Let {X k , k ≥ 1} be a sequence of independently and identically distributed random variables with common subexponential distribution function concentrated on (−∞, ∞), and let τ be a nonnegative and integer-valued random variable with a finite mean and which is independent of the sequence {X k , k ≥ 1}. This paper investigates asymptotic behavior of the tail probabilities P(· > x) and the local probabilities P(x < · ≤ x + h) of the quantities and for n ≥ 1, and their randomized versions X (τ), S τ and S (τ), where X 0 = 0 by convention and h > 0 is arbitrarily fixed.