Subexponential Tails and Long-Range Dependence: Clustered Behaviors of Extreme Values

Abstract

I study extreme values from certain stationary infinitely divisible (SID) processes with subexponential tails. These processes are represented as stochastic integrals of a nonrandom function composed with iterated transformations with respect to some random measure. In this ergodic theory approach, I concentrate on conservative dynamics which generate long memories of various strength. I prove functional theorems both in the space of sup measures and in the space of C\`{a}dl\`{a}g functions for such SID processes. In a subexponential stochastic system, extremal events are often caused by a unique dominating movement. However, under long range dependence, I demonstrate that extremes in a SID process may come from multiple large values. Extremes for a class of a symmetric stable random fields are thoroughly studied. For such tails, only under moderate long range dependence, a unique large value determines the extremal behavior and the limits have the Fr\'{e}chet distribution, and the non-Fr\'{e}chet limits under stronger long memory are also characterized. However, a moderately long range dependent setting is not sufficient to guarantee ``heuristic of a single big value". In this moderate zone, as the subexponential tails become light enough, i.e. dropping into the Gumbel maximum domain of attraction, extremes may automatically depend on multiple large values of the driving noise. In addition, I show that extremal clusters can be quite intricate and they result from the interactions between the flows and the noise. In the extremal limit theorems of sup-measures, the limits are based on random upper continuous functions supported on stable regenerative sets. If the tails are heavy enough, e.g. power-law tails and lognormal tails, extremal clusters share the common feature that each stable regenerative set supports one value. However, for semiexponential tails, a new shape arises, where each stable regenerative set supports a random panoply of varying extremes. In the presence of long memory, fine heterogeneities within the subexponential distributions can be manifested.