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Hidden regular variation of moving average processes with heavy-tailed innovations

Part of: Approximation methods and numerical treatment of dynamical systems Classical measure theory Stochastic processes

Published online by Cambridge University Press:  30 March 2016

Sidney I. Resnick  and
Joyjit Roy
Sidney I. Resnick
Affiliation:
School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853, USA. Email address: [email protected].
Joyjit Roy
Affiliation:
School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853, USA. Email address: [email protected].
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Abstract

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We look at joint regular variation properties of MA(∞) processes of the form X = (Xk, k ∈ Z), where Xk = ∑j=0∞ψjZk-j and the sequence of random variables (Zi, i ∈ Z) are independent and identically distributed with regularly varying tails. We use the setup of MO-convergence and obtain hidden regular variation properties for X under summability conditions on the constant coefficients (ψj: j ≥ 0). Our approach emphasizes continuity properties of mappings and produces regular variation in sequence space.

Keywords

Regular variationmultivariate heavy tailhidden regular variationmoving average process

MSC classification

Primary: 28A33: Spaces of measures, convergence of measures 60G70: Extreme value theory; extremal processes
Secondary: 37M10: Time series analysis
Type
Part 6. Heavy tails
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014 , pp. 267 - 279
Copyright
Copyright © Applied Probability Trust 2014 

References

Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.Google Scholar
Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Brockwell, P. J., and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Cline, D. B. H. (1983). {Estimation and linear prediction for regression, autoregression and {ARMA} with infinite variance data}. , Colorado State University.Google Scholar
Cline, D. B. H. (1983). Infinite series of random variables with regularly varying tails. Tech. Rep., Institute of Applied Mathematics and Statistics, University British Columbia.Google Scholar
Daley, D. J., and Vere-Jones, D. (2007). An Introduction to the Theory of Point Processes. Vol I. Elementary Theory and Methods. Springer, New York.Google Scholar
Das, B., Mitra, A., and Resnick, S. (2013). Living on the multidimensional edge: seeking hidden risks using regular variation. Adv. Appl. Prob. 45, 139–163.Google Scholar
Davis, R., and Resnick, S. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Prob. 13, 179–195.CrossRefGoogle Scholar
Hult, H., and Lindskog, F. (2005). Extremal behavior of regularly varying stochastic processes. Stoch. Process. Appl. 115, 249–274.Google Scholar
Hult, H., and Lindskog, F. (2006). Regular variation for measures on metric spaces. Publ. Inst. Math. (Beograd)(N.S.) 80, 121–140.Google Scholar
Hult, H., and Lindskog, F. (2007). Extremal behavior of stochastic integrals driven by regularly varying Lévy processes. Ann. Prob. 35, 309–339.CrossRefGoogle Scholar
Hult, H., and Samorodnitsky, G. (2008). Tail probabilities for infinite series of regularly varying random vectors. Bernoulli 14, 838–864.Google Scholar
Kallenberg, O. (1983). Random Measures, 3rd edn. Akademie-Verlag, Berlin.Google Scholar
Lindskog, F., Resnick, S. I., and Roy, J. (2014). Regularly varying measures on metric spaces: hidden regular variation and hidden jumps. Prob. Surveys 11, 270–314.Google Scholar
Maulik, K., and Resnick, S. (2004). Characterizations and examples of hidden regular variation. Extremes 7, 31–67.Google Scholar
Mikosch, T., and Samorodnitsky, G. (2000). The supremum of a negative drift random walk with dependent heavy-tailed steps. Ann. Appl. Prob. 10, 1025–1064.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes (Appl. Prob. Trust Ser. 4). Springer, New York.Google Scholar
Resnick, S. (2002). Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5, 303–336.Google Scholar
Resnick, S. I. (2007). Heavy-Tail Phenomena. Springer, New York.Google Scholar
Resnick, S. I. (2008). Extreme Values, Regular Variation and Point Processes. Springer, New York.Google Scholar
Resnick, S. I., and Willekens, E. (1991). Moving averages with random coefficients and random coefficient autoregressive models. Commun. Statist. Stoch. Models 7, 511–525.CrossRefGoogle Scholar
Rootzén, H. (1978). Extremes of moving averages of stable processes. Ann. Prob. 6, 847–869.CrossRefGoogle Scholar
Rootzén, H. (1986). Extreme value theory for moving average processes. Ann. Prob. 14, 612–652.Google Scholar
Wang, D., and Tang, Q. (2006). Tail probabilities of randomly weighted sums of random variables with dominated variation. Stoch. Models 22, 253–272.Google Scholar