# Large deviations for independent random variables – Application to Erdös-Renyi’s functional law of large numbers

ESAIM: Probability and Statistics (2005)

- Volume: 9, page 116-142
- ISSN: 1292-8100

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topNajim, Jamal. "Large deviations for independent random variables – Application to Erdös-Renyi’s functional law of large numbers." ESAIM: Probability and Statistics 9 (2005): 116-142. <http://eudml.org/doc/245785>.

@article{Najim2005,

abstract = {A Large Deviation Principle (LDP) is proved for the family $\frac\{1\}\{n\}\sum _1^n \mathbf \{f\}(x_i^n) \cdot Z^n_i$ where the deterministic probability measure $\frac\{1\}\{n\}\sum _1^n \delta _\{x_i^n\}$ converges weakly to a probability measure $R$ and $(Z^n_i)_\{i\in \mathbb \{N\}\}$ are $\mathbb \{R\}^d$-valued independent random variables whose distribution depends on $x_i^n$ and satisfies the following exponential moments condition:\[\hspace*\{-56.9055pt\}\sup \_\{i,n\} \{\mathbb \{E\}\}\{\rm e\}^\{\alpha ^* |Z\_i^n|\}< +\infty \quad \textrm \{for some\}\quad 0<\alpha ^*<+\infty .\]In this context, the identification of the rate function is non-trivial due to the absence of equidistribution. We rely on fine convex analysis to address this issue. Among the applications of this result, we extend Erdös and Rényi’s functional law of large numbers.},

author = {Najim, Jamal},

journal = {ESAIM: Probability and Statistics},

keywords = {large deviations; epigraphical convergence; Erdös-Rényi’s law of large numbers; Erdős-Rényi’s law of large numbers},

language = {eng},

pages = {116-142},

publisher = {EDP-Sciences},

title = {Large deviations for independent random variables – Application to Erdös-Renyi’s functional law of large numbers},

url = {http://eudml.org/doc/245785},

volume = {9},

year = {2005},

}

TY - JOUR

AU - Najim, Jamal

TI - Large deviations for independent random variables – Application to Erdös-Renyi’s functional law of large numbers

JO - ESAIM: Probability and Statistics

PY - 2005

PB - EDP-Sciences

VL - 9

SP - 116

EP - 142

AB - A Large Deviation Principle (LDP) is proved for the family $\frac{1}{n}\sum _1^n \mathbf {f}(x_i^n) \cdot Z^n_i$ where the deterministic probability measure $\frac{1}{n}\sum _1^n \delta _{x_i^n}$ converges weakly to a probability measure $R$ and $(Z^n_i)_{i\in \mathbb {N}}$ are $\mathbb {R}^d$-valued independent random variables whose distribution depends on $x_i^n$ and satisfies the following exponential moments condition:\[\hspace*{-56.9055pt}\sup _{i,n} {\mathbb {E}}{\rm e}^{\alpha ^* |Z_i^n|}< +\infty \quad \textrm {for some}\quad 0<\alpha ^*<+\infty .\]In this context, the identification of the rate function is non-trivial due to the absence of equidistribution. We rely on fine convex analysis to address this issue. Among the applications of this result, we extend Erdös and Rényi’s functional law of large numbers.

LA - eng

KW - large deviations; epigraphical convergence; Erdös-Rényi’s law of large numbers; Erdős-Rényi’s law of large numbers

UR - http://eudml.org/doc/245785

ER -

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