On Probabilities of Large Deviations for Random Walks. I. Regularly Varying Distribution Tails

We establish first-order approximations and asymptotic expansions for probabilities of crossing arbitrary curvilinear boundaries in the large deviations range by random walks with regularly varying distribution tails. In particular, we study the large deviations probabilities for the sums and maxima of partial sums of independent and identically distributed random variables, including the asymptotic behavior of the densities when they exist. Extensions to the "regular exponential" case (when the distribution tail differs from the exponential one by a regularly varying factor) are considered in part II of the paper.