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

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 $(\Omega, \mathscr{H}, \widehat{\mathbb E})$. 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 applications, the self-normalized laws of the iterated logarithm are obtained.