On Billard's theorem for random Fourier series

• Autorzy: Cohen G. , Cuny C.

Identyfikatory

DOI

10.4064/ba53-1-5

• Języki publikacji: EN

Abstrakty

EN

We show that Billard's theorem on a.s. uniform convergence of random Fourier series with independent symmetric coefficients is not true when the coefficients are only assumed to be centered independent. We give some necessary or sufficient conditions to ensure the validity of Billard's theorem in the centered case.

Słowa kluczowe

EN

Banach valued random variables; random Fourier series; almost sure uniform convergence; Billard's theorem

PL

zmienne losowe o wartościach z przestrzeni Banacha; szeregi Fouriera losowe; zbieżność jednostajna prawie pewna; twierdzenie Billarda

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2005
• Tom: Vol. 53, no 1
• Strony: 39--53
• Opis fizyczny: Bibliogr. 19 poz.

Twórcy

autor

Cohen G.

• Department of Mathematics, Ben Gurion University, P.O.B. 653, 841105 Beer Sheva, Israel
• Erwin Schrödinger Institute, Boltzmanngasse 9, A-1090 Wien, Austria

autor

Cuny C.

• Department of Mathematics, University of New Caledonia, Equipe ERIM, B.P. 4477, F-98847 Noumea Cedex, France

Bibliografia

• [1] P. Billard, Séries de Fourier aléatoirement bornées, continues, uniformément convergentes, Ann. Sci. École Norm. Sup. (3) 82 (1965), 131-179.
• [2] J. Cuzick and T. L. Lai, On random Fourier series, Trans. Amer. Math. Soc. 261 (1980), 53-80.
• [3] W. Feller, An Introduction to Probability Theory and its Applications, Vol. II, 2nd ed., Wiley, New York, 1971.
• [4] X. Fernique, Séries de Fourier aléatoires, C. R. Acad. Sci. Paris Sér. I 322 (1996), 485-488.
• [5] J. Hoffman-Jørgensen, Sums of independent Banach space valued random variables, Studia Math. 52 (1974), 159-186.
• [6] K. Itô and M. Nisio, On the convergence of sums of independent Banach space valued random variables, Osaka Math. J. 5 (1968), 35-48.
• [7] N. C. Jain and M. B. Marcus, Integrability of infinite sums of independent vector-valued random variables, Trans. Amer. Math. Soc. 212 (1975), 1-36.
• [8] J-P. Kahane, Some Random Series of Functions, 2nd ed., Cambridge Univ. Press, 1985 (1st ed.: D. C. Heath, Lexington, MA, 1968).
• [9] M. Ledoux and M. Talagrand, Probability in Banach Spaces, Ergeb. Math. Grenzgeb. 23, Springer, Berlin, 1991.
• [10] J. Marcinkiewicz et A. Zygmund, Sur les fonctions indépendantes, Fund. Math. 29 (1937), 60-90.
• [11] M. B. Marcus, Uniform convergence of random Fourier series, Ark. Mat. 13 (1975), 107-122.
• [12] M. B. Marcus and G. Pisier, Random Fourier Series with Applications to Harmonie Analysis, Ann. of Math. Stud. 101, Princeton Univ. Press, Princeton, 1981.
• [13] —, —, Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes, Acta Math. 152 (1984), 245-301.
• [14] R. E. A. C. Paley and A. Zygmund, On some series of functions, part I, Proc. Cambridge Philos. Soc. 26 (1930), 337-357; part II, ibid. 26 (1930), 458-474; part III, ibid. 28 (1932), 190-205.
• [15] F. Riesz and B. Sz.-Nagy, Functional Analysis, translated from the 2nd French edition by Leo F. Boron, Dover, New York, 1990.
• [16] R. Salem and A. Zygmund, Some properties of trigonometric series whose terms have random signs, Acta Math. 91 (1954), 245-301.
• [17] M. Talagrand, A borderline random Fourier series, Ann. Probab. 23 (1995), 776-785.
• [18] M. Weber, Estimating random polynomials by means of metric entropy methods, Math. Inequal. Appl. 3 (2000), 443-457.
• [19] A. Zygmund, Trigonometric Series, corrected 2nd ed., Cambridge Univ. Press, 1968.