Given k ∈ L¹ (0,1) satisfying certain smoothness and growth conditions at 0, we consider the Volterra convolution operator V_k defined on L^p (0,1) by

(Vku)(t) = ò0t k(t-s)u(s)\textds,(V_{k}u)(t)= \int_{0}^{t} {k(t-s)u(s){\text{d}}s},

and its iterates (V_kⁿ)_{n Î \mathbbN}.(V_{k}^{n})_{n \in {\mathbb{N}}}. We construct some much simpler sequences which, as n → ∞, are asymptotically equal in the operator norm to V_kⁿ. This leads to a simple asymptotic formula for ||V_kⁿ|| and to a simple ‘asymptotically extremal sequence’; that is, a sequence (u_n) in L^p (0, 1) with ||u_n||_p=1 and ||V_kⁿ u_n|| ~ ||V_kⁿ||||V_{k}^{n} u_{n}|| \sim ||V_{k}^{n}|| as n → ∞. As an application, we derive a limit theorem for large deviations, which appears to be beyond the established theory.