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.
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