A note on a strong renewal theorem

Let { S n , n ≥ 1} be a sequence of partial sums of independent and identically distributed non‐negative random variables with a common distribution function F. Let F belong to the domain of attraction of a stable law with exponent α, 0 < α < 1. Suppose H(t) = ϵ N(t), t ⩾ 0, where N(t) = max(n : S n ≥ t ). Under some additional assumptions on F , the difference between H(t) and its asymptotic value as t → ∞ is estimated.