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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Fernique, X.: Fonctions aléatoires gaussiennes, vecteurs aléatoires gaussiens. Université de Montréal, Centre de Recherches Mathématiques, Montreal (1977). iv+217 pp
Ledoux, M., Talagrand, M.: Probability in a Banach Space: Isoperimetry and Processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 23. Springer, Berlin (1991). xii+480 pp. ISBN: 3-540-52013-9
Marcus, M.B.: ξ-Radial Processes and Random Fourier Series. Mem. Amer. Math. Soc., vol. 68(368) (1987). viii+181 pp
Marcus, M.B., Pisier, G.: Random Fourier Series with Applications to Harmonic Analysis. Annals of Mathematics Studies, vol. 101. Princeton University Press, Princeton; University of Tokyo Press, Tokyo (1981). v+151 pp. ISBN: 0-691-08289-8; 0-691-08292-8
Marcus, M.B., Pisier, G.: Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes. Acta Math. 152(3–4), 245–301 (1984)
Paley, R., Zygmund, A.: On some series of functions (1). Proc. Camb. Philos. Soc. 26, 337–357 (1930)
Paley, R., Zygmund, A.: On some series of functions (2). Proc. Camb. Philos. Soc. 26, 458–474 (1930)
Paley, R., Zygmund, A.: On some series of functions (3). Proc. Camb. Philos. Soc. 28, 190–205 (1932)
Salem, R., Zygmund, A.: Some properties of trigonometric series whose terms have random signs. Acta Math. 91, 245–301 (1954)
Talagrand, M.: Necessary and sufficient conditions for sample continuity of random Fourier series and of harmonic infinitely divisible processes. Ann. Probab. 20, 1–28 (1992)
Talagrand, M.: A borderline random Fourier series. Ann. Probab. 23, 776–785 (1995)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-642-54075-2_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-54074-5
Online ISBN: 978-3-642-54075-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)