Abstract
G-Brownian motion has a very rich and interesting new structure that nontrivially generalizes the classical Brownian motion. Its quadratic variation process is also a continuous process with independent and stationary increments. We prove a self-normalized functional central limit theorem for independent and identically distributed random variables under the sub-linear expectation with the limit process being a G-Brownian motion self-normalized by its quadratic variation. To prove the self-normalized central limit theorem, we also establish a new Donsker’s invariance principle with the limit process being a generalized G-Brownian motion.
Highlights
Let {Xn; n ≥ 1} be a sequence of independent and identically distributed random variables on a probability space (, F, P )
The purpose of this paper is to establish the self-normalized central limit theorem under the sub-linear expectation
The central limit theorem under the sub-linear expectation was first established by
Summary
Keywords Sub-linear expectation · G-Brownian motion · Central limit theorem · Invariance principle · Self-normalization Let {Xn; n ≥ 1} be a sequence of independent and identically distributed random variables on a probability space ( , F , P ). The purpose of this paper is to establish the self-normalized central limit theorem under the sub-linear expectation. The central limit theorem under the sub-linear expectation was first established by
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