Uniform bounds under increment conditions

Author:
Michel Weber

Journal:
Trans. Amer. Math. Soc. **358** (2006), 911-936

MSC (2000):
Primary 60F99; Secondary 28D99

Published electronically:
June 9, 2005

MathSciNet review:
2177045

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We apply a majorizing measure theorem of Talagrand to obtain uniform bounds for sums of random variables satisfying increment conditions of the type considered in Gál-Koksma Theorems. We give some applications.

**[B]**Berkes I. (*private communication*).**[BW]**Fakhreddine Bouhkari and Michel Weber,*Almost sure convergence of weighted series of contractions*, Illinois J. Math.**46**(2002), no. 1, 1–21. MR**1936072****[CL]**Guy Cohen and Michael Lin,*Laws of large numbers with rates and the one-sided ergodic Hilbert transform*, Illinois J. Math.**47**(2003), no. 4, 997–1031. MR**2036987****[DL]**Yves Derriennic and Michael Lin,*Fractional Poisson equations and ergodic theorems for fractional coboundaries*, Israel J. Math.**123**(2001), 93–130. MR**1835290**, 10.1007/BF02784121**[Ga]**V. F. Gaposhkin,*The dependence of the rate of convergence in the strong law of large numbers for stationary processes on the rate of diminution of the correlation function*, Teor. Veroyatnost. i Primenen.**26**(1981), no. 4, 720–733 (Russian, with English summary). MR**636767****[GK]**I. S. Gál and J. F. Koksma,*Sur l’ordre de grandeur des fonctions sommables*, Nederl. Akad. Wetensch., Proc.**53**(1950), 638–653 = Indagationes Math. 12, 192–207 (1950) (French). MR**0036291****[K]**Ulrich Krengel,*Ergodic theorems*, de Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. MR**797411****[LW]**Mikhail Lifshits and Michel Weber,*Spectral regularization inequalities*, Math. Scand.**86**(2000), no. 1, 75–99. MR**1738516****[S]**Dominique Schneider,*Théorèmes ergodiques perturbés*, Israel J. Math.**101**(1997), 157–178 (French, with English summary). MR**1484874**, 10.1007/BF02760927**[Tal1]**Talagrand M. [1996]*Convergence of Orthogonal Series Using Stochastic Processes*(http: //www.math.ohio-state.edu/ talagran/preprints).**[Tal2]**Michel Talagrand,*Sample boundedness of stochastic processes under increment conditions*, Ann. Probab.**18**(1990), no. 1, 1–49. MR**1043935****[Tan]**Tandori K. [1957]*Zur Divergenz der Orthogonal Reihe*, Acta Sci. Math. Szeged**18**p. 57-130.**[W1]**Michel Weber,*Estimating random polynomials by means of metric entropy methods*, Math. Inequal. Appl.**3**(2000), no. 3, 443–457. MR**1768824**, 10.7153/mia-03-44**[W2]**Michel Weber,*Some theorems related to almost sure convergence of orthogonal series*, Indag. Math. (N.S.)**11**(2000), no. 2, 293–311. MR**1813729**, 10.1016/S0019-3577(00)89085-0**[W3]**M. Weber,*Some examples of application of the metric entropy method*, Acta Math. Hungar.**105**(2004), no. 1-2, 39–83. MR**2093929**, 10.1023/B:AMHU.0000045531.17229.d4

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Additional Information

**Michel Weber**

Affiliation:
Mathématique (IRMA), Université Louis-Pasteur et CNRS, 7, rue René Descartes, 67084 Strasbourg Cedex, France

Email:
weber@math.u-strasbg.fr

DOI:
https://doi.org/10.1090/S0002-9947-05-03805-5

Keywords:
Almost sure convergence,
majorizing measure,
maximal inequality,
G\'al-Koksma inequality,
ergodic sums

Received by editor(s):
March 29, 2003

Received by editor(s) in revised form:
April 21, 2004

Published electronically:
June 9, 2005

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.