Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T19:31:08.836Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Distribution of the Nearly Unstable AR(1) Process with Heavy Tails

Part of: Limit theorems Distribution theory Approximations and expansions Stochastic processes Approximation methods and numerical treatment of dynamical systems

Published online by Cambridge University Press:  01 July 2016

Mariana Olvera-Cravioto
Mariana Olvera-Cravioto*
Affiliation:
Columbia University
*
∗ Postal address: Department of Industrial Engineering and Operations Research, Columbia University, 306 S. W. Mudd Building, 500 W. 120th Street, New York, NY 10027, USA. Email address: molvera@ieor.columbia.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a nearly unstable, or near unit root, AR(1) process with regularly varying innovations. Two different approximations for the stationary distribution of such processes exist: a Gaussian approximation arising from the nearly unstable nature of the process and a heavy-tail approximation related to the tail asymptotics of the innovations. We combine these two approximations to obtain a new uniform approximation that is valid on the entire real line. As a corollary, we obtain a precise description of the regions where each of the Gaussian and heavy-tail approximations should be used.

Keywords

AR processnear unit rootheavy tailweighted sumuniform approximationlarge deviation

MSC classification

Primary: 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 37M10: Time series analysis
Secondary: 60F10: Large deviations 60G10: Stationary processes 60G50: Sums of independent random variables; random walks 60G70: Extreme value theory; extremal processes 62E17: Approximations to distributions (nonasymptotic)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 1 , March 2010 , pp. 106 - 136
Copyright
Copyright © Applied Probability Trust 2010 

References

Barbe, P. and McCormick, W. P. (2009). Asymptotic expansions for infinite weighted convolutions of heavy tail distributions and applications. Mem. Amer. Math. Soc. 197, 1–117.Google Scholar
Basak, G. K. and Ho, K.-W. R. (2004). Level-crossing probabilities and first-passage times for linear processes. Adv. Appl. Prob. 36, 643–666.CrossRefGoogle Scholar
Billingsley, P. (1995). Probability and Measure. John Wiley, New York.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Borovkov, A. A. and Borovkov, K. A. (2008). Asymptotic Analysis of Random Walks. Cambridge University Press.CrossRefGoogle Scholar
Chan, N. H. and Wei, C. Z. (1987). Asymptotic inference for nearly nonstationary AR(1) processes. Ann. Statist. 15, 1050–1063.CrossRefGoogle Scholar
Chen, Y., Ng, K. N. and Tang, Q. (2005). Weighted sums of subexponential random variables and their maxima. Adv. Appl. Prob. 37, 510–522.CrossRefGoogle Scholar
Cline, D. (1983). Estimation and linear prediction for regression, autoregression and arma with infinite variance data. , Colorado State University.Google Scholar
Cline, D. (1983). Infinite series of random variables with regularly varying tails. Tech. Rep. 83-24, Institute of Applied Mathematics and Statistics, University of British Columbia.Google Scholar
Cox, D. D. (1991). Gaussian likelihood estimation for nearly nonstationary AR(1) processes. Ann. Statist. 19, 1129–1142.CrossRefGoogle Scholar
Cumberland, W. G. and Sykes, Z. M. (1982). Weak convergence of an autoregressive process used in modeling population growth. J. Appl. Prob. 19, 450–455.CrossRefGoogle Scholar
Davis, R. and Resnick, S. (1988). Extremes of moving averages of random variables from the domain of attraction of the double exponential distribution. Stoch. Process. Appl. 30, 41–68.CrossRefGoogle Scholar
Finster, M. (1982). The maximum term and first passage times for autoregressions. Ann. Prob. 10, 737–744.CrossRefGoogle Scholar
Geluk, J. L. and De Vries, C. G. (2006). Weighted sums of subexponential random variables and asymptotic dependence between returns on reinsurance equities. Insurance Math. Econom. 38, 39–56.CrossRefGoogle Scholar
Jessen, A. H. and Mikosch, T. (2006). Regularly varying functions. Publ. Inst. Math. 80, 171–192.CrossRefGoogle 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.CrossRefGoogle Scholar
Novikov, A. (1990). On the first passage time of an autoregressive process over a level and an application to a ‘disorder’ problem. Theory Prob. Appl. 35, 269–279.CrossRefGoogle Scholar
Olvera-Cravioto, M. and Glynn, P. (2009). On the transition from heavy traffic to heavy tails for the M/G/1 queue: the semiexponential case. In preparation.Google Scholar
Olvera-Cravioto, M., Blanchet, J. and Glynn, P. (2009). On the transition from heavy traffic to heavy tails for the M/G/1 queue: the regularly varying case. Submitted.Google Scholar
Petrov, V. V. (1995). Limit Theorems of Probability Theory. Oxford University Press.Google Scholar
Resnick, S. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.CrossRefGoogle Scholar
Rootzén, H. (1986). Extreme value theory for moving average processes. Ann. Prob. 14, 612–652.CrossRefGoogle Scholar
Rozovskiıˇ, L. V. (1989). Probabilities of large deviations of sums of independent random variables with a common distribution function that belongs to the domain of attraction of a normal law. Theory Prob. Appl. 34, 625–644.CrossRefGoogle Scholar
Van der Meer, T., Pap, G. and van Zuijlen, M. C. A. (1999). Asymptotic inference for nearly unstable AR(p) processes. Econometric Theory 15, 184–217.CrossRefGoogle Scholar
Zeevi, A. and Glynn, P. W. (2004). Recurrence properties of autoregressive processes with super-heavy-tailed innovations. J. Appl. Prob. 41, 639–653.CrossRefGoogle Scholar