On random Fourier series

Cuzick, Jack; Lai, Tze Leung

doi:10.1090/S0002-9947-1980-0576863-8

On random Fourier series
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by Jack Cuzick and Tze Leung Lai PDF

Trans. Amer. Math. Soc. 261 (1980), 53-80 Request permission

Abstract:

Motivated by Riemann’s ${R_1}$ summation method for i.i.d. random variables ${X_1}, {X_2}, \ldots$, this paper studies random Fourier series of the form $\sum \nolimits _1^\infty {{a_n}{X_n} \sin (nt + {\Phi _n})}$, where $\{ {a_n}\}$ is a sequence of constants and $\{ {\Phi _n}\}$ is a sequence of independent random variables which are independent of $\{ {X_n}\}$. Questions of continuity and of unboundedness are analyzed through the interplay between the asymptotic properties of $\{ {a_n}\}$ and the tail distribution of ${X_1}$. A law of the iterated logarithm for the local behavior of the series is also obtained and extends the classical result for Brownian motion to a general class of random Fourier series.

References

Leo Breiman, Probability, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1968. MR 0229267
Lennart Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135–157. MR 199631, DOI 10.1007/BF02392815

Tables of integral transforms

G. H. Hardy and W. W. Rogosinski, Notes on Fourier series. V. Summability $(R_1)$, Proc. Cambridge Philos. Soc. 45 (1949), 173–185. MR 28979, DOI 10.1017/s0305004100024695
G. A. Hunt, Random Fourier transforms, Trans. Amer. Math. Soc. 71 (1951), 38–69. MR 51340, DOI 10.1090/S0002-9947-1951-0051340-3
Naresh C. Jain, Kumar Jogdeo, and William F. Stout, Upper and lower functions for martingales and mixing processes, Ann. Probability 3 (1975), 119–145. MR 368130, DOI 10.1214/aop/1176996453
Naresh C. Jain and M. B. Marcus, Sufficient conditions for the continuity of stationary Gaussian processes and applications to random series of functions, Ann. Inst. Fourier (Grenoble) 24 (1974), no. 2, vi, 117–141 (English, with French summary). MR 413239
Naresh C. Jain and Michael B. Marcus, Integrability of infinite sums of independent vector-valued random variables, Trans. Amer. Math. Soc. 212 (1975), 1–36. MR 385995, DOI 10.1090/S0002-9947-1975-0385995-7
J.-P. Kahane, Propriétés locales des fonctions à séries de Fourier aléatoires, Studia Math. 19 (1960), 1–25 (French). MR 117506, DOI 10.4064/sm-19-1-1-25

Some random series of functions

M. J. Klass, Toward a universal law of the iterated logarithm. I, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36 (1976), no. 2, 165–178. MR 415742, DOI 10.1007/BF00533999

The relation between Riemann and Cesaro summability

38

Some relations between different kinds of Riemann summability

40

Tze Leung Lai, Summability methods for independent identically distributed random variables, Proc. Amer. Math. Soc. 45 (1974), 253–261. MR 356194, DOI 10.1090/S0002-9939-1974-0356194-4

Probability theory

Sur les fonctions indépendantes

29

M. B. Marcus, A comparison of continuity conditions for Gaussian processes, Ann. Probability 1 (1973), no. 1, 123–130. MR 346885, DOI 10.1214/aop/1176997028
Michael B. Marcus, Uniform convergence of random Fourier series, Ark. Mat. 13 (1975), 107–122. MR 372994, DOI 10.1007/BF02386200
Michael B. Marcus, Continuity and the central limit theorem for random trigonometric series, Z. Wahrsch. Verw. Gebiete 42 (1978), no. 1, 35–56. MR 486035, DOI 10.1007/BF00534206
M. B. Marcus and G. Pisier, Necessary and sufficient conditions for the uniform convergence of random trigonometric series, Lecture Notes Series, vol. 50, Aarhus Universitet, Matematisk Institut, Aarhus, 1978. Lecture 1977/78. MR 507061
M. B. Marcus and G. Pisier, Random Fourier series on locally compact abelian groups, Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78) Lecture Notes in Math., vol. 721, Springer, Berlin, 1979, pp. 72–89. MR 544781
Makiko Nisio, On the extreme values of Gaussian processes, Osaka Math. J. 4 (1967), 313–326. MR 226722

On some series of functions

26

28

Clifford Qualls and Hisao Watanabe, An asymptotic $0-1$ behavior of Gaussian processes, Ann. Math. Statist. 42 (1971), 2029–2035. MR 307317, DOI 10.1214/aoms/1177693070
R. Salem and A. Zygmund, Some properties of trigonometric series whose terms have random signs, Acta Math. 91 (1954), 245–301. MR 65679, DOI 10.1007/BF02393433
Tunekiti Sirao and Hisao Watanabe, On the upper and lower class for stationary Gaussian processes, Trans. Amer. Math. Soc. 147 (1970), 301–331. MR 256455, DOI 10.1090/S0002-9947-1970-0256455-4
Volker Strassen, Almost sure behavior of sums of independent random variables and martingales, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 315–343. MR 0214118
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776