Abstract
Consider a pure recurrent positive renewal process generated by some inter-arrival waiting time. If the waiting time has power-law fall-off exponent (i.e. tail index) in (1, 2), and if the jump's amplitude has non-zero but finite mean and finite variance, the cumulative amplitude process is long-range dependent with Hurst exponent in (1/2, 1). (Results in this direction have been obtained by Daley under the sole assumptions that the waiting time has moment index in (1, 2)). If the amplitude has zero mean, up to a Brownian trend, the cumulative amplitude process exhibits a negative-dependence property with Hurst exponent in (0, 1/2). The case of delayed stationary renewal processes is also investigated, together with two classes of limiting renewal processes: the compound exponential and the Levy classes. These are of some interest when the average time between consecutive jumps tends to zero jointly with the probability mass of the jumps' height concentrating about zero in some precise way. Under suitable hypothesis, the Hurst effect is maintained in the limit.
Published Version
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call