A LIMIT LAW FOR FUNCTIONALS OF MULTIPLE INDEPENDENT FRACTIONAL BROWNIAN MOTIONS

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doi:10.1007/s10473-020-0311-6

Abstract: Let $B=\{B^H(t)\}_{t\geq0}$ be a $d$ -dimensional fractional Brownian motion with Hurst parameter $H\in (0,1)$ . Consider the functionals of $k$ independent $d$ -dimensional fractional Brownian motions

\frac{1}{\sqrt{n}} \int_{0}^{e^{n t_{1}}} \dots \int_{0}^{e^{n t_{k}}} f (B^{H, 1} (s_{1}) + \dots + B^{H, k} (s_{k})) d s_{1} \dots d s_{k},

$\frac{1}{\sqrt{n}} \int^{e^{nt_1}}_0\cdots\int^{e^{nt_k}}_0 f(B^{H,1}(s_1)+\cdots +B^{H,k}(s_k)){\rm d}s_1\cdots{\rm d}s_k,$ where the Hurst index

H = k / d

$H=k/d$ . Using the method of moments, we prove the limit law and extending a result by Xu \cite{xu} of the case

k = 1

$k=1$ . It can also be regarded as a fractional generalization of Biane \cite{biane} in the case of Brownian motion.

Key words: Limit theorem, fractional Brownian motion, method of moments, chaining argument

CLC Number:

60F17

Qian YU. A LIMIT LAW FOR FUNCTIONALS OF MULTIPLE INDEPENDENT FRACTIONAL BROWNIAN MOTIONS[J].Acta mathematica scientia,Series B, 2020, 40(3): 734-754.

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[1] Bi J, Xu F. A first-order limit law for functionals of two independent fractional Brownian motions in the critical case. J Theoret Probab, 2016, 29:941-957
[2] Biagini F, Hu Y. Øksendal B, Zhang T. Stochastic calculus for fractional Brownian motion and applications. Probability and its application. Berlin:Springer, 2008
[3] Biane P. Comportement asymptotique de certaines fonctionnelles additives de plusieurs mouvements browniens. Séminaire de Probabilités XXIII. Lect Notes Math. Berlin:Springer, 1989, 1372:198-233
[4] Hu Y, Wang B. Convergence rate of an approximation to multiple integral of fBm. Acta Mathematica Scientia, 2010, 30B(3):975-992
[5] Kallianpur G, Robbins H. Ergodic property of the Brownian motion process. Proc Nat Acad Sci U S A, 1953, 39:525-533
[6] Kasahara Y, Kotani S. On limit processes for a class of additive functionals of recurrent diffusion processes. Z Wahrsch Verw Gebiete, 197949:133-153
[7] LeGall J F. Propriétés d'intersection des marches aléatoires II. Etude des cas critiques. Commun Math Phys, 1986, 104:509-528
[8] Mandelbrot B B, Van Ness J W. Fractional Brownian motions, fractional noises and applications. SIAM Rev, 1968, 10:422-437
[9] Nualart D. Malliavin Calculus and Related Topics. Berlin:Springer, 2006
[10] Nualart D, Xu F. Central limit theorem for an additive functional of the fractional Brownian motion II. Electron Comm Probab, 2013, 18(74):1-10
[11] Nualart D, Xu F. Central limit theorem for functionals of two independent fractional Brownian motions. Stochastic Process Appl, 2014, 124(11):3782-3806
[12] Samorodnitsky G, Taqqu M S. Stable non-Gaussian Random Processes. Stochastic models with infinite variance. New York:Chapman & Hall, 1994
[13] Sato K I. Lévy Processes and Infinitely Divisible Distributions. New York:Cambridge University Press, 1999
[14] Shen G, Yan L, Liu J. Power variation of subfractional Brownian motion and application. Acta Mathematica Scientia, 2013, 33B(4):901-912
[15] Shen G, Yan L. An approximation of sub-fractional Brownian motion. Comm Statist Theory Methods, 2014, 43(9):1873-1886
[16] Song J, Xu F, Yu Q. Limit theorems for functionals of two independent Gaussian processes. Stochastic Process Appl, 2019, 129(11):4791-4836
[17] Tudor C. Some properties of the sub-fractional Brownian motion. Stochastics, 2007, 79(5):431-448
[18] Wang B. Convergence rate of multiple fractional stratonovich type integral for Hurst parameter less than 1/2. Acta Mathematica Scientia, 2011, 31B(5):1694-1708
[19] Xu F. Second-order limit laws for occupation times of the fractional Brownian motion. J Appl Prob, 2017, 54:444-461
[20] Yan L, Shen G. On the collision local time of sub-fractional Brownian motions. Statist Probab Lett, 2010, 80(5):296-308

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