A Cramér Type Theorem for Weighted Random Variables
Jamal Najim, Université Paris 10-Nanterre
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 $(Z_i)_{iin N}$ are $R^d$-valued independent and identically
distributed random variables having some exponential moments, i.e.,
Eea |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.
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