A Large Deviation Principle (LDP) is proved for the family $\frac{1}{n}{\sum}_{1}^{n}\mathbf{f}\left({x}_{i}^{n}\right)\xb7{Z}_{i}^{n}$ where the deterministic probability measure $\frac{1}{n}{\sum}_{1}^{n}{\delta}_{{x}_{i}^{n}}$ converges weakly to a probability measure $R$ and ${\left({Z}_{i}^{n}\right)}_{i\in \mathbb{N}}$ are ${\mathbb{R}}^{d}$-valued independent random variables whose distribution depends on ${x}_{i}^{n}$ and satisfies the following exponential moments condition:

$$\phantom{\rule{-56.9055pt}{0ex}}\underset{i,n}{sup}\mathbb{E}{\mathrm{e}}^{{\alpha}^{*}\left|{Z}_{i}^{n}\right|}<+\infty \phantom{\rule{1em}{0ex}}\mathrm{forsome}\phantom{\rule{1em}{0ex}}0<{\alpha}^{*}<+\infty .$$ |

Keywords: large deviations, epigraphical convergence, Erdös-Rényi's law of large numbers

@article{PS_2005__9__116_0, author = {Najim, Jamal}, title = {Large deviations for independent random variables - {Application} to {Erd\"os-Renyi's} functional law of large numbers}, journal = {ESAIM: Probability and Statistics}, pages = {116--142}, publisher = {EDP-Sciences}, volume = {9}, year = {2005}, doi = {10.1051/ps:2005006}, zbl = {1136.60323}, mrnumber = {2148963}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps:2005006/} }

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 SP - 116 EP - 142 VL - 9 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2005006/ DO - 10.1051/ps:2005006 LA - en ID - PS_2005__9__116_0 ER -

%0 Journal Article %A Najim, Jamal %T Large deviations for independent random variables - Application to Erdös-Renyi's functional law of large numbers %J ESAIM: Probability and Statistics %D 2005 %P 116-142 %V 9 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2005006/ %R 10.1051/ps:2005006 %G en %F PS_2005__9__116_0

Najim, Jamal. Large deviations for independent random variables - Application to Erdös-Renyi's functional law of large numbers. ESAIM: Probability and Statistics, Volume 9 (2005), pp. 116-142. doi : 10.1051/ps:2005006. http://www.numdam.org/articles/10.1051/ps:2005006/

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