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September 2012 A functional limit theorem for dependent sequences with infinite variance stable limits
Bojan Basrak, Danijel Krizmanić, Johan Segers
Ann. Probab. 40(5): 2008-2033 (September 2012). DOI: 10.1214/11-AOP669


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 $\operatorname{GARCH}(1,1)$ processes and stochastic volatility models.


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Bojan Basrak. Danijel Krizmanić. Johan Segers. "A functional limit theorem for dependent sequences with infinite variance stable limits." Ann. Probab. 40 (5) 2008 - 2033, September 2012.


Published: September 2012
First available in Project Euclid: 8 October 2012

zbMATH: 1295.60041
MathSciNet: MR3025708
Digital Object Identifier: 10.1214/11-AOP669

Primary: 60F17 , 60G52
Secondary: 60G55 , 60G70

Keywords: Convergence in distribution , Functional limit theorem , GARCH , Mixing , moving average , partial sum , Point processes , regular variation , spectral processes , Stable processes , stochastic volatility

Rights: Copyright © 2012 Institute of Mathematical Statistics


Vol.40 • No. 5 • September 2012
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