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

Najim, Jamal

ESAIM: Probability and Statistics, Tome 9 (2005), pp. 116-142

A Large Deviation Principle (LDP) is proved for the family $\frac{1}{n}{\sum }_1^n𝐟\left(x_i^n\right)·Z_i^n$ where the deterministic probability measure $\frac{1}{n}{\sum }_1^n{\delta }_{x_i^n}$ converges weakly to a probability measure $R$ and ${\left(Z_i^n\right)}_{i\in \mathbb{N}}$ are ${ℝ}^d$-valued independent random variables whose distribution depends on $x_i^n$ and satisfies the following exponential moments condition:

$$\underset{i,n}{sup}𝔼{\mathrm{e}}^{{\alpha }^{*}\left|Z_i^n\right|}<+\infty \mathrm{forsome}0<{\alpha }^{*}<+\infty .$$

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.

Classification : 46E30, 60F10, 60G57
Keywords: large deviations, epigraphical convergence, Erdös-Rényi's law of large numbers

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

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

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

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

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

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

[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

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

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

[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

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

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

[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

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

[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

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

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

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

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

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

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

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

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

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

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

[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

[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

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

[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