A multiple stochastic integral criterion for almost sure limit theorems

In this paper, we study almost sure central limit theorems for multiple stochastic integrals and provide a criterion based on the kernel of these multiple integrals. We apply our result to normalized partial sums of Hermite polynomials of increments of fractional Brownian motion. We obtain almost sure central limit theorems for these normalized sums when they converge in law to a normal distribution.