Iterates of Volterra

Given k ∈ L(0, 1) satisfying certain smoothness and growth conditions at 0, we consider the Volterra convolution operator Vk defined on L(0, 1) by (Vku)(t) = ∫ t 0 k(t − s)u(s) ds, and its iterates (V n k )n∈N . We construct some much simpler sequences which, as n → ∞, are asymptotically equal in the operator norm to V n k . This leads to a simple asymptotic formula for ‖V n k ‖ and to a simple ‘asymptotically extremal sequence’; that is, a sequence (un) in L(0, 1) with ‖un‖p = 1 and ‖V n k un‖ ∼ ‖V n k ‖ as n → ∞. As an application, we derive a limit theorem for large deviations, which appears to be beyond the established theory.