## Electronic Communications in Probability

### Exponential inequalities for self-normalized processes with applications

#### Abstract

We prove the following exponential inequality for a pair of random variables $(A,B)$ with $B >0$ satisfying the canonical assumption, $E[\exp(\lambda A - \frac{\lambda^2}{2} B^2)]\leq 1$ for $\lambda \in R$, $$P\left( \frac{|A|}{\sqrt{ \frac{2q-1}{q} \left(B^2+ (E[|A|^p])^{2/p} \right) }} \geq x \right) \leq \left(\frac{q}{2q-1} \right)^{\frac{q}{2q-1}} x^{-\frac{q}{2q-1}} e^{-x^2/2}$$ for $x>0$, where $1/p+ 1/q =1$ and $p\geq1$. Applying this inequality, we obtain exponential bounds for the tail probabilities for self-normalized martingale difference sequences. We propose a method of hypothesis testing for the $L^p$-norm $(p \geq 1)$ of $A$ (in particular, martingales) and some stopping times. We apply this inequality to the stochastic TSP in $[0,1]^d$ ($d\geq 2$), connected to the CLT.

#### Article information

Source
Electron. Commun. Probab., Volume 14 (2009), paper no. 37, 372-381.

Dates
Accepted: 8 September 2009
First available in Project Euclid: 6 June 2016

https://projecteuclid.org/euclid.ecp/1465234746

Digital Object Identifier
doi:10.1214/ECP.v14-1490

Mathematical Reviews number (MathSciNet)
MR2545288

Zentralblatt MATH identifier
1189.60042

Rights

#### Citation

de la Peña, Victor; Pang, Guodong. Exponential inequalities for self-normalized processes with applications. Electron. Commun. Probab. 14 (2009), paper no. 37, 372--381. doi:10.1214/ECP.v14-1490. https://projecteuclid.org/euclid.ecp/1465234746

#### References

• B. Bercu and A. Touati. Exponential inequalities for self-normalized martingales with applications. Ann. Appl. Probab. (2008), 18, 1848–1869.
• N.J. Cerf, J. Boutet de Monvel, O. Bohigas, O.C. Martin and A.G. Percus. The random link approximation for the Euclidean Traveling Salesman Problem. Journal de Physique I. (1997), 7, 117–136.
• V.H. de la Peña. A general class of exponential inequalities for martingales and ratios. Ann. Probab. (1999), 27, 537–564.
• V.H. de la Peña, M.J. Klass and T.L. Lai. Self-normalized processes: exponential inequalities, moment bounds and iterated logarithm laws. Ann. Probab.. (2004), Vol. 32, No.3A, 1902–1933.
• V.H. de la Peña, M.J. Klass and T.L. Lai. Pseudo-maximization and self-normalized processes. Probability Surveys. (2007), Vol. 4,172–192.
• V.H. de la Peña, T.L. Lai and Q.M. Shao. Self-Normalized Processes: Limit Theory and Statistical Applications. Springer. (2009).
• V. Egorov. On the Growth Rate of Moments of Random Sums. Preprint. (1998).
• E. Giné, F. G ötze and D. Mason. When is the Student $t$-statistic asymptotically standard normal? Ann. Probab. (1997), 25, 1514–1531.
• B. Efron. Student's $t$-test under symmetry conditions. J. Amer. Statist. Assoc. (1969), 64, 1278–1302.
• I. Karatzas and S.E. Shreve. Brownian Motion and Stochastic Calculus. 2nd ed. Springer. (1991).
• B.F. Logan, C.L. Mallows, S.O. Rice and L.A. Shepp. Limit Distributions of Self-Normalized Sums. Ann. Probab. (1973), 1, 788–809.
• W.T. Rhee and M. Talagrand. Martingale inequalities and NP-complete problems. Mathematics of Operations Research. (1987), 12, 177–181.
• W.T. Rhee and M. Talagrand. Martingale inequalities, interpolation and NP-complete problems. Mathematics of Operations Research. (1989a), 13, 91–96.
• W.T. Rhee and M. Talagrand. A sharp deviation inequality for the stochastic Traveling Salesman Problem. Ann. Probab. (1989b), 17, 1–8.
• J.M. Steele. Complete convergence of short paths and Karp's algorithm for the TSP. Mathematics of Operations Research. (1981), 6, 374–378.
• J.M. Steele. Probability Theory and Combinatorial Optimization. CBMS-NSF regional conference series in applied mathematics. (1997).