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Large deviations for weighted sums of stretched exponential random variables

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Large deviations for weighted sums of stretched exponential random variables
2.	Creator	Author's name, affiliation, country	Nina Gantert; Technische Universität München; Germany
2.	Creator	Author's name, affiliation, country	Kavita Ramanan; Brown University; United States
2.	Creator	Author's name, affiliation, country	Franz Rembart; University of Oxford; United Kingdom
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	large deviations, weighted sums, subexponential random variables, self-normalized weights, quenched and annealed large deviations, random projections, kernels, nonparametric regression
3.	Subject	Subject classification	60F10, 62G32
4.	Description	Abstract	We consider the probability that a weighted sum of n i.i.d. random variables $X_j, j = 1,\ldots,n$, with stretched exponential tails is larger than its expectation and   determine the rate of its decay, under suitable conditions on the weights. We show that the decay is subexponential, and  identify the rate function in terms of the tails of $X_j$ and the weights. Our result generalizes the large deviation principle given by Kiesel and Stadtmüller as well as the tail asymptotics for sums of i.i.d. random variables provided by Nagaev. As an application of our result, motivated by random projections of high-dimensional vectors, we consider the case of random, self-normalized weights that are independent of the sequence $X_j$, identify the decay rate for both the quenched and annealed large deviations in this case, and show that they coincide. As another example we consider weights derived from kernel functions that arise in nonparametric regression.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	NSF, ARO
7.	Date	(YYYY-MM-DD)	2014-07-12
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/3266
10.	Identifier	Digital Object Identifier	10.1214/ECP.v19-3266
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 19
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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