Asymptotic Behaviour of Iterates of Volterra Operators on L p (0, 1)

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 and its iterates 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 as n → ∞. As an application, we derive a limit theorem for large deviations, which appears to be beyond the established theory.