A note on strong limit theorems for arbitrary stochastic sequences

The purpose of this paper is to establish a strong limit theorem of the Dubins–Freedman type for arbitrary stochastic sequences. This generalizes a result by Isaac. Our theorem on growth rate for arbitrary stochastic sequences generalizes a result by Freedman and a result by Petrov. We also establish a theorem for a sequence of independent, symmetric random variables which generalizes another result by Freedman.