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Exponential inequalities for self-normalized processes with applications

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Exponential inequalities for self-normalized processes with applications
2.	Creator	Author's name, affiliation, country	Victor H de la Peña; Columbia University
2.	Creator	Author's name, affiliation, country	Guodong Pang; Columbia University
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	self-normalization, exponential inequalities, martingales, hypothesis testing, stochastic Traveling Salesman Problem
3.	Subject	Subject classification	Primary: {60E15, 60G42, 60G44, 68M20; } Secondary: {62F03, 60G40}
4.	Description	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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2009-09-08
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ecp.ejpecp.org/article/view/1490
10.	Identifier	Digital Object Identifier	10.1214/ECP.v14-1490
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 14
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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