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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Research supported by Grants from the National Natural Science Foundation of China (No. 11225104), 973 Program (No. 2015CB352302), and the Fundamental Research Funds for the Central Universities.
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Zhang, LX. Self-Normalized Moderate Deviation and Laws of the Iterated Logarithm Under G-Expectation. Commun. Math. Stat. 4, 229–263 (2016). https://doi.org/10.1007/s40304-015-0084-8
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DOI: https://doi.org/10.1007/s40304-015-0084-8