Self-normalized moderate deviation and laws of the iterated logarithm under G-expectation

Abstract

The sub-linear expectation or called G-expectation is a nonlinear expectation having advantage of modeling non-additive probability problems and the volatility uncertainty in finance. Let $\{X_n;n\ge 1\}$ be a sequence of independent random variables in a sub-linear expectation space $(\Omega ,\mathcal{H},\hat{\mathbb{E}})$. Denote $S_n={\sum }_{k=1}^n{X}_k$ and $V_n^2={\sum }_{k=1}^n{X}_k^2$. In this paper, a moderate deviation for self-normalized sums, that is, the asymptotic capacity of the event $\{S_n/V_n\ge x_n\}$ for $x_n=o(\sqrt{n})$, is found both for identically distributed random variables and independent but not necessarily identically distributed random variables. As an applications, the self-normalized laws of the iterated logarithm are obtained.

arXiv admin note: text overlap with arXiv:1507.07600

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• Rosenthal's inequalities for independent and negatively dependent random variables under sub-linear expectations with applications
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Cited by in corpus (1)

• Exponential inequalities under sub-linear expectations with applications to laws of the iterated logarithm