Open Access
2009 Exponential inequalities for self-normalized processes with applications
Victor de la Peña, Guodong Pang
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Electron. Commun. Probab. 14: 372-381 (2009). DOI: 10.1214/ECP.v14-1490


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.


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Victor de la Peña. Guodong Pang. "Exponential inequalities for self-normalized processes with applications." Electron. Commun. Probab. 14 372 - 381, 2009.


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

zbMATH: 1189.60042
MathSciNet: MR2545288
Digital Object Identifier: 10.1214/ECP.v14-1490

Primary: 60E15 , 60G42 , 60G44 , 68M20
Secondary: 60G40 , 62F03

Keywords: Exponential inequalities , Hypothesis testing , Martingales , self-normalization , stochastic traveling salesman problem

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