Journal of Theoretical Probability

, Volume 26, Issue 3, pp 676–696

# Convergence in Law to Operator Fractional Brownian Motions

Article

## Abstract

In this paper, we provide two approximations in law of operator fractional Brownian motions. One is constructed by Poisson processes, and the other generalizes a result of Taqqu (Z. Wahrscheinlichkeitstheor. Verw. Geb. 31:287–302, 1975).

## Keywords

Operator fractional Brownian motion Poisson processes Vector-valued Gaussian sequence Weak convergence

60F17 60G15

## Notes

### Acknowledgements

The author thanks Professor Yimin Xiao, Michigan State University, USA, and Professor Yuqiang Li, East China Normal University, China, for stimulating discussions. I also would like to thank the reviewer for helpful comments to improve this work. This work was supported by the Scientific Research Foundation of Guangxi University (No. XBZ110398).

## References

1. 1.
Biermé, H., Meerschaert, M.M., Scheffler, H.P.: Operator scaling stable random fields. Stoch. Process. Appl. 117, 312–332 (2007)
2. 2.
Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)
3. 3.
Chung, C.F.: Sample means, sample autocovariances, and linear regression of stationary multivariate long memory processes. Econom. Theory 18, 51–78 (2002)
4. 4.
Dai, H., Li, Y.: A weak limit theorem for generalized multifractional Brownian motion. Stat. Probab. Lett. 80, 348–356 (2010)
5. 5.
Davidson, J., de Jong, R.M.: The functional central limit theorem and weak convergence to stochastic integrals II. Econom. Theory 16, 643–666 (2000)
6. 6.
Davidson, J., Hashimzade, N.: Alternative frequency and time domain versions of fractional Brownian motion. Econom. Theory 24, 256–293 (2008)
7. 7.
Davydov, Y.: The invariance principle for stationary processes. Teor. Verojatnost. Primenen. 15, 498–509 (1970)
8. 8.
Delgado, R.: A reflected fBm limit for fluid models with ON/OFF sources under heavy traffic. Stoch. Process. Appl. 117, 188–201 (2007)
9. 9.
Delgado, R., Jolis, M.: Weak approximation for a class of Gaussian process. J. Appl. Probab. 37, 400–407 (2000)
10. 10.
Didier, G., Pipiras, V.: Integral representations and properties of operator fractional Brownian motions. Bernoulli 17, 1–33 (2011)
11. 11.
Didier, G., Pipiras, V.: Exponents, symmetry groups and classification of operator fractional Brownian motions. J. Theor. Probab. doi: (2011) Google Scholar
12. 12.
Dolado, J., Marmol, F.: Asymptotic inference results for multivariate long-memory processes. Econom. J. 7, 168–190 (2004)
13. 13.
Enriquez, N.: A simple construction of the fractional Brownian motion. Stoch. Process. Appl. 109, 203–223 (2004)
14. 14.
Ethier, S., Kurtz, T.: Markov Processes: Characterization and Convergence. Wiley, New York (1986)
15. 15.
Feller, W.: An Introduction to Probability Theory and its Applications, 2nd edn. Wiley, New York (1971)
16. 16.
Hudson, W.N., Mason, J.D.: Operator-self-similar processes in a finite-dimensional space. Trans. Am. Math. Soc. 273, 281–297 (1982)
17. 17.
Jurek, Z.J., Mason, J.D.: Operator Limit Distributions in Probability Theory. Wiley, New York (1993)
18. 18.
Konstantopoulos, T., Lin, S.J.: Fractional Brownian approximations of queuing networks. In: Stochastic Networks. Lecture Notes in Statistics, vol. 117, pp. 257–273. Springer, New York (1996)
19. 19.
Laha, T.L., Rohatgi, V.K.: Operator self-similar processes in ℝd. Stoch. Process. Appl. 12, 73–84 (1982)
20. 20.
Lamperti, L.: Semi-stable stochastic processes. Trans. Am. Math. Soc. 104, 62–78 (1962)
21. 21.
Maejima, M., Mason, J.D.: Operator-self-similar stable processes. Stoch. Process. Appl. 54, 139–163 (1994)
22. 22.
Marinucci, D., Robinson, P.: Weak convergence of multivariate fractional processes. Stoch. Process. Appl. 86, 103–120 (2000)
23. 23.
Mason, J.D., Xiao, Y.: Sample path properties of operator-self-similar Gaussian random fields. Theory Probab. Appl. 46, 58–78 (2002)
24. 24.
Meerschaert, M.M., Scheffler, H.P.: Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice. Wiley, New York (2001) Google Scholar
25. 25.
Robinson, P.: Multiple local whittle estimation in stationary systems. Ann. Stat. 36, 2508–2530 (2008)
26. 26.
Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman and Hall, New York (1994)
27. 27.
Sato, K.: Self-similar processes with independent increments. Probab. Theory Relat. Fields 89, 285–300 (1991)
28. 28.
Stroock, D.: In: Topics in Stochastic Differential Equations, Tata Institute of Fundamental Research, Bomaby. Springer, New York (1982) Google Scholar
29. 29.
Taqqu, M.S.: Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitstheor. Verw. Geb. 31, 287–302 (1975)
30. 30.
Vervaat, W.: Sample path properties of self-similar processes with stationary increments. Ann. Probab. 13, 1–27 (1985)
31. 31.
Whitt, W.: Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York (2002) Google Scholar

## Copyright information

© Springer Science+Business Media, LLC 2012

## Authors and Affiliations

1. 1.College of Mathematics and Information SciencesGuangxi UniversityNanningChina