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

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

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
*
1 Postal address: Department of Statistics, Colorado State University, Fort Collins, CO 80523–1877, USA. Email: rdavis@stat.colostate.edu
2 Postal 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

Information

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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References

[1] Basrak, B., Davis, R. A. and Mikosch, T. (1999). The sample ACF of a simple bilinear process. Stoch. Proc. Appl. 83, 1–14.Google Scholar
[2] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.Google Scholar
[3] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
[4] Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theory Probab. Appl. 10, 323–331.Google Scholar
[5] Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods , 2nd Edn. Springer, New York.Google Scholar
[6] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
[7] Davis, R. A. and Hsing, T. (1995). Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Probab. 23, 879–917.Google Scholar
[8] Davis, R. A. and Mikosch, T. (1998). The sample autocorrelations of heavy-tailed processes with applications to ARCH. Ann. Statist. 26, 2049–2080.Google Scholar
[9] Davis, R. A. and Mikosch, T. (2000). The sample autocorrelations of financial time series models. In Nonlinear and Nonstationary Signal Processing , eds Fitzgerald, W. et al., Cambridge University Press, 247–274.Google Scholar
[10] Davis, R. A., Mikosch, T. and Basrak, B. (1999). Limit theory for the sample autocorrelations of solutions to stochastic recurrence equations with applications to GARCH processes. Preprint.Google Scholar
[11] Davis, R. A. and Resnick, S. I. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Probab. 13, 179–195.Google Scholar
[12] Davis, R. A. and Resnick, S. I. (1985). More limit theory for the sample correlation function of moving averages. Stoch. Proc. Appl. 20, 257–279.Google Scholar
[13] Davis, R. A. and Resnick, S. I. (1986). Limit theory for the sample covariance and correlation functions of moving averages. Ann. Statist. 14, 533–558.CrossRefGoogle Scholar
[14] Davis, R. A. and Resnick, S. I. (1988). Extremes of moving averages of random variables from the domain of attraction of the double exponential distribution. Stoch. Proc. Appl. 30, 41–68.CrossRefGoogle Scholar
[15] Davis, R. A. and Resnick, S. I. (1996). Limit theory for bilinear processes with heavy tailed noise. Ann. Appl. Probab. 6, 1191–1210.CrossRefGoogle Scholar
[16] Doukhan, P. (1994). Mixing. Properties and Examples (Lecture Notes Statist. 85). Springer, New York.Google Scholar
[17] Embrechts, P., Kluppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.Google Scholar
[18] Engle, R.F., (Ed.) (1995). ARCH Selected Readings. Oxford University Press.Google Scholar
[19] Gourieroux, C. (1997). ARCH Models and Financial Applications. Springer, New York.Google Scholar
[20] Kallenberg, O. (1983). Random Measures , 3rd edition. Akademie, Berlin.Google Scholar
[21] Mikosch, T. and Starica, C. (2000). Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process. Ann. Statist. 28. 1427–1451.Google Scholar
[22] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.Google Scholar
[23] Resnick, S. I. (1997). Heavy tail modeling and teletraffic data (with discussion and a rejoinder by the author). Ann. Statist. 25, 1805–1869.Google Scholar
[24] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance. Chapman and Hall, London.Google Scholar
[25] Shephard, N. (1996). Statistical aspects of ARCH and stochastic volatility. In Time Series Models, in Econometrics, Finance and other Fields , eds Cox, D. R., Hinkley, D. V. and Barndorff-Nielsen, O. E., Chapman and Hall, London, 1–67.Google Scholar
[26] Taylor, S. J. (1986). Modelling Financial Time Series. Wiley, Chichester.Google Scholar
[27] Willinger, W., Taqqu, M. S., Leland, M. and Wilson, D. (1995). Self-similarity through high variability: statistical analysis of ethernet LAN traffic at the source level. Computer Comm. Review 25, 100–113.Google Scholar