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

Marcin Dudziński

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 71 (2017), / Harvested from The Polish Digital Mathematics Library

Résumé

Let: $\mathbf {Y=}\left( \mathbf {Y}_{i}\right)$, where $\mathbf {Y}_{i}=\left( Y_{i,1},...,Y_{i,d}\right)$, $i=1,2,\dots $, be a $d$-dimensional, identically distributed, stationary, centered process with uniform marginals and a joint cdf $F$, and $F_{n}\left( \mathbf {x}\right) :=\frac{1}{n}\sum _{i=1}^{n}\mathbb {I}\left(Y_{i,1}\le x_{1},\dots ,Y_{i,d}\le x_{d}\right)$ denote the corresponding empirical cdf. In our work, we prove the almost sure central limit theorem for an empirical process $B_{n}=\sqrt{n}\left( F_{n}-F\right)$ under some weak dependence conditions due to Doukhan and Louhichi. Some application of the established result to copula processes is also presented.

Détail

Publié le : 2017-01-01
EUDML-ID : urn:eudml:doc:289767

Bibliographie

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

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

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

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

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

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

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.

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

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

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.

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.

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

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

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

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

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.

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

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.

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

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

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.

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.

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

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

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

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

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

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

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

Probab. 9 (1996), 371-384.

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

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

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.

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.