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Large deviation exponential inequalities for supermartingales
- Source :
- Electron. Commun. Probab. 17 (2012), no 59, 1-8
- Publication Year :
- 2011
-
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 $P(\max_{1\leq k \leq n} S_k \geq n)=O(\exp\{-C_1 n^{\alpha}\}),$ $n\rightarrow \infty,$ 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 condition. In particular, when $\alpha=1/3$, we recover an inequality of Lesigne and Voln\'{y}.<br />Comment: 9 pages
- Subjects :
- Mathematics - Probability
60F10, 60G42
Subjects
Details
- Database :
- arXiv
- Journal :
- Electron. Commun. Probab. 17 (2012), no 59, 1-8
- Publication Type :
- Report
- Accession number :
- edsarx.1111.1407
- Document Type :
- Working Paper