Contribution to an open problem of Harkness and Shantaram.

Here, we solve an open problem raised by Harkness and Shantaram in 1969 who obtained, under sufficient conditions, a limit theorem in law for sequences of nonnegative random variables built with the iterated stationary excess operator. More precisely, they considered the distribution function (d.f.) F of a nonnegative random variable X having all moments μn=∫0∞undF(u) finite. The stationary excess operator corresponds to the d.f. F1(x)=μ1−1∫0x1−F(u)du,x>0, so that the iterated stationary excess operator (ISEO) at the order n ≥ 2 corresponds to the d.f. Fn(x)=μ1,n−1−1∫0x1−Fn−1(u)du, where μ1,n−1=∫0∞1−Fn−1(u)du. Let the r.v. En(X) denotes a realization of Fn. Harkness and Shantaram provided sufficient conditions for the existence of a normalizing sequence cn such that the sequence En(X)/cn converges in distribution or, equivalently, Fn(cnx) → G(x) for some d.f. G. This raises a problem of identification the class of possible limits G. We give here a complete answer to this problem through the convergence of families built by the continuous time version of the ISEO and also by size biasing. In this context, we show that (i)the conditions of Harkness and Shantaram are actually necessary;(ii)continuous time convergence is equivalent to discrete time convergence; and(iii)the only possible limits in distribution G are mixture of exponential with log‐normal distributions. [ABSTRACT FROM AUTHOR]