Large deviations for independent random variables - Application to Erdös-Renyi's functional law of large numbers
ESAIM: Probability and Statistics, Tome 9 (2005), pp. 116-142.

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

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

DOI : https://doi.org/10.1051/ps:2005006
Classification : 46E30,  60F10,  60G57
Mots clés : 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},
mrnumber = {2148963},
zbl = {1136.60323},
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
DA  - 2005///
SP  - 116
EP  - 142
VL  - 9
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2005006/
UR  - https://www.ams.org/mathscinet-getitem?mr=2148963
UR  - https://zbmath.org/?q=an%3A1136.60323
UR  - https://doi.org/10.1051/ps:2005006
DO  - 10.1051/ps:2005006
LA  - en
ID  - PS_2005__9__116_0
ER  - 
Najim, Jamal. Large deviations for independent random variables - Application to Erdös-Renyi's functional law of large numbers. ESAIM: Probability and Statistics, Tome 9 (2005), pp. 116-142. doi : 10.1051/ps:2005006. http://www.numdam.org/articles/10.1051/ps:2005006/

[1] G. Ben Arous, A. Dembo and A. Guionnet, Aging of spherical spin glasses. Probab. Theory Related Fields 120 (2001) 1-67. | Zbl 0993.60055

[2] K.A. Borovkov, The functional form of the Erdős-Rényi law of large numbers. Teor. Veroyatnost. i Primenen. 35 (1990) 758-762. | Zbl 0717.60042

[3] Z. Chi, The first-order asymptotic of waiting times with distortion between stationary processes. IEEE Trans. Inform. Theory 47 (2001) 338-347. | Zbl 0996.60049

[4] Z. Chi, Stochastic sub-additivity approach to the conditional large deviation principle. Ann. Probab. 29 (2001) 1303-1328. | Zbl 1018.60026

[5] I. Csiszár, Sanov property, generalized $I$-projection and a conditionnal limit theorem. Ann. Probab. 12 (1984) 768-793. | Zbl 0544.60011

[6] D.A. Dawson and J. Gärtner, Multilevel large deviations and interacting diffusions. Probab. Theory Related Fields 98 (1994) 423-487. | Zbl 0794.60015

[7] D.A. Dawson and J. Gärtner, Analytic aspects of multilevel large deviations, in Asymptotic methods in probability and statistics (Ottawa, ON, 1997). North-Holland, Amsterdam (1998) 401-440. | Zbl 0931.60068

[8] P. Deheuvels, Functional Erdős-Rényi laws. Studia Sci. Math. Hungar. 26 (1991) 261-295. | Zbl 0767.60029

[9] A. Dembo and I. Kontoyiannis, The asymptotics of waiting times between stationary processes, allowing distortion. Ann. Appl. Probab. 9 (1999) 413-429. | Zbl 0940.60033

[10] A. Dembo and T. Zajic, Large deviations: from empirical mean and measure to partial sums process. Stochastic Process. Appl. 57 (1995) 191-224. | Zbl 0838.60026

[11] A. Dembo and O. Zeitouni, Large Deviations Techniques And Applications. Springer-Verlag, New York, second edition (1998). | MR 1619036 | Zbl 0896.60013

[12] J. Dieudonné, Calcul infinitésimal. Hermann, Paris (1968). | MR 226971 | Zbl 0155.10001

[13] H. Djellout, A. Guillin and L. Wu, Large and moderate deviations for quadratic empirical processes. Stat. Inference Stoch. Process. 2 (1999) 195-225. | Zbl 1059.60501

[14] R.M. Dudley, Real Analysis and Probability. Wadsworth and Brooks/Cole (1989). | MR 982264 | Zbl 0686.60001

[15] R.S. Ellis, J. Gough and J.V. Pulé, The large deviation principle for measures with random weights. Rev. Math. Phys. 5 (1993) 659-692. | Zbl 0797.60027

[16] P. Erdős and A. Rényi, On a new law of large numbers. J. Anal. Math. 23 (1970) 103-111. | Zbl 0225.60015

[17] F. Gamboa and E. Gassiat, Bayesian methods and maximum entropy for ill-posed inverse problems. Ann. Statist. 25 (1997) 328-350. | Zbl 0871.62010

[18] N. Gantert, Functional Erdős-Renyi laws for semiexponential random variables. Ann. Probab. 26 (1998) 1356-1369. | Zbl 0945.60026

[19] G. Högnäs, Characterization of weak convergence of signed measures on $\left[0,1\right]$. Math. Scand. 41 (1977) 175-184. | Zbl 0368.28007

[20] C. Léonard and J. Najim, An extension of Sanov's theorem: application to the Gibbs conditioning principle. Bernoulli 8 (2002) 721-743. | Zbl 1013.60018

[21] J. Lynch and J. Sethuraman, Large deviations for processes with independent increments. Ann. Probab. 15 (1987) 610-627. | Zbl 0624.60045

[22] J. Najim, A Cramér type theorem for weighted random variables. Electron. J. Probab. 7 (2002) 32 (electronic). | MR 1887624 | Zbl 1011.60005

[23] R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton (1970). | MR 274683 | Zbl 0193.18401

[24] R.T. Rockafellar, Integrals which are convex functionals, II. Pacific J. Math. 39 (1971) 439-469. | Zbl 0236.46031

[25] R.T. Rockafellar and R.J-B. Wets, Variational Analysis. Springer (1998). | MR 1491362 | Zbl 0888.49001

[26] G.R. Sanchis, Addendum: “A functional limit theorem for Erdős and Rényi's law of large numbers”. Probab. Theory Related Fields 99 (1994) 475. | Zbl 0803.60031

[27] G.R. Sanchis, A functional limit theorem for Erdős and Rényi's law of large numbers. Probab. Theory Related Fields 98 (1994) 1-5. | Zbl 0794.60018

[28] P.H. Schuette, Large deviations for trajectories of sums of independent random variables. J. Theoret. Probab. 7 (1994) 3-45. | Zbl 0808.60032

[29] S.L. Zabell, Mosco convergence and large deviations, in Probability in Banach spaces, 8 (Brunswick, ME, 1991). Birkhäuser Boston, Boston, MA, Progr. Probab. 30 (1992) 245-252. | Zbl 0795.60023

Cité par Sources :