Abstract
Under certain mild conditions, limit theorems for additive functionals of some d-dimensional self-similar Gaussian processes are obtained. These limit theorems work for general Gaussian processes including fractional Brownian motions, subfractional Brownian motions and bi-fractional Brownian motions. To prove these results, we use the method of moments and an enhanced chaining argument. The Gaussian processes under consideration are required to satisfy certain strong local nondeterminism property. A tractable sufficient condition for the strong local nondeterminism property is given and it only relays on the covariance functions of the Gaussian processes. Moreover, we give a sufficient condition for the distribution function of a random vector to be determined by its moments.
Funding Statement
The third author was partially supported by National Natural Science Foundation of China (Grant No.11871219 and Grant No. 12371156).
Acknowledgments
The authors would like to thank Professor Yimin Xiao for very helpful discussions on the local nondeterminism and the strong local nondeterminism for Gaussian processes. The authors would also like to thank Editors and anonymous referees for making very valuable comments.
The third author is the corresponding author and also the affiliated faculty for NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai.
Citation
Minhao Hong. Heguang Liu. Fangjun Xu. "Limit theorems for additive functionals of some self-similar Gaussian processes." Ann. Appl. Probab. 34 (6) 5462 - 5497, December 2024. https://doi.org/10.1214/24-AAP2096
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