Self-Normalized Moderate Deviation and Laws of the Iterated Logarithm Under G-Expectation

Abstract

The sub-linear expectation or called G-expectation is a non-linear 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 $(\Omega , \mathscr {H}, \widehat{\mathbb {E}})$. 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 application, the self-normalized laws of the iterated logarithm are obtained. A Bernstein’s type inequality is also established for proving the law of the iterated logarithm.

Keywords

Non-linear expectation / Capacity / Self-normalization / Law of the iterated logarithm / Moderate deviation

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Funding

National Natural Science Foundation of China(11225104)