Central limit theorem for linear processes generated by IID random variables under the sub-linear expectation

Abstract

In this paper, we investigate the central limit theorem and the invariance principle for linear processes generated by a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng [19]. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov’s central limit theorem and invariance principle to the case where probability measures are no longer additive.

Moderate Deviation Principle for Linear Processes Generated by Dependent Sequences under Sub-Linear Expectation

“…We also refer to refs. [21][22][23][24][25][26][27][28] for the other limit properties under the sub-linear expectation. Most work on the LDP (MDP) assumes that the random variables under discussion are independent despite the different definitions of independence.…”

Note on Precise Rates in the Law of Iterated Logarithm for the Moment Convergence of I.I.D.: Random Variables under Sublinear Expectations

“…Zhang [3][4][5] investigated the convergence of the sums of independent random variables, Lindeberg's central limit theorems for martingale like sequences, and Heyde's theorem under sublinear expectations. Liu and Zhang [6,7] proved the law of the iterated logarithm for linear processes generated by a sequence of stationary independent random variables, central limit theorem for linear processes generated by independent, and identically distributed (i. i. d.) random variables under sublinear expectations. For more relevant works under sublinear expectations, the reader could refer to Gao and Xu [8], Zhang [9][10][11][12], Wu [13], Xu and Cheng [14][15][16], Zhong and Wu [17], Xu and Zhang [18,19], Wu and Jiang [20], Chen [21], Fang et al [22], Hu et al [23], Hu and Yang [24], Kuczmaszewska [25], Ding [26], and references therein.…”

The Law of the Iterated Logarithm for Linear Processes Generated by a Sequence of Stationary Independent Random Variables under the Sub-Linear Expectation

“…Philips and Solo [15] prove the strong law of numbers and the law of iterated logarithm for linear processes, Zhang [16] gives the limit law of the iterated logarithm for linear processes. Recently, Liu and Zhang [17] obtained the central limit theorem and invariance principle for linear processes generated by independent and identically distributed (IID for short) random variables under sub-linear expectation.…”