Extremal limit theorems for observations separated by random power law waiting times

Abstract

This paper develops extreme value theory for random observations separated by random waiting times whose exceedence probability falls off like a power law. In the case where the waiting times between observations have an infinite mean, a limit theorem is established, where the limit is comprised of an extremal process whose time index is randomized according to the non-Markovian hitting time process for a stable subordinator. The resulting limit distributions are shown to be solutions of fractional differential equations, where the order of the fractional time derivative coincides with the power law index of the waiting time. The probability that the limit process remains below a threshold is also computed. For waiting times with finite mean but infinite variance, a two-scale argument yields a fundamentally different limit process. The resulting limit is an extremal process whose time index is randomized according to the first passage time of a positively skewed stable Lévy motion with positive drift. This two-scale limit provides a second-order correction to the usual limit behavior.

On randomly spaced observations and continuous-time random walks

“…In section 3, we present the limiting distribution of all extreme observations until a given time, discussing asymptotic tail independence and asymptotic full tail dependence between the observations and interarrival times in detail. Our main result extends previously mentioned results in this context, as well as some more recent results in Meerschaert and Stoev [17] and Pancheva et al [19] for instance. In the asymptotic full tail dependence case.…”

Limit theorems for extremal processes generated by a point process with correlated time and space components

“…Comment 2 Meerschaert and Stoev (2007) considered similar processes in the case where the random variables X k and Y k in each pair are independent. Under this condition they proved the weak convergence in Skorohod J 1 topology in D(0, ∞) × (−∞, ∞) under similar normalization as above.…”

“…Different functional limits are obtained in rather general settings, like multivariate space components and non-linear normalization, see e.g. Balkema and Pancheva (1996), Silvestrov and Teugels (1998), Pancheva (1998), Pancheva et al (2006), Meerschaert and Stoev (2007) and references therein. In most of the studies concerning the extremal processes the main assumption is the independence of the time process and the space (or magnitude) variables.…”

Extremes of random variables observed in renewal times

“…For the renewal process with infinite mean, but with regularly varying steps, one can still determine the asymptotic distribution of the maximum, see [4]. In such a setting, the convergence of (M τ (t)) was shown at the level of stochastic processes, see [12,13]. In the rest of the paper we show how one can move beyond the maxima and extend those results to all upper order statistic in both finite and infinite mean case.…”

“…The earliest references are [10,4,3], see also [17,2] for extensions and applications in engineering. More recently, [12] and [13] studied the convergence of the process (M(t)) towards an appropriate extremal process. A general treatment of extremal processes with random sample size can be found in [18,19].…”