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Large deviation exponential inequalities for supermartingales

   
@article{ECP2318,
	author = {Xiequan Fan and Ion Grama and Quansheng Liu},
	title = {Large deviation exponential inequalities for supermartingales},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {17},
	year = {2012},
	keywords = {Large deviation; martingales; exponential inequality; Bernstein type inequality},
	abstract = {Let $(X_{i}, \mathcal{F}_{i})_{i\geq 1}$ be a sequence of supermartingale differences and let $S_k=\sum_{i=1}^k X_i$. We give an exponential moment condition under which $\mathbb{P}( \max_{1\leq k \leq n} S_k \geq n)=O(\exp\{-C_1 n^{\alpha}\}),$ $n\rightarrow \infty, $ where $\alpha \in (0, 1)$ is given and $C_{1}>0$ is a constant. We also show that the power $\alpha$ is optimal under the given moment condition.},
	pages = {no. 59, 1-8},
	issn = {1083-589X},
	doi = {10.1214/ECP.v17-2318},    
        url = {http://ecp.ejpecp.org/article/view/2318}}