## Electronic Communications in Probability

### Large deviation exponential inequalities for supermartingales

#### 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 $\alpha \in$ is given and $C_{1}>0$ is a constant. We also show that the power $\alpha$ is optimal under the given moment condition.

#### Article information

Source
Electron. Commun. Probab., Volume 17 (2012), paper no. 59, 8 pp.

Dates
Accepted: 12 December 2012
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465263192

Digital Object Identifier
doi:10.1214/ECP.v17-2318

Mathematical Reviews number (MathSciNet)
MR3005732

Zentralblatt MATH identifier
1312.60016

Rights

#### Citation

Fan, Xiequan; Grama, Ion; Liu, Quansheng. Large deviation exponential inequalities for supermartingales. Electron. Commun. Probab. 17 (2012), paper no. 59, 8 pp. doi:10.1214/ECP.v17-2318. https://projecteuclid.org/euclid.ecp/1465263192

#### References

• Dedecker, Jérôme. Exponential inequalities and functional central limit theorems for a random fields. ESAIM Probab. Statist. 5 (2001), 77–104 (electronic).
• Delyon, Bernard. Exponential inequalities for sums of weakly dependent variables. Electron. J. Probab. 14 (2009), no. 28, 752–779.
• Dzhaparidze, K.; van Zanten, J. H. On Bernstein-type inequalities for martingales. Stochastic Process. Appl. 93 (2001), no. 1, 109–117.
• Fan, Xiequan; Grama, Ion; Liu, Quansheng. Hoeffding's inequality for supermartingales. Stochastic Process. Appl. 122 (2012), no. 10, 3545–3559.
• Freedman, David A. On tail probabilities for martingales. Ann. Probability 3 (1975), 100–118.
• Hall, P.; Heyde, C. C. Martingale limit theory and its application. Probability and Mathematical Statistics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. xii+308 pp. ISBN: 0-12-319350-8
• Laib, NaÃ¢mane. Exponential-type inequalities for martingale difference sequences. Application to nonparametric regression estimation. Comm. Statist. Theory Methods 28 (1999), no. 7, 1565–1576.
• Lanzinger, H.; StadtmÃ¼ller, U. Maxima of increments of partial sums for certain subexponential distributions. Stochastic Process. Appl. 86 (2000), no. 2, 307–322.
• Lesigne, Emmanuel; VolnÃ½, Dalibor. Large deviations for martingales. Stochastic Process. Appl. 96 (2001), no. 1, 143–159.
• Liu, Quansheng; Watbled, Frédérique. Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment. Stochastic Process. Appl. 119 (2009), no. 10, 3101–3132.
• Merlevède, Florence; Peligrad, Magda; Rio, Emmanuel. Bernstein inequality and moderate deviations under strong mixing conditions. High dimensional probability V: the Luminy volume, 273–292, Inst. Math. Stat. Collect., 5, Inst. Math. Statist., Beachwood, OH, 2009.
• Merlevède, Florence; Peligrad, Magda; Rio, Emmanuel. A Bernstein type inequality and moderate deviations for weakly dependent sequences. Probab. Theory Related Fields 151 (2011), no. 3-4, 435–474.