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

Abstract

In this paper, we obtain the law of iterated logarithm for linear processes in sub-linear expectation space. It is established for strictly stationary independent random variable sequences with finite second-order moments in the sense of non-additive capacity.

The Law of the Iterated Logarithm

“…We suggest a paper N.H. Bingham (Bingham, 1986) to know about various other contexts where a LIL has been established. For LIL in Banach space, please refer (Ledoux & Talagrand, 1991)For the various results on the LIL for random vectors and for some open problems in LIL, please refer (Liu & Zhang, 2021).…”

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.…”