Deviation inequalities for Banach space valued martingales differences sequences and random fields

Abstract

We establish deviation inequalities for the maxima of partial sums of a martingale differences sequence, and of an orthomartingale differences random field. These inequalities can be used to give rates for linear regression and the law of large numbers.

Limit theorems for U-statistics of Bernoulli data

“…We end the Appendix by two lemmas on conditional expectation. Proposition A.1 (Theorem 1.3 in Giraudo, 2019). Let 1 < p < 2 and q > p. Then there exists constants c 1 and c 2 depending only on {p, q} such that if (d i ) n i=1 is a martingale differences sequence with respect to a filtration (F i ) n i=1 , then for each integer n and each positive x,…”

Weak invariance principle in some Besov spaces for stationary martingale differences

“…Lemma 3.5 (Proposition 3.5 in [5] ). For any q > 2, there exists a constant c(q) such that if f • T i i 0 is a martingale differences sequence with respect to the filtration T −i F 0 i 0 then for each integer n 1,…”

An exponential inequality for orthomartingale differences random fields and some applications

“…However, for rates in the law of large numbers and the functional central limit theorem in Hölder spaces, only a few results are available in the literature. Indeed, rates on the law of large number for orthomartingales with polynomial moment have been given in [14,18,19], but it seems that the question of exponential moments was only addressed for martingale difference sequences. For the functional central limit theorem in Hölder spaces for random fields, the i.i.d.…”