Asymptotic results for runs and empirical cumulative entropies

• We prove large and moderate deviations for sequences of independent Bernoulli distributed random variables, and we illustrate their connection with the existing literature on runs based on order statistics.
• We prove large and moderate deviations for sequences of independent exponentially distributed random variables which appear in the literature as empirical cumulative entropies.
• In both cases we provide an upper bound of the rate function in terms of an integral of suitable relative entropies.

We prove large and moderate deviations for two sequences of estimators based on the order statistics and, more precisely, on spacings. In the first case we deal with runs, and we have sums of independent Bernoulli distributed random variables. In the second case we deal with empirical cumulative entropies, and we have linear combinations of independent exponentially distributed random variables.