Let ξ 1 , ξ 2 , ... be independent, identically distributed random variables with infinite mean E [ | ξ 1 | ] = ∞. Consider a random walk S n = ξ 1 + ⋯ + ξ n , a stopping time τ = min { n ≥ 1 : S n ≤ 0 } and let M τ = max 0 ≤ i ≤ τ S i . We study the asymptotics for P (M τ > x) , as x → ∞ .