On moderate deviations for martingales
The paper considers square-integrable martingales \((X_t\), \(0\leq t\leq1)\) and investigates the large deviation problem \(P(X_1\geq r)=(1-\Phi(r))(1+\theta(r)R)\) for each \(|r|\leq C\), where \(|\theta(r)|\leq1\). The author receives a precise description of the remainder \(R\) and of the range \(C\) by means of \(L_{2\delta}=E\;\sum_{0<s\leq1}|\Delta X_s|^{2\delta}\) and \(N_{2\delta}=E|\langle X\rangle_1-1|^{1+\delta}\), where \(\Delta X\) is the jumping part and \(\langle X\rangle\) is the quadratic characteristic of the martingale \(X\). The result gives a rate for large deviations in martingale CLT and even improves the result known for the case of independent random variables.
scientific article; zbMATH DE number 1018385