Weak Convergence to the Two-Parameter Volterra Multifractional Process in Besov Spaces
Received:October 06, 2016  Revised:February 27, 2017
Key Word: multifractional Brownian sheet   Poisson process   weak convergence
Fund ProjectL:Junfeng LIU is partially supported by National Natural Science Foundation of China (Grant Nos.11401313; 11771209), Natural Science Foundation of Jiangsu Province (Grant No.BK20161579), China Postdoctoral Science Foundation (Grant Nos.2014M560368; 2015T80475) and 2014 Qing Lan Project. Xichao SUN is partially supported by National Natural Science Foundation of China (Grant No.11426036), Natural Science Foundation of Anhui Province (Grant No.1408085QA10) and Key Natural Science Foundation of Anhui Education Commission (Grant No.KJ2016A453).
 Author Name Affiliation Junfeng LIU Department of Statistics, Nanjing Audit University, Jiangsu 211815, P. R. China Xichao SUN Department of Mathematics and Physics, Bengbu College, Anhui 233030, P. R. China
Hits: 202
In this paper, we prove that two-parameter Volterra multifractional process can be approximated in law in the topology of the anisotropic Besov spaces by the family of processes $\{B_{n}(s,t)\}_{n\in \mathbb{N}}$ defined by $$B_{n}(s,t)=\int_{0}^{s}\int_{0}^{t}K_{\alpha(s)}(s,u)K_{\beta(t)}(t,v)\theta_{n}(u,v)\d u\d v,$$ where $\{\theta_{n}(u,v)\}_{n\in \mathbb{N}}$ is a family of processes, converging in law to a Brownian sheet as $n\rightarrow\infty$, based on the well known Donsker's theorem.