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Limit Theorems for Moving Averages with Random Coefficients and Heavy-Tailed Noise

Part of: Stochastic processes Distribution theory - Probability Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

Rafał Kulik
Rafał Kulik*
Affiliation:
University of Wrocław and University of Ottawa
*
∗ Postal address: Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, ON, K1N 6N5, Canada. Email address: [email protected]
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Abstract

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We consider a stationary moving average process with random coefficients, , generated by an array, {Ct,k, t ∈ Z, k ≥ 0}, of random variables and a heavy-tailed sequence, {Zt, t ∈ Z}. We analyze the limit behavior using a point process analysis. As applications of our results we compare the limiting behavior of the moving average process with random coefficients with that of a standard MA(∞) process.

Keywords

Nonlinear processpoint processself-normalized sum

MSC classification

Primary: 62M10: Time series, auto-correlation, regression, etc.
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60G10: Stationary processes 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 1 , March 2006 , pp. 245 - 256
Copyright
© Applied Probability Trust 2006 

References

Bougerol, P. and Picard, N. (1992). Strict stationarity of generalized autoregressive processes. Ann. Prob. 20, 1714–1730.Google Scholar
Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theory Prob. Appl. 10, 351–360.Google Scholar
Dabrowski, A. R., Dehling, H. G., Mikosch, T. and Sharipov, O. (2002). Poisson limits for U-statistics. Stoch. Process. Appl. 99, 137–157.Google Scholar
Davis, R. A. and Hsing, T. (1995). Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Prob. 23, 879–917.Google Scholar
Davis, R. A. and Mikosch, T. (1998). The sample autocorrelations of heavy-tailed processes with applications to ARCH. Ann. Statist. 26, 2049–2080.CrossRefGoogle Scholar
Davis, R. A. and Resnick, S. I. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Prob. 13, 179–195.Google Scholar
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. Process. Appl. 30, 41–68.Google Scholar
Davis, R. A. and Resnick, S. I. (1996). Limit theory for bilinear processes with heavy-tailed noise. Ann. Appl. Prob. 6, 1191–1210.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1987). Modelling Extremal Events for Insurance and Finance (Appl. Math. (New York) 33). Springer, Berlin.Google Scholar
Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1, 126–166.CrossRefGoogle Scholar
Konstantinides, D. G. and Mikosch, T. (2005). Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations. Ann. Prob. 33, 1992–1992.Google Scholar
Mikosch, T. (2004). Modelling dependence and tails of financial time series. In Extreme Values in Finance, Telecommunications, and the Environment, eds Finkenstädt, B. and Rootzén, H., Chapman and Hall, Boca Raton, FL, pp. 185–286.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.Google Scholar
Resnick, S. I. (1998). Why non-linearities can ruin the heavy-tailed modeler's day. In A Practical Guide to Heavy Tails, eds Adler, R. J., Feldman, R. E. and Taqqu, M. S., Birkhäuser, Boston, MA, pp. 219–239.Google Scholar
Resnick, S. I. and van den Berg, E. (2000). A test for nonlinearity of time series with infinite variance. Extremes 3, 145–172.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.Google Scholar