An empirical almost sure central limit theorem under the weak dependence assumptions and its application to copula processes

Dudziński, Marcin

Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 71 (2017) no. 1.

Résumé

Let: 𝐘=( 𝐘_i), where 𝐘_i=( Y_i,1,...,Y_i,d), i=1,2,…, be a d-dimensional, identically distributed, stationary, centered process with uniform marginals and a joint cdf F, and F_n( 𝐱) :=1/n∑_i=1^n𝕀(Y_i,1≤ x_1,… ,Y_i,d≤ x_d) denote the corresponding empirical cdf. In our work, we prove the almost sure central limit theorem for an empirical process B_n=√(n)( F_n-F) under some weak dependence conditions due to Doukhan and Louhichi. Some application of the established result to copula processes is also presented.

Détail

Keywords: Almost sure central limit theorem, weak dependence, empirical processes, copulas

Bibliographie

[1] Berkes, I., Csaki, E., A universal result in almost sure central limit theory, Stoch. Proc. Appl. 94 (2001), 105-134.

[2] Brosamler, G., An almost everywhere central limit theorem, Math. Proc. Cambridge Philos. Soc. 104 (1988), 561-574.

[3] Chen, S., Lin, Z., Almost sure max-limits for nonstationary Gaussian sequence, Statist. Probab. Lett. 76 (2006), 1175-1184.

[4] Cheng, S., Peng, L., Qi, Y., Almost sure convergence in extreme value theory, Math. Nachr. 190 (1998), 43-50.

[5] Csaki, E., Gonchigdanzan, K., Almost sure limit theorems for the maximum of stationary Gaussian sequences, Statist. Probab. Lett. 58 (2002), 195-203.

[6] Doukhan, P., Louhichi, S., A new weak dependence condition and applications to moment inequalities, Stochastic Process. Appl. 84 (1999), 314-342.

[7] Doukhan, P., Fermanian, J. D., Lang, G., An empirical central limit theorem with applications to copulas under weak dependence, Stat. Infer. Stoch. Process. 12 (2009), 65-87.

[8] Dudley, R. M., Central limit theorems for empirical measures, Ann. Probability 6 (1978), 899-929 (Correction, ibid. 7 (1979), 909-911).

[9] Dudziński, M., A note on the almost sure central limit theorem for some dependent random variables, Statist. Probab. Lett. 61 (2003), 31-40.

[10] Dudziński, M., The almost sure central limit theorems in the joint version for the maxima and sums of certain stationary Gaussian sequences, Statist. Probab. Lett. 78 (2008), 347-357.

[11] Dudziński, M., Górka , P., The almost sure central limit theorems for the maxima of sums under some new weak dependence assumptions, Acta Math. Sin., English Series 29, (2013), 429-448.

[12] Fazekas, I., Rychlik, Z., Almost sure functional limit theorems, Ann. Univ. Mariae Curie-Skłodowska Sect. A 56 (2002), 1-18.

[13] Ganssler, P., Stute, W., Empirical Processes: A survey of results for independent and identically distributed random variables, Ann. Probab. 7 (1979), 193-243.

[14] Ganssler, P., Empirical Processes, IMS Lecture Notes - Monograph Series, vol. 3, Hayward, 1983.

[15] Gine, E., Zinn, J., Some limit theorems for empirical processes, Ann. Probab. 12 (1984), 929-989.

[16] Gine, E., Zinn, J., Lectures on the central limit theorem for empirical processes, in: Probability and Banach spaces (Zaragoza, 1985), vol. 1221 of Lecture Notes in Math., 50-113, Springer, Berlin, 1986.

[17] Gine, E., Empirical processes and applications: An overview, Bernoulli 2 (1996), 1-28.

[18] Gonchigdanzan, K., Rempała, G., A note on the almost sure limit theorem for the product of partial sums, Appl. Math. Lett. 19 (2006), 191-196.

[19] Lacey, M., Philipp, W., A note on the almost sure central limit theorem, Statist. Probab. Lett. 9 (1990), 201-205.

[20] Matuła, P., Convergence of weighted averages of associated random variables, Probab. Math. Statist. 16 (1996), 337-343.

[21] Mielniczuk, J., Some remarks on the almost sure central limit theorem for dependent sequences. In: Limit theorems in Probability and Statistics II (I. Berkes, E. Csaki, M. Csorgo, eds.), Bolyai Institute Publications, Budapest, 2002, 391-403.

[22] Peligrad, M., Shao, Q., A note on the almost sure central limit theorem for weakly dependent random variables, Statist. Probab. Lett. 22 (1995), 131-136.

[23] Pollard, D., Limit theorems for empirical processes, Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete 57 (1981), 181-195.

[24] Pollard, D., A central limit theorem for empirical processes, J. Austral. Math. Soc. (Series A) 33 (1982), 235-248.

[25] Pollard, D., Empirical Processes: Theory and Applications, vol. 2 of NSF-CBMS Regional Conference Series in Probability and Statistics, IMS, Hayward, 1990.

[26] Schatte, P., On strong versions of the central limit theorem, Math. Nachr. 137 (1988), 249-256.

[27] Schatte, P., On the central limit theorem with almost sure convergence, Probab. Math. Statist. 11 (1991), 237-246.

[28] Stadtmuller, U., Almost sure versions of distributional limit theorems for certain order statistics, Statist. Probab. Lett. 58 (2002), 413-426.

[29] Talagrand, M., The Glivenko–Cantelli problem. Ten years later, J. of Theoret.

[30] Probab. 9 (1996), 371-384.

[31] van de Geer, S., Empirical Process Theory and Applications, ETH, Zurich, 2006.

[32] van der Vaart, A. W., Wellner, J. A., Weak Convergence and Empirical Processes (With Applications to Statistics), Springer, New York, 1996.

[33] Vapnik, V. N., Chervonenkis, A. Y., On the uniform convergence of relative frequencies of events to their probabilities, Theory Probab. Appl. 16 (1971), 264-280.

[34] Zhao, S., Peng, Z., Wu, S., Almost sure convergence for the maximum and the sum of nonstationary Gaussian sequences, J. Inequal. Appl. 2010 (2010), Art. ID 856495, 14 pp.