Large deviations for martingales and derivatives

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:366

Article

Butler, S.¹  Pavlov, S.²  Rosenblatt, J.¹ 

[1] Univ Illinois, Dept Math, Urbana, IL 61801 USA

[2] Renaissance Technol LLC, E Setauket, NY 11733 USA

关键词: Martingales;    Lebesque derivatives;    Large deviations;   

DOI  :  10.1016/j.jmaa.2009.12.054

来源: Elsevier

【 摘 要 】

Fix a sequence of positive integers (m(n)) and a sequence of positive real numbers (w(n)). Two closely related sequences of linear operators (T(n)) are considered. One sequence has T(n) : L(1)(R) -> L(1) (R) given by the Lebesgue derivatives T(n)f (x) = D(n)f (x) = 2(n) integral(1/2n)(0) f(x + t)dt. The other sequence has T(n) : L(1) inverted right perpendicular0, 1) -> L(1)inverted right perpendicular0, 1) given by the dyadic martingale T(n)f (x) = E(f vertical bar beta(n))(x) = 2(n) integral(l/2n)((l-1)/2n) f (t)dt when (l - 1)/2(n) <= x < 1/2(n) for l = 1, ... , 2(n). We prove both positive and negative results concerning the convergence of Sigma(infinity)(n=1)m{vertical bar T(mn) f(x)vertical bar >= w(n)}. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

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