A LIMIT LAW FOR FUNCTIONALS OF MULTIPLE INDEPENDENT FRACTIONAL BROWNIAN MOTIONS

doi:10.1007/s10473-020-0311-6

数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (3): 734-754.doi: 10.1007/s10473-020-0311-6

A LIMIT LAW FOR FUNCTIONALS OF MULTIPLE INDEPENDENT FRACTIONAL BROWNIAN MOTIONS

余迁

School of Statistics, East China Normal University, Shanghai 200241, China

收稿日期:2018-12-12 修回日期:2019-10-14 出版日期:2020-06-25 发布日期:2020-07-17
作者简介:Qian YU,E-mail:qyumath@163.com
基金资助:
Q. Yu is partially supported by ECNU Academic Innovation Promotion Program for Excellent Doctoral Students (YBNLTS2019-010) and the Scientific Research Innovation Program for Doctoral Students in Faculty of Economics and Management (2018FEM-BCKYB014).

A LIMIT LAW FOR FUNCTIONALS OF MULTIPLE INDEPENDENT FRACTIONAL BROWNIAN MOTIONS

Qian YU

School of Statistics, East China Normal University, Shanghai 200241, China

Received:2018-12-12 Revised:2019-10-14 Online:2020-06-25 Published:2020-07-17
Supported by:
Q. Yu is partially supported by ECNU Academic Innovation Promotion Program for Excellent Doctoral Students (YBNLTS2019-010) and the Scientific Research Innovation Program for Doctoral Students in Faculty of Economics and Management (2018FEM-BCKYB014).

摘要/Abstract

摘要： 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 $$ \frac{1}{\sqrt{n}} \int^{e^{nt_1}}_0\cdots\int^{e^{nt_k}}_0 f(B^{H,1}(s_1)+\cdots +B^{H,k}(s_k)){\rm d}s_1\cdots{\rm d}s_k, $$ where the Hurst index $H=k/d$. Using the method of moments, we prove the limit law and extending a result by Xu \cite{xu} of the case $k=1$. It can also be regarded as a fractional generalization of Biane \cite{biane} in the case of Brownian motion.

关键词: Limit theorem, fractional Brownian motion, method of moments, chaining argument

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 $$ \frac{1}{\sqrt{n}} \int^{e^{nt_1}}_0\cdots\int^{e^{nt_k}}_0 f(B^{H,1}(s_1)+\cdots +B^{H,k}(s_k)){\rm d}s_1\cdots{\rm d}s_k, $$ where the Hurst index $H=k/d$. Using the method of moments, we prove the limit law and extending a result by Xu \cite{xu} of the case $k=1$. 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

中图分类号:

60F17

余迁. A LIMIT LAW FOR FUNCTIONALS OF MULTIPLE INDEPENDENT FRACTIONAL BROWNIAN MOTIONS[J]. 数学物理学报(英文版), 2020, 40(3): 734-754.

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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A LIMIT LAW FOR FUNCTIONALS OF MULTIPLE INDEPENDENT FRACTIONAL BROWNIAN MOTIONS

A LIMIT LAW FOR FUNCTIONALS OF MULTIPLE INDEPENDENT FRACTIONAL BROWNIAN MOTIONS

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