A Note on the Almost Sure Central Limit Theorem in the Joint Version for the Maxima and Partial Sums of Certain Stationary Gaussian Sequences

DOI: 10.4236/am.2014.510153   PDF   HTML     2,865 Downloads   3,630 Views   Citations


Considering a sequence of standardized stationary Gaussian random variables, a universal result in the almost sure central limit theorem for maxima and partial sum is established. Our result generalizes and improves that on the almost sure central limit theory previously obtained by Marcin Dudzinski [1]. Our result reaches the optimal form.

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Wang, Y. and Wu, Q. (2014) A Note on the Almost Sure Central Limit Theorem in the Joint Version for the Maxima and Partial Sums of Certain Stationary Gaussian Sequences. Applied Mathematics, 5, 1598-1608. doi: 10.4236/am.2014.510153.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Dudzinski, M. (2008) The Almost Sure Central Limit Theorems in The Joint Version for The Maxima and Sums of Certain Stationary Gaussian Sequence. Statistics & Probability Letters, 78, 347-357.
[2] Brosamler, G.A. (1988) An Almost Everywhere Central Limit Theorem. Mathematical Proceedings of the Cambridge Philosophical Society, 104, 561-574. http://dx.doi.org/10.1017/S0305004100065750
[3] Schatte, P. (1988) On Strong Versions of the Central Limit Theorem. Mathematische Nachrichten, 137, 249-256. http://dx.doi.org/10.1002/mana.19881370117
[4] Fahrner, I. and Stadtmuller, U. (1998) On Almost Sure Central Max-Limit Theorems. Statistics & Probability Letters, 37, 229-236. http://dx.doi.org/10.1016/S0167-7152(97)00121-1
[5] Cheng, H., Peng, L. and Qi, Y.C. (1998) Almost Sure Con-vergence in Extreme Value Theory. Mathematische Nachrichten, 190, 43-50. http://dx.doi.org/10.1002/mana.19981900104
[6] Chandrasekharan, K. and Minakshisundaram, S. (1952) Typical Means. Oxford University Press, Oxford.
[7] Leadbetter, M.R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer, New York. http://dx.doi.org/10.1007/978-1-4612-5449-2
[8] Csaki, E. and Gonchigdanzan, K. (2002) Almost Sure Limit Theo-rems for The Maximum of Stationary Gaussian Sequences. Statistics & Probability Letters, 58, 195-203.
[9] Mielniczuk, J. (2002) Some Remarks on the Almost Sure Central Limit Theorem for Dependent Sequences. Limit Theorems in Probability and Statistics, 2, 391-403.

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