Strong Approximations for Nonconventional Sums with Applications to Law of Iterated Logarithm and Almost Sure Central Limit Theorem

We obtain a strong invariance principle for nonconventional sums and applying this result we derive for them a version of the law of iterated logarithm, 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.