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A general class of exponential inequalities for martingales and ratios. (English) Zbl 0942.60004
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 S. Kwapien and 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.

MSC:
60E15 Inequalities; stochastic orderings
60G42 Martingales with discrete parameter
60G44 Martingales with continuous parameter
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