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A Cramér Type Theorem for Weighted Random Variables

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1.	Title	Title of document	A Cramér Type Theorem for Weighted Random Variables
2.	Creator	Author's name, affiliation, country	Jamal Najim; Université Paris 10-Nanterre
3.	Subject	Discipline(s)	Mathematics
3.	Subject	Keyword(s)	Large Deviations, empirical means, empirical measures, maximum entropy on the means
3.	Subject	Subject classification	60F10, 60G57
4.	Description	Abstract	A Large Deviation Principle (LDP) is proved for the family $(1/n)\sum_1^n f(x_i^n) Z_i$ where $(1/n)\sum_1^n \delta_{x_i^n}$ converges weakly to a probability measure on $R$ and $(Z_i)_{i\in N}$ are $R^d$-valued independent and identically distributed random variables having some exponential moments, i.e., $$E e^{a |Z|}< \infty$$ for some $0< a< \infty$. The main improvement of this work is the relaxation of the steepness assumption concerning the cumulant generating function of the variables $(Z_i)_{i \in N}$. In fact, Gärtner-Ellis' theorem is no longer available in this situation. As an application, we derive a LDP for the family of empirical measures $(1/n) \sum_1^n Z_i \delta_{x_i^n}$. These measures are of interest in estimation theory (see Gamboa et al., Csiszar et al.), gas theory (see Ellis et al., van den Berg et al.), etc. We also derive LDPs for empirical processes in the spirit of Mogul'skii's theorem. Various examples illustrate the scope of our results.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2001-10-12
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/103
10.	Identifier	Digital Object Identifier	10.1214/EJP.v7-103
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 7
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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