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Strong invariance principles with rate for “reverse” martingale differences and applications. (English) Zbl 1353.60031

This paper is concerned with ‘almost sure invariance principles’ of the form: Given a sequence \((X_k)\) of random variables, establish the existence of a sequence \((Z_k)\) of i.i.d. Gaussian random variables such that \[ \sup_{1\leq m\leq n}\Bigl|\sum_{k=0}^{m-1}X_k-Z_k\Bigr|=o(b_n) \quad\text{almost surely}, \] where ‘the rate of order’ \((b_n)\) is a non-decreasing sequence tending to infinity ‘as slow as possible’ (\(b_n=(n\log\log n)^{1/2}\) is the classical Strassen invariance principle). This is investigated for \((X_k)\) consisting of square integrable “reverse martingale differences”, yielding rates of order of the form \(n^{1/p}\log^\beta n\) for suitable \(2<p\leq 4\) and \(\beta>0\). Applications include certain classes of piecewise expanding maps of the interval and of uniformly expanding maps on compact Riemannian manifolds.

MSC:

60F17 Functional limit theorems; invariance principles
30C30 Schwarz-Christoffel-type mappings
37E05 Dynamical systems involving maps of the interval
60G48 Generalizations of martingales
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