## The Annals of Probability

### Ergodic properties of Poissonian ID processes

Emmanuel Roy

#### Abstract

We show that a stationary IDp process (i.e., an infinitely divisible stationary process without Gaussian part) can be written as the independent sum of four stationary IDp processes, each of them belonging to a different class characterized by its Lévy measure. The ergodic properties of each class are, respectively, nonergodicity, weak mixing, mixing of all order and Bernoullicity. To obtain these results, we use the representation of an IDp process as an integral with respect to a Poisson measure, which, more generally, has led us to study basic ergodic properties of these objects.

#### Article information

Source
Ann. Probab., Volume 35, Number 2 (2007), 551-576.

Dates
First available in Project Euclid: 30 March 2007

https://projecteuclid.org/euclid.aop/1175287754

Digital Object Identifier
doi:10.1214/009117906000000692

Mathematical Reviews number (MathSciNet)
MR2308588

Zentralblatt MATH identifier
1146.60031

#### Citation

Roy, Emmanuel. Ergodic properties of Poissonian ID processes. Ann. Probab. 35 (2007), no. 2, 551--576. doi:10.1214/009117906000000692. https://projecteuclid.org/euclid.aop/1175287754

#### References

• Aaronson, J. (1997). An Introduction to Infinite Ergodic Theory. Amer. Math. Soc., Providence, RI.
• Cornfeld, I. P., Fomin, S. V. and Sinaï, Y. G. (1982). Ergodic Theory. Springer, New York.
• Derriennic, Y., Frączek, K., Lemańczyk, M. and Parreau, F. (2004). Ergodic automorphisms whose weak closure of off-diagonal measures consists of ergodic self-joinings. Preprint.
• Grabinsky, G. (1984). Poisson process over $\sigma$-finite Markov chains. Pacific J. Math. 2 301--315.
• Gruher, K., Hines, F., Patel, D., Silva, C. E. and Waelder, R. (2003). Power mixing does not imply multiple recurence in infinite measure and other counterexamples. New York J. Math. 9 1--22.
• Krengel, U. and Sucheston, L. (1969). On mixing in infinite measure spaces. Z. Wahrsch. Verw. Gebiete 13 150--164.
• Lemańczyk, M., Parreau, F. and Thouvenot, J.-P. (2000). Gaussian automorphisms whose ergodic self-joinings are Gaussian. Fund. Math. 164 253--293.
• Maruyama, G. (1970). Infinitely divisible processes. Theory Probab. Appl. 15 1--22.
• Matthes, K., Kerstan, J. and Mecke, J. (1978). Infinitely Divisible Point Processes. Wiley, New York.
• Neretin, Yu. A. (1996). Categories of Symmetries and Infinite-Dimensional Groups. Oxford Univ. Press.
• Petersen, K. (1983). Ergodic Theory. Cambridge Univ. Press.
• Pipiras, V. and Taqqu, M. S. (2004). Stable stationary processes related to cyclic flows. Ann. Probab. 32 2222--2260.
• Rosiński, J. (1995). On the structure of stationary stable processes. Ann. Probab. 23 1163--1187.
• Rosiński, J. (2000). Decomposition of $S\alphaS$-stable random fields. Ann. Probab. 28 1797--1813.
• 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 and weak mixing for infinitely divisible processes. J. Theoret. Probab. 10 73--86.
• Samorodnitsky, G. (2005). Null flows, positive flows and the structure of stationary symmetric stable processes. Ann. Probab. 33 1782--1803.
• Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press.