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Second-order limit laws for occupation times of fractional Brownian motion

  • Fangjun Xu (a1) (a2)

Abstract

We prove a second-order limit law for additive functionals of a d-dimensional fractional Brownian motion with Hurst index H = 1 / d, using the method of moments and extending the Kallianpur–Robbins law, and then give a functional version of this result. That is, we generalize it to the convergence of the finite-dimensional distributions for corresponding stochastic processes.

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Corresponding author

* Postal address: School of Statistics, East China Normal University, Shanghai, 200241, China. Email address: fjxu@finance.ecnu.edu.cn
** Postal address: NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, Shanghai 200062 China

References

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[1] Hu, Y., Nualart, D. and Xu, F. (2014). Central limit theorem for an additive functional of the fractional Brownian motion. Ann. Prob. 42, 168203.
[2] Kasahara, Y. (1982). A limit theorem for slowly increasing occupation times. Ann. Prob. 10, 728736.
[3] Kasahara, Y. (1984). Another limit theorem for slowly increasing occupation times. J. Math. Kyoto Univ. 24, 507520.
[4] Kasahara, Y. and Kosugi, N. (1997). A limit theorem for occupation times of fractional Brownian motion. Stoch. Process. Appl. 67, 161175.
[5] Kasahara, Y. and Kotani, S. (1979). On limit processes for a class of additive functionals of recurrent diffusion processes. Z. Wahrscheinlichkeitsth. 49, 133153.
[6] Kallianpur, G. and Robbins, H. (1953). Ergodic property of the Brownian motion process. Proc. Nat. Acad. Sci. USA. 39, 525533.
[7] Kôno, N. (1996). Kallianpur–Robbins law for fractional Brownian motion. In Probability Theory and Mathematical Statistics (Tokyo, 1995), World Scientific, River Edge, NJ, pp. 229236.
[8] Nualart, D. and Xu, F. (2013). Central limit theorem for an additive functional of the fractional Brownian motion II. Electron. Commun. Prob. 18, 10pp.
[9] Nualart, D. and Xu, F. (2014). A second order limit law for occupation times of the Cauchy process. Stochastics 86, 967974.

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Second-order limit laws for occupation times of fractional Brownian motion

  • Fangjun Xu (a1) (a2)

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