Strong approximations for nonconventional sums and almost sure limit theorems

We improve, first, a strong invariance principle from Kifer (2013) [10] for nonconventional sums of the form ∑ n = 1 [ N t ] F ( X ( n ) , X ( 2 n ) , … , X ( ℓ n ) ) (normalized by 1 / N ) where X ( n ) , n ≥ 0 ’s is a sufficiently fast mixing vector process with some moment conditions and stationarity properties and F satisfies some regularity conditions. Applying this result we obtain next a version of the law of iterated logarithm for such sums, as well as an almost sure central limit theorem. Among motivations for such results are their applications to multiple recurrence for stochastic processes and dynamical systems.