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On random Fourier series


Authors: Jack Cuzick and Tze Leung Lai
Journal: Trans. Amer. Math. Soc. 261 (1980), 53-80
MSC: Primary 60G17; Secondary 42A20, 60F15
DOI: https://doi.org/10.1090/S0002-9947-1980-0576863-8
MathSciNet review: 576863
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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.


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  • [1] L. Breiman, Probability, Addison-Wesley, Reading, Mass., 1968. MR 0229267 (37:4841)
  • [2] L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135-157. MR 0199631 (33:7774)
  • [3] A. Erdélyi, Tables of integral transforms, Vol. 1, McGraw Hill, New York, 1953.
  • [4] G. H. Hardy and W. W. Rogonsinski, Notes on Fourier series. V: Summability $ ({R_1})$, Proc. Cambridge Philos. Soc. 45 (1949), 173-185. MR 0028979 (10:528f)
  • [5] G. A. Hunt, Random Fourier transforms, Trans. Amer. Math. Soc. 71 (1951), 38-69. MR 0051340 (14:465b)
  • [6] N. C. Jain, K. Jogdeo and W. F. Stout, Upper and lower functions for martingales and mixing processes, Ann. Probability 3 (1975), 119-145. MR 0368130 (51:4372)
  • [7] N. 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), 117-141. MR 0413239 (54:1356)
  • [8] -, Integrability of infinite sums of independent vector-valued random variables, Trans. Amer. Math. Soc. 212 (1975), 1-36. MR 0385995 (52:6854)
  • [9] J. P. Kahane, Propriétés locales des fonctions à séries de Fourier aléatories, Studia Math. 19 (1960), 1-25. MR 0117506 (22:8285)
  • [10] -, Some random series of functions, Heath, Lexington, Mass., 1968.
  • [11] M. Klass, Toward a universal law of the iterated logarithm, Part I, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36 (1976), 165-178. MR 0415742 (54:3822)
  • [12] B. Kuttner, The relation between Riemann and Cesaro summability, Proc. London Math. Soc. 38 (1935), 273-283.
  • [13] -, Some relations between different kinds of Riemann summability, Proc. London Math. Soc. 40 (1936), 524-540.
  • [14] T. L. Lai, Summability methods for independent, identically distributed random variables, Proc. Amer. Math. Soc. 45 (1974), 253-261. MR 0356194 (50:8665)
  • [15] M. Loève, Probability theory, Van Nostrand, Princeton, N. J., 1962.
  • [16] J. Marcinkiewicz and A. Zygmund, Sur les fonctions indépendantes, Fund. Math. 29 (1937), 60-90.
  • [17] M. B. Marcus, A comparison of continuity conditions for Gaussian processes, Ann. Probability 1 (1973), 123-130. MR 0346885 (49:11606)
  • [18] -, Uniform convergence of random Fourier series, Ark. Mat. 13 (1975), 107-122. MR 0372994 (51:9196)
  • [19] -, Continuity and the central limit theorem for random trigonometric series, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 42 (1978), 35-56. MR 486035 (80c:60037)
  • [20] M. B. Marcus and G. Pisier, Necessary and sufficient conditions for the uniform convergence of random trigonometric series, Lecture Notes Series No. 50, Aarhus Univ., Denmark, 1978. MR 507061 (80m:42009)
  • [21] -, Random Fourier series on locally compact Abelian groups, Séminaire de Probabilités (P. A. Meyer, Ed.), Lecture Notes in Math., vol. 709, Springer-Verlag, Berlin and New York, 1979, pp. 72-89. MR 544781 (81m:60063)
  • [22] M. Nisio, On the extreme values of Gaussian processes, Osaka J. Math. 4 (1967), 313-326. MR 0226722 (37:2309)
  • [23] R. E. A. C. Paley and A. Zygmund, On some series of functions. I, II, III, Proc. Cambridge Philos. Soc. 26 (1930), 337-357; 26 (1930), 458-474; 28 (1932), 190-205.
  • [24] C. Qualls and H. Watanabe, An asymptotic 0-1 behavior of Gaussian processes, Ann. Math. Statist. 42 (1971), 2029-2035. MR 0307317 (46:6437)
  • [25] R. Salem and A. Zygmund, Some properties of trigonometric series whose terms have random signs, Acta Math. 91 (1954), 245-301. MR 0065679 (16:467b)
  • [26] T. Sirao and H. Watanabe, On the upper and lower class for stationary Gaussian processes, Trans. Amer. Math. Soc. 147 (1970), 301-331. MR 0256455 (41:1111)
  • [27] V. Strassen, Almost sure behavior of sums of independent random variables and martingales, Proc. Fifth Berkeley Sympos. Math. Statist. Probability, Vol. 2, Univ. of California Press, Berkeley, Calif., 1965, pp. 315-343. MR 0214118 (35:4969)
  • [28] A. Zygmund, Trigonometric series. Vol. 1, Cambridge Univ. Press, Cambridge, Mass., 1959. MR 0107776 (21:6498)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0576863-8
Keywords: Riemann's $ {R_1}$ summability, convergence, continuity, unboundedness, law of the iterated logarithm, random Fourier series
Article copyright: © Copyright 1980 American Mathematical Society

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