A general class of exponential inequalities for martingales and ratios
The author derives several new exponential inequalities for a martingale difference sequence \((d_i,F_i)\), which satisfies either \(E[|d_j|^k\mid F_{j-1}]\leq (k!/2)\sigma^2_j c^{k-2}\) or \(P(|d_j|\leq c\mid F_{j-1})= 1\), for \(k>2\), \(0< c<\infty\), where \(\sigma^2_j= E[d^2_j\mid F_{j-1}]\). For example, put \(V^2_n= \sum^n_{j=1} \sigma^2_j\) and \(M_m= \sum^n_{j=1} d_j\), and then for all \(x,y>0\), \[ P(M_n\geq x, V^2_n\text{ for some }n)\leq \exp\Biggl\{-{x^2\over 2(y+ cx)}\Biggr\}. \] The proof of the above and related results is based on decoupling techniques introduced by \textit{S. Kwapien} and \textit{W. A. Woyczynski} [in: Almost everywhere convergence, 237-265 (1989; Zbl 0693.60033)] and further developed by the author and others. The author gives also results for continuous time square integrable martingales. He also shows that in the special case of conditionally symmetric random variables the integrability conditions on the sequence \(d_j\) can be relaxed.
scientific article; zbMATH DE number 1359640
scientific article