How to Estimate the Rate Function of a Cumulative Process

Consider two sequences of bounded random variables, a value and a timing process, that satisfy the large deviation principle (LDP) with rate function J (·, ·) and whose cumulative process satisfies the LDP with rate function I (·). Under mixing conditions, an LDP for estimates of I constructed by transforming an estimate of J is proved. For the case of a cumulative renewal process it is demonstrated that this approach is favourable to a more direct method, as it ensures that the laws of the estimates converge weakly to a Dirac measure at I .