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Representations of hermite processes using local time of intersecting stationary stable regenerative sets

Part of: Stochastic processes

Published online by Cambridge University Press:  23 November 2020

Shuyang Bai
Shuyang Bai*
Affiliation:
University of Georgia, US
*
*Postal address: Department of Statistics, University of Georgia, 310 Herty Drive, Athens, GA 30602, USA. Email address: [email protected]
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Abstract

Hermite processes are a class of self-similar processes with stationary increments. They often arise in limit theorems under long-range dependence. We derive new representations of Hermite processes with multiple Wiener–Itô integrals, whose integrands involve the local time of intersecting stationary stable regenerative sets. The proof relies on an approximation of regenerative sets and local times based on a scheme of random interval covering.

Keywords

Hermite processmultiple Wiener–Itô integralstable regenerative setlocal timelong-range dependencerandom interval covering

MSC classification

Primary: 60G18: Self-similar processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 4 , December 2020 , pp. 1234 - 1251
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Copyright
© Applied Probability Trust 2020

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References

Bai, S., Owada, T. and Wang, Y. (2020). A functional non-central limit theorem for multiple-stable processes with long-range dependence. Stoch. Process. Appl. 130, 5768–5801.10.1016/j.spa.2020.04.007CrossRefGoogle Scholar
Bai, S. and Taqqu, M. S. (2018). How the instability of ranks under long memory affects large-sample inference. Statist. Sci. 33, 96–116.10.1214/17-STS633CrossRefGoogle Scholar
Bai, S. and Taqqu, M. S. (2020). Limit theorems for long-memory flows on Wiener chaos. Bernoulli 26, 1473–1503.10.3150/19-BEJ1168CrossRefGoogle Scholar
Bertoin, J. (1999). Subordinators: Examples and Applications. Springer, New York.Google Scholar
Bertoin, J. and Pitman, J. (2000). Two coalescents derived from the ranges of stable subordinators. Electron. J. Prob. 5, 7.10.1214/EJP.v5-63CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation . Encyclopedia of Mathematics and Its Applications. Cambridge University Press.Google Scholar
Dobrushin, R. L. and Major, P. (1979). Non-central limit theorems for non-linear functional of Gaussian fields. Prob. Theory Relat. Fields 50, 27–52.Google Scholar
Fitzsimmons, P. J., Fristedt, B. and Maisonneuve, B. (1985). Intersections and limits of regenerative sets. Z. Wahrscheinlichkeitsth. 70, 157–173.10.1007/BF02451426CrossRefGoogle Scholar
Fitzsimmons, P. J., Fristedt, B. and Shepp, L. A. The set of real numbers left uncovered by random covering intervals. Z. Wahrscheinlichkeitsth. 70, 175–189.10.1007/BF02451427CrossRefGoogle Scholar
Fitzsimmons, P. J. and Taksar, M. (1988). Stationary regenerative sets and subordinators. Ann. Prob. 16, 1299–1305.10.1214/aop/1176991692CrossRefGoogle Scholar
Ho, H. and Hsing, T. (1997). Limit theorems for functionals of moving averages. Ann. Prob. 25, 1636–1669.10.1214/aop/1023481106CrossRefGoogle Scholar
Janson, S. (1997), Gaussian Hilbert Spaces. Cambridge University Press.10.1017/CBO9780511526169CrossRefGoogle Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.10.1007/978-1-4757-4015-8CrossRefGoogle Scholar
Kingman, J. F. C. (1973). An intrinsic description of local time. J. London Math. Soc. 2, 725–731.10.1112/jlms/s2-6.4.725CrossRefGoogle Scholar
Lacaux, C. and Samorodnitsky, G. (2016). Time-changed extremal process as a random sup measure. Bernoulli, 22, 1979–2000.10.3150/15-BEJ717CrossRefGoogle Scholar
Lindgren, G. (2012). Stationary Stochastic Processes: Theory and Applications. CRC Press, Boca Raton.10.1201/b12171CrossRefGoogle Scholar
Maisonneuve, B. (1987). Subordinators regenerated. In Seminar on Stochastic Processes, 1986, eds E. Çinlar, K. L. Chung, R. K. Getoor, and J. Glover, pp. 155–161.10.1007/978-1-4684-6751-2_11CrossRefGoogle Scholar
Major, P. (2014). Multiple Wiener–Itô Integrals, with Applications to Limit Theorems, 2nd ed. Springer, New York.10.1007/978-3-319-02642-8CrossRefGoogle Scholar
Molchanov, I. S. (2017) Theory of Random Sets, 2nd ed., Vol. 87. Springer, New York.10.1007/978-1-4471-7349-6CrossRefGoogle Scholar
Nourdin, I., Nualart, D. and Tudor, C. (2010). Central and non-central limit theorems for weighted power variations of fractional Brownian motion. Ann. Inst. H. Poincaré Prob. Statist. 46, 1055–1079.10.1214/09-AIHP342CrossRefGoogle Scholar
Peccati, G. and Taqqu, M. S. (2011). Wiener Chaos: Moments, Cumulants and Diagrams: A Survey With Computer Implementation. Springer, New York.10.1007/978-88-470-1679-8CrossRefGoogle Scholar
Pipiras, V. and Taqqu, M. S. (2010). Regularization and integral representations of Hermite processes. Statist. Prob. Lett. 80, 2014–2023.10.1016/j.spl.2010.09.008CrossRefGoogle Scholar
Pipiras, V. and Taqqu, M. S. (2017). Long-Range Dependence and Self-Similarity, Vol. 45. Cambridge University Press.10.1017/CBO9781139600347CrossRefGoogle Scholar
Rosenblatt, M. (1961). Independence and dependence. In Proc. 4th Berkeley Symp. Math. Statist. Prob., Vol. 2. University of California Press, Berkeley, pp. 431–443.Google Scholar
Samorodnitsky, G. and Wang, Y. (2019). Extremal theory for long range dependent infinitely divisible processes. Ann. Prob. 47, 2529–2562.10.1214/18-AOP1318CrossRefGoogle Scholar
Sethuraman, J. (2002). Some extensions of the Skorohod representation theorem. Sankhyā A 64, 884–893.Google Scholar
Slud, E. V. (1993). The moment problem for polynomial forms in normal random variables. Ann. Prob. 21, 2200–2214.10.1214/aop/1176989017CrossRefGoogle Scholar
Surgailis, D. (1982). Zones of attraction of self-similar multiple integrals. Lithuanian Math. J. 22, 327–340.10.1007/BF00966427CrossRefGoogle Scholar
Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Prob. Theory Relat. Fields 31, 287–302.Google Scholar
Taqqu, M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Prob. Theory Relat. Fields 50, 53–83.Google Scholar
Tudor, C. (2013). Analysis of Variations for Self-Similar Processes: A Stochastic Calculus Approach. Springer, New York.10.1007/978-3-319-00936-0CrossRefGoogle Scholar
Tudor, C. A. (2008). Analysis of the Rosenblatt process. ESIAM Prob. Statist. 12, 230–257.10.1051/ps:2007037CrossRefGoogle Scholar
Williams, D. (1991). Probability with Martingales. Cambridge University Press.10.1017/CBO9780511813658CrossRefGoogle Scholar