Let ( Y n , N n ) n ≥1 be independent and identically distributed bivariate random variables such that the N n are positive with finite mean ν and the Y n have a common heavy-tailed distribution F . We consider the process ( Z n ) n ≥1 defined by Z n = Y n - Σ n -1 , where It is shown that the probability that the maximum M = max n ≥1 Z n exceeds x is approximately as x → ∞, where F ' := 1 - F . Then we study the integrated tail of the maximum of a random walk with long-tailed increments and negative drift over the interval [0, σ], defined by some stopping time σ, in the case in which the randomly stopped sum is negative. Finally, an application to risk theory is considered.