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The strongp-variation of martingales and orthogonal series
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  • Published: December 1988

The strongp-variation of martingales and orthogonal series

  • Gilles Pisier1,2 &
  • Quanhua Xu1,2 

Probability Theory and Related Fields volume 77, pages 497–514 (1988)Cite this article

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Summary

Let 1≦p<∞ and letx=(x n)n≧0 be a sequence of scalars. The strongp-variation ofx, denoted byW p (x), is defined as

$$W_p (x) = \sup \left\{ {\left( {|x_0 |^p + \sum\limits_{k = 1}^\infty {|x_{n_k } - x_{n_{k - 1} } |^p } } \right)^{1/p} } \right\}$$

where the supremum runs over all increasing sequences of integers 0=n 0 ≦n 1 ≦n 2 ≦...

Let 1≦p<2 and letM=(M n ) n≧0 be a martingale inL p . Our main results are as follows: If\(\Sigma \mathbb{E}|M_n - M_{n - 1} |^p< \infty \), thenW p (M) is finite a.s. and we have

$$\mathbb{E}W_p (M)^p \leqq C(\mathbb{E}|M_0 |^p + \sum\limits_{n \geqq 1} {\mathbb{E}|M_n - M_{n - 1} |^p )} $$

for some constantC depending only onp. On the other hand, let (ϕ n be an arbitrary orthonormal system of functions inL 2, considerx=(x n ) n≧0 inl 2 and letS n =Σ n0 x i ϕ i andS=(S n ) n≧0. We prove that ifΣ|x n |p<∞ (1≦p<2) thenW p (S(t))<∞ for a.e.t and ∥W p (S)∥2≦C(Σ|x n |p)1/p for some constantC. Each of these results is an extension of a result proved by Bretagnolle for sums of independent mean zero r.v.'s. The casep>2 in also discussed. Our proofs use the real interpolation method of Lions-Peetre. They admit extensions in the Banach space valued case, provided suitable assumptions are imposed on the Banach space.

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Authors and Affiliations

  1. Texas A & M University, 77843, College Station, TX

    Gilles Pisier & Quanhua Xu

  2. Equipe D'analyse, Wuhan University and Université Paris VI, Tour 46, 4ème Etage, 4, Place Jussieu, F-75252, Paris Cedex 05, France

    Gilles Pisier & Quanhua Xu

Authors
  1. Gilles Pisier
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  2. Quanhua Xu
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Additional information

Partially supported by N.S.F. Grant No. DMS-8500764

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Pisier, G., Xu, Q. The strongp-variation of martingales and orthogonal series. Probab. Th. Rel. Fields 77, 497–514 (1988). https://doi.org/10.1007/BF00959613

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  • Received: 20 March 1987

  • Revised: 19 November 1987

  • Issue Date: December 1988

  • DOI: https://doi.org/10.1007/BF00959613

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Keywords

  • Banach Space
  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Interpolation Method
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