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

ESAIM: Probability and Statistics (2010)

- 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 (2010): 116-142. <http://eudml.org/doc/104326>.

@article{Najim2010,

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:
$$ \sup\_\{i,n\} \{\mathbb E\}\{\rm e\}^\{\alpha^* |Z\_i^n|\}< +\infty \quad\textrm\{for some\}\quad 0<\alpha^*<+\infty.$$
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. ; large deviations; Erdős-Rényi’s law of large numbers},

language = {eng},

month = {3},

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/104326},

volume = {9},

year = {2010},

}

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

DA - 2010/3//

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:
$$ \sup_{i,n} {\mathbb E}{\rm e}^{\alpha^* |Z_i^n|}< +\infty \quad\textrm{for some}\quad 0<\alpha^*<+\infty.$$
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. ; large deviations; Erdős-Rényi’s law of large numbers

UR - http://eudml.org/doc/104326

ER -

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