Electronic Communications in Probability

Exponential inequalities for self-normalized processes with applications

Victor de la Peña and Guodong Pang

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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

Permanent link to this document
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

Subjects
Primary: 60E15: Inequalities; stochastic orderings 60G42: Martingales with discrete parameter 60G44: Martingales with continuous parameter 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx]
Secondary: 62F03: Hypothesis testing 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Keywords
self-normalization exponential inequalities martingales hypothesis testing stochastic Traveling Salesman Problem

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

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


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