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Point process convergence of stochastic volatility processes with application to sample autocorrelation

Part of: Stochastic processes Inference from stochastic processes Limit theorems Applications

Published online by Cambridge University Press:  14 July 2016

Richard A. Davis  and
Thomas Mikosch
Richard A. Davis*
Affiliation:
Colorado State University
Thomas Mikosch*
Affiliation:
University of Groningen and EURANDOM
*
1Postal address: Department of Statistics, Colorado State University, Fort Collins, CO 80523–1877, USA. Email: rdavis@stat.colostate.edu
2Postal address: Laboratory of Actuarial Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark. Email: mikosch@math.ku.dk
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Abstract

The paper considers one of the standard processes for modeling returns in finance, the stochastic volatility process with regularly varying innovations. The aim of the paper is to show how point process techniques can be used to derive the asymptotic behavior of the sample autocorrelation function of this process with heavy-tailed marginal distributions. Unlike other non-linear models used in finance, such as GARCH and bilinear models, sample autocorrelations of a stochastic volatility process have attractive asymptotic properties. Specifically, in the infinite variance case, the sample autocorrelation function converges to zero in probability at a rate that is faster the heavier the tails of the marginal distribution. This behavior is analogous to the asymptotic behavior of the sample autocorrelations of independent identically distributed random variables.

Keywords

Point processvague convergenceregular variationmixing conditionstationary processheavy tailsample autocovariancesample autocorrelationslowly varying modelfinancial time series

MSC classification

Primary: 60G55: Point processes
Secondary: 60F05: Central limit and other weak theorems 62M10: Time series, auto-correlation, regression, etc. 62P05: Applications to actuarial sciences and financial mathematics
Type
Time series analysis
Information
Journal of Applied Probability , Volume 38 , Issue A: Probability, Statistics and Seismology , 2001 , pp. 93 - 104
Copyright
Copyright © Applied Probability Trust 2001 

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