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Large deviations for martingales and derivatives. (English) Zbl 1214.60014

The authors consider large deviations principle for two closely related sequences of linear operators \(T_n\) on \(L_1(\mathbb R):\) one is given by the Lebesgue derivatives and the other one by the dyadic martingale.
They prove both positive and negative results concerning the convergence of
\[ \sum_{n=1}^{\infty}m \big\{|T_{m_n}f(x)|\geq w_n\big\}, \]
where \(m_n\) are positive integer numbers, \(w_n\) are positive real numbers.

MSC:

60G42 Martingales with discrete parameter
47B39 Linear difference operators
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References:

[1] Hare, K.; Stokolos, A., On weak type inequalities for rare maximal functions, Colloq. Math., 83, 2, 173-182 (2000) · Zbl 1030.42017
[2] Jones, R.; Kaufman, R.; Rosenblatt, J.; Wierdl, M., Oscillation in ergodic theory, Ergodic Theory Dynam. Systems, 18, 889-935 (1998) · Zbl 0924.28009
[3] Rosenblatt, J.; Wierdl, M., A new maximal inequality and its applications, Ergodic Theory Dynam. Systems, 12, 509-558 (1992) · Zbl 0757.28015
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