A functional limit theorem for dependent sequences with infinite variance stable limits

Abstract

Under an appropriate regular variation condition, the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non-Gaussian stable random variable. A functional version of this is known to be true as well, the limit process being a stable Lévy process. The main result in the paper is that for a stationary, regularly varying sequence for which clusters of high-threshold excesses can be broken down into asymptotically independent blocks, the properly centered partial sum process still converges to a stable Lévy process. Due to clustering, the Lévy triple of the limit process can be different from the one in the independent case. The convergence takes place in the space of càdlàg functions endowed with Skorohod's $M_1$ topology, the more usual $J_1$ topology being inappropriate as the partial sum processes may exhibit rapid successions of jumps within temporal clusters of large values, collapsing in the limit to a single jump. The result rests on a new limit theorem for point processes which is of independent interest. The theory is applied to moving average processes, squared GARCH(1,1) processes and stochastic volatility models.

Published in at http://dx.doi.org/10.1214/11-AOP669 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (6)

• The extremogram: A correlogram for extreme events
• A new maximal inequality and invariance principle for stationary sequences
• Convergence to Lévy stable processes under some weak dependence conditions
• Functional limit theorems for linear processes in the domain of attraction of stable laws
• Nonstandard limit theorem for infinite variance functionals
• Selection from a stable box

Cited by in corpus (5)

• Boundary regularity for fully nonlinear integro-differential equations
• A large deviations approach to limit theory for heavy-tailed time series
• Complete convergence theorem for stationary heavy tailed sequences
• A Fourier analysis of extreme events
• Convergence to a Lévy process in the Skorohod $M_1$ and $M_2$ topologies for nonuniformly hyperbolic systems, including billiards with cusps