A strong invariance principle for nonconventional sums

In Kifer and Varadhan (Nonconventional limit theorems in discrete and continuous time via martingales, 2010) we obtained a functional central limit theorem (known also as a weak invariance principle) for sums of the form ${\sum_{n=1}^{Nt]} F\big(X(n), X(2n), .\, .\, .\, .\, X(kn), X(q_{k+1}(n)), X(q_{k+2}(n)), .\, .\, .\, , X(q_\ell(n))\big)}$ (normalized by ${1/\sqrt N}$ ) where X(n), n ≥ 0 is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, F is a continuous function with polynomial growth and certain regularity properties and q _i , i > k are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when q _i ’s are polynomials of growing degrees. This paper deals with strong invariance principles (known also as strong approximation theorems) for such sums which provide their uniform in time almost sure approximation by processes built out of Brownian motions with error terms growing slower than ${\sqrt N}$ . This yields, in particular, an invariance principle in the law of iterated algorithm for the above sums. Among motivations for such results are their applications to multiple recurrence for stochastic processes and dynamical systems as well, as to some questions in metric number theory and they can be considered as a natural follow up of a series of papers dealing with nonconventional ergodic averages.