Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-18T18:35:14.670Z Has data issue: false hasContentIssue false
  • English
  • Français

On a random-coefficient AR(1) process with heavy-tailed renewal switching coefficient and heavy-tailed noise

Part of: Stochastic processes Special processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Remigijus Leipus ,
Vygantas Paulauskas  and
Donatas Surgailis
Remigijus Leipus*
Affiliation:
Vilnius University and Institute of Mathematics and Informatics, Vilnius
Vygantas Paulauskas*
Affiliation:
Vilnius University and Institute of Mathematics and Informatics, Vilnius
Donatas Surgailis*
Affiliation:
Vilnius University and Institute of Mathematics and Informatics, Vilnius
*
∗Postal address: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania.
∗Postal address: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania.
∗∗∗∗Postal address: Institute of Mathematics and Informatics, Akademijos str. 4, LT-08663 Vilnius, Lithuania. Email address: [email protected]
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 discuss the limit behavior of the partial sums process of stationary solutions to the (autoregressive) AR(1) equation Xt = atXt−1 + εt with random (renewal-reward) coefficient, at, taking independent, identically distributed values Aj ∈ [0,1] on consecutive intervals of a stationary renewal process with heavy-tailed interrenewal distribution, and independent, identically distributed innovations, εt, belonging to the domain of attraction of an α-stable law (0 < α ≤ 2, α ≠ 1). Under suitable conditions on the tail parameter of the interrenewal distribution and the singularity parameter of the distribution of Aj near the unit root a = 1, we show that the partial sums process of Xt converges to a λ-stable Lévy process with index λ < α. The paper extends the result of Leipus and Surgailis (2003) from the case of finite-variance Xt to that of infinite-variance Xt.

Keywords

AR(1) modelregime switchingrenewal-reward processstable Lévy process

MSC classification

Primary: 60F05: Central limit and other weak theorems 60G52: Stable processes
Secondary: 60K05: Renewal theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 2 , June 2006 , pp. 421 - 440
Copyright
© Applied Probability Trust 2006 

Footnotes

Supported by bilateral Lithuania-France research project Gilibert and the Lithuanian State Science and Studies Foundation grant T-10/06.

References

Blanchard, O. J. (1979). Speculative bubbles, crashes and rational expectations. Econom. Lett. 3, 387–389.CrossRefGoogle Scholar
Brandt, A. (1986). The stochastic equation Y_{n+1}=A_n Y_n+B_n with stationary coefficients. Adv. Appl. Prob. 18, 211–220.Google Scholar
Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theory Prob. Appl. 10, 323–331.Google Scholar
Chistyakov, G. P. and Götze, F. (2004). Limit distributions of Studentized means. Ann. Prob. 32, 28–77.CrossRefGoogle Scholar
Christoph, G. and Wolf, W. (1992). Convergence Theorems with a Stable Limit Law (Math. Res. 70). Akademie-Verlag, Berlin.Google Scholar
Cox, D. R. (1984). Long-range dependence: a review. In Statistics: an Appraisal, eds David, H. A. and David, H. T., Iowa State University Press, pp. 55–74.Google Scholar
Davidson, J. and Sibbertsen, P. (2005). Generating schemes for long memory processes: regimes, aggregation and linearity. J. Econometrics 128, 253–282.Google Scholar
Dehling, H. and Philipp, W. (2002). Empirical process techniques for dependent data. In Empirical Process Techniques for Dependent Data, eds Dehling, H., Mikosch, T. and Sørensen, M., Birkhäuser, Boston, MA, pp. 3–113.Google Scholar
Diebold, F. X. and Inoue, A. (2001). Long memory and regime switching. J. Econometrics 105, 131–159.CrossRefGoogle Scholar
Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. 2. John Wiley, New York.Google Scholar
Gourieroux, C. and Jasiak, J. (2001). Memory and infrequent breaks. Econom. Lett. 70, 29–41.CrossRefGoogle Scholar
Granger, C. W. J. and Hyung, N. (2004). Occasional structural breaks and long memory, with application to the S&P 500 absolute stock returns. J. Emp. Finance 11, 399–421.CrossRefGoogle Scholar
Heyde, C. C. and Yang, Y. (1997). On defining long-range dependence. J. Appl. Prob. 34, 939–944.CrossRefGoogle Scholar
Jensen, M. J. and Liu, M. (2006). Do long swings in the business cycle lead to strong persistence in output? J. Monetary Econom. 53, 597–611.Google Scholar
Karlsen, H. A. (1990). Existence of moments in a stationary difference equation. Adv. Appl. Prob. 22, 129–146.CrossRefGoogle Scholar
Leipus, R. and Surgailis, D. (2003). Random coefficient autoregression, regime switching and long memory. Adv. Appl. Prob. 35, 737–754.CrossRefGoogle Scholar
Leipus, R. and Viano, M.-C. (2003). Long memory and stochastic trend. Statist. Prob. Lett. 61, 177–190.Google Scholar
Leipus, R., Paulauskas, V. and Surgailis, D. (2005). Renewal regime switching and stable limit laws. J. Econometrics 129, 299–327.Google Scholar
Liu, M. (2000). Modeling long memory in stock market volatility. J. Econometrics 99, 139–171.Google Scholar
Mikosch, T. and Stărică, C. (2004). Nonstationarities in financial time series, the long-range dependence and the IGARCH effects. Rev. Econom. Statist. 86, 378–390.Google Scholar
Mikosch, T., Resnick, S., Rootzén, H. and Stegeman, A. (2002). Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Ann. Appl. Prob. 12, 23–68.Google Scholar
Parke, W. R. (1999). What is fractional integration? Rev. Econom. Statist. 81, 632–638.Google Scholar
Paulauskas, V. (1974). The estimation of the remainder term in limit theorem with a limiting stable distribution. Lithuanian Math. J. 14, 165–187.Google Scholar
Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, New York.Google Scholar
Pipiras, V., Taqqu, M. S. and Levy, J. B. (2004). Slow, fast and arbitrary growth conditions for renewal reward processes when both the renewals and the rewards are heavy-tailed. Bernoulli 10, 121–163.Google Scholar
Pourahmadi, M. (1988). Stationarity of the solution of X_t=A_t X_{t-1}+veps_t and analysis of non-Gaussian dependent variables. J. Time Ser. Anal. 9, 225–239.Google Scholar
Rosiński, J. (1980). Remarks on Banach spaces of stable type. Prob. Math. Statist. 1, 67–71.Google Scholar
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.Google Scholar
Taqqu, M. S. and Levy, J. B. (1986). Using renewal processes to generate long-range dependence and high variability. In Dependence in Probability and Statistics, eds Eberlein, E. and Taqqu, M. S., Birkhäuser, Boston, MA, pp. 73–89.Google Scholar
Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.Google Scholar
Von Bahr, B. and Esseen, C.-G. (1965). Inequalities for the rth absolute moment of a sum of random variables, 1≤ r≤ 2. Ann. Math. Statist. 36, 299–303.Google Scholar