Electronic Journal of Probability

A Cramér Type Theorem for Weighted Random Variables

Jamal Najim

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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., $$E e^{a |Z|} \lt \infty$$ for some $0 \lt a \lt \infty$. 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.

Article information

Electron. J. Probab., Volume 7 (2002), paper no. 4, 32 pp.

Accepted: 12 October 2001
First available in Project Euclid: 16 May 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F10: Large deviations
Secondary: 60G57: Random measures

Large Deviations empirical means empirical measures maximum entropy on the means

This work is licensed under aCreative Commons Attribution 3.0 License.


Najim, Jamal. A Cramér Type Theorem for Weighted Random Variables. Electron. J. Probab. 7 (2002), paper no. 4, 32 pp. doi:10.1214/EJP.v7-103. https://projecteuclid.org/euclid.ejp/1463434877

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