relaxed and study the corresponding class P(ξ) of continuous functions f on T. We obtain sufficient conditions (based on some spectral properties of the covariance matrix of (ξn )) for each of the relations P ⊂ P(ξ), P(ξ) ⊂ P, and P = P(ξ). We illustrate these results by the classical fractional Gaussian noise. Whether in general P(ξ) is also a Banach algebra is an open problem. MSC: 46J10, 42A20, 60B15, 60G10 Keywords: random Fourier series, fractional Gaussian noise, dependent random variables, Pisier algebra
1 Introduction Probability methods have rendered great services to harmonic analysis in the sense that, as explained by Kahane, “in many circumstances it is hard or even impossible to find a mathematical object with some prescribed properties, and pretty easy to exhibit a random object which enjoys these properties almost surely.” In many instances, such random objects are constructed using a sequence of independent identically distributed (i.i.d) random variables, and in return the original mathematical object appears as a property of the sequence of i.i.d random variables used in its construction. It is therefore interesting to know if indeed such a property only holds for sequences of i.i.d random variables or, equivalently, if dependent random variables can also be used to build the mathematical object. Consider the following old problem raised by Katznelson (see Kahane [5, p. 504]): find a homogeneous Banach algebra B strictly contained in C(T) (the class of all continuous functions on the unit circle T = R/Z) and strictly containing the classical Wiener algebra A(T) (the class of continuous functions on T with absolutely convergent Fourier series) such that not all continuous functions on T operate in B and not only analytic functions on T operate in B. Initially, Zafran [12] provided a solution using some difficult constructions, but the Banach algebra obtained by Pisier [9] in 1979 using a sequence of i.i.d standard Gaussian random variables is striking by its simplicity and elegance. It is commonly referred to as the Pisier algebra and denoted thereafter as P. To defi
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