Limit theorems for additive functionals of the fractional Brownian motion

• 成果类型: Article
• 署名作者: Jaramillo, Arturo; Nourdin, Ivan; Nualart, David; Peccati, Giovanni
• 署名单位: CIMAT - Centro de Investigacion en Matematicas; University of Luxembourg; University of Kansas
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1612
• 发表日期: 2023
• 页码: 1066-1111
• 关键词: local-times; zero-energy

摘要

We investigate first and second order fluctuations of additive functionals of a fractional Brownian motion (fBm) of the form [GRAPHICS] . where B={B-t; t >= 0} is a fBm with Hurst parameter H is an element of(0,1), f is a suitable test function and lambda is an element of R. We develop our study by distinguishing two regimes which exhibit different behaviors. When H is an element of(0,1/3), we show that a suitable renormalization of (0.1), compensated by a multiple of the local time of B, converges toward a constant multiple of the derivative of the local time of B. In contrast, in the case H is an element of[1/3,1) we show that (0.1) converges toward an independent Brownian motion subordinated to the local time of B. Our results refine and complement those from (Ann. Appl. Probab. 31 (2021) 2143-2191), (Jeganathani (2006)), (Ann. Probab. 42 (2014) 168-203), (Electron. Commun. Probab. 74 (2013) 18) and solve at the same time the critical case H=1/3 which had remained open until now.