Abstract
In Chapter 7 we return to the study of random Fourier series, but now without making any assumption of integrability on the random coefficients, which we simply assumed to be independent symmetric r.v.s. This chapter also develops one of the fundamental ideas of this work: many processes can be exactly controlled, not by using one or two distances, but by using an entire family of distances. With these tools, we are able to give in full generality necessary and sufficient conditions for convergence of random Fourier series. These conditions can be formulated in words by saying that convergence is equivalent to the finiteness of (a proper generalization of) a certain “entropy integral”. We then give examples of application of the abstract theorems to the case of ordinary random Fourier series.
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Talagrand, M. (2014). Random Fourier Series and Trigonometric Sums, II. In: Upper and Lower Bounds for Stochastic Processes. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 60. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54075-2_7
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DOI: https://doi.org/10.1007/978-3-642-54075-2_7
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