## The Annals of Probability

### Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows

#### Abstract

We establish a new class of functional central limit theorems for partial sum of certain symmetric stationary infinitely divisible processes with regularly varying Lévy measures. The limit process is a new class of symmetric stable self-similar processes with stationary increments that coincides on a part of its parameter space with a previously described process. The normalizing sequence and the limiting process are determined by the ergodic-theoretical properties of the flow underlying the integral representation of the process. These properties can be interpreted as determining how long the memory of the stationary infinitely divisible process is. We also establish functional convergence, in a strong distributional sense, for conservative pointwise dual ergodic maps preserving an infinite measure.

#### Article information

Source
Ann. Probab., Volume 43, Number 1 (2015), 240-285.

Dates
First available in Project Euclid: 12 November 2014

https://projecteuclid.org/euclid.aop/1415801557

Digital Object Identifier
doi:10.1214/13-AOP899

Mathematical Reviews number (MathSciNet)
MR3298473

Zentralblatt MATH identifier
1320.60090

#### Citation

Owada, Takashi; Samorodnitsky, Gennady. Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows. Ann. Probab. 43 (2015), no. 1, 240--285. doi:10.1214/13-AOP899. https://projecteuclid.org/euclid.aop/1415801557

#### References

• Aaronson, J. (1981). The asymptotic distributional behaviour of transformations preserving infinite measures. J. Anal. Math. 39 203–234.
• Aaronson, J. (1997). An Introduction to Infinite Ergodic Theory. Mathematical Surveys and Monographs 50. Amer. Math. Soc., Providence, RI.
• Avram, F. and Taqqu, M. S. (1992). Weak convergence of sums of moving averages in the $\alpha$-stable domain of attraction. Ann. Probab. 20 483–503.
• Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
• Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.
• Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
• Bingham, N. H. (1971). Limit theorems for occupation times of Markov processes. Z. Wahrsch. Verw. Gebiete 17 1–22.
• Cohen, S. and Samorodnitsky, G. (2006). Random rewards, fractional Brownian local times and stable self-similar processes. Ann. Appl. Probab. 16 1432–1461.
• Darling, D. A. and Kac, M. (1957). On occupation times for Markoff processes. Trans. Amer. Math. Soc. 84 444–458.
• Davis, R. and Resnick, S. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Probab. 13 179–195.
• Dobrushin, R. L. and Major, P. (1979). Noncentral limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50 27–52.
• Dombry, C. and Guillotin-Plantard, N. (2009). Discrete approximation of a stable self-similar stationary increments process. Bernoulli 15 195–222.
• Ehm, W. (1981). Sample function properties of multiparameter stable processes. Z. Wahrsch. Verw. Gebiete 56 195–228.
• Getoor, R. K. and Kesten, H. (1972). Continuity of local times for Markov processes. Compos. Math. 24 277–303.
• Gradshteyn, I. S. and Ryzhik, I. M. (1994). Table of Integrals, Series, and Products, 5th ed. Academic Press, Boston, MA.
• Harris, T. E. and Robbins, H. (1953). Ergodic theory of Markov chains admitting an infinite invariant measure. Proc. Natl. Acad. Sci. USA 39 860–864.
• Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Probability and Its Applications (New York). Springer, New York.
• Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
• Lamperti, J. (1962). Semi-stable stochastic processes. Trans. Amer. Math. Soc. 104 62–78.
• Maejima, M. (1983). On a class of self-similar processes. Z. Wahrsch. Verw. Gebiete 62 235–245.
• Marcus, M. B. and Rosen, J. (2006). Markov Processes, Gaussian Processes, and Local Times. Cambridge Studies in Advanced Mathematics 100. Cambridge Univ. Press, Cambridge.
• Maruyama, G. (1970). Infinitely divisible processes. Theory Probab. Appl. 15 3–22.
• Meerschaert, M. M. and Scheffler, H.-P. (2004). Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Probab. 41 623–638.
• Merlevède, F., Peligrad, M. and Utev, S. (2006). Recent advances in invariance principles for stationary sequences. Probab. Surv. 3 1–36.
• Paulauskas, V. and Surgailis, D. (2008). On the rate of approximation in limit theorems for sums of moving averages. Theory Probab. Appl. 52 361–370.
• Pratt, J. W. (1960). On interchanging limits and integrals. Ann. Math. Statist. 31 74–77.
• Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes. Probab. Theory Related Fields 82 451–487.
• Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
• Resnick, S., Samorodnitsky, G. and Xue, F. (2000). Growth rates of sample covariances of stationary symmetric $\alpha$-stable processes associated with null recurrent Markov chains. Stochastic Process. Appl. 85 321–339.
• Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42 43–47.
• Rosiński, J. (1995). On the structure of stationary stable processes. Ann. Probab. 23 1163–1187.
• Rosiński, J. and Samorodnitsky, G. (1993). Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Probab. 21 996–1014.
• Rosiński, J. and Samorodnitsky, G. (1996). Classes of mixing stable processes. Bernoulli 2 365–377.
• Rosiński, J. and Żak, T. (1996). Simple conditions for mixing of infinitely divisible processes. Stochastic Process. Appl. 61 277–288.
• Rosiński, J. and Żak, T. (1997). The equivalence of ergodicity of weak mixing for infinitely divisible processes. J. Theoret. Probab. 10 73–86.
• Roy, E. (2007). Ergodic properties of Poissonian ID processes. Ann. Probab. 35 551–576.
• Samorodnitsky, G. (2004). Extreme value theory, ergodic theory and the boundary between short memory and long memory for stationary stable processes. Ann. Probab. 32 1438–1468.
• Samorodnitsky, G. (2005). Null flows, positive flows and the structure of stationary symmetric stable processes. Ann. Probab. 33 1782–1803.
• Samorodnitsky, G. (2006). Long range dependence. Found. Trends Stoch. Syst. 1 163–257.
• Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman & Hall, New York.
• Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press, Cambridge.
• Taqqu, M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrsch. Verw. Gebiete 50 53–83.
• Thaler, M. (2001). Infinite ergodic theory. The Dynamic Odyssey course, CIRM.
• Thaler, M. and Zweimüller, R. (2006). Distributional limit theorems in infinite ergodic theory. Probab. Theory Related Fields 135 15–52.
• Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York.
• Zolotarev, V. M. (1986). One-Dimensional Stable Distributions. Amer. Math. Soc., Providence, RI. Translated from the Russian by H. H. McFaden, Translation edited by Ben Silver.
• Zweimüller, R. (2000). Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points. Ergodic Theory Dynam. Systems 20 1519–1549.
• Zweimüller, R. (2007a). Infinite measure preserving transformations with compact first regeneration. J. Anal. Math. 103 93–131.
• Zweimüller, R. (2007b). Mixing limit theorems for ergodic transformations. J. Theoret. Probab. 20 1059–1071.
• Zweimüller, R. (2009). Surrey notes on infinite ergodic theory. Lecture notes, Surrey Univ.