A note on the almost sure limit theorem for self-normalized partial sums of random variables in the domain of attraction of the normal law

Wu

Description

Let X , X 1 , X 2 ,. be a sequence of independent and identically distributed random variables in the domain of attraction of a normal distribution. A universal result in almost sure limit theorem for the self-normalized partial sums S n / V n is established, where S n = ∑ i = 1 n X i , V n 2 = ∑ i = 1 n X i 2 . Mathematical Scientific Classification : 60F15.

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• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 4
• Langue: English

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Ding and KarapınarJournal of Inequalities and Applications2012,2012:170 http://www.journalofinequalitiesandapplications.com/content/2012/1/170

R E S E A R C H

A note on some coupled fixed-point theorems onG-metric spaces 1* 2 Hui-Sheng Dingand Erdal Karapınar

* Correspondence: [e-mail] 1 College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, People’s Republic of China Full list of author information is available at the end of the article

Open Access

Abstract The purpose of this paper is to extend some recent coupled fixed-point theorems in the context ofG-metric space by essentially different and more natural way. We state some examples to illustrate our results. MSC:46N40; 47H10; 54H25; 46T99 Keywords:coupled fixed point; coincidence point; mixedg-monotone property; ordered set;G-metric space

1 Introduction In nonlinear functional analysis, one of the most productive tools is the fixed-point the- ory, which has numerous applications in many quantitative disciplines such as biology, chemistry, computer science, and additionally in many branches of engineering. In this theory, the Banach contraction principle can be considered as a cornerstone pioneering result which in elementary terms states that each contraction has a unique fixed point in a complete metric space. Due to its potential of applications in the fields above mentioned and many more, the fixed-point theory, in particular, the Banach contraction principle, attracts considerable attention from many authors (see,e.g., [–]). Especially, it is con- sidered very natural and curious to investigate the existence and uniqueness of a fixed point for several contraction type mappings in various abstract spaces. A major example in this direction is the work of Mustafa and Sims [] in which they introduced the con- cept ofG-metric spaces as a generalization of (usual) metric spaces in . After this remarkable paper, a number of papers have appeared on this topic in the literature (see, e.g., [–, , , –]). For the sake of completeness, we recall some basic definitions and elementary results * from the literature. Throughout this paper,Nis the set of nonnegative integers, andN is the set of positive integers.

+ Definition (See [])LetXbe a nonempty set,G:X×X×X→Rbe a function satisfying the following properties: (G)G(x,y,z) = ifx=y=z, (G) <G(x,x,y)for allx,y∈Xwithx=y, (G)G(x,x,y)≤G(x,y,z)for allx,y,z∈Xwithy=z, (G)G(x,y,z) =G(x,z,y) =G(y,z,x) =∙ ∙ ∙(symmetry in all three variables), (G)G(x,y,z)≤G(x,a,a) +G(a,y,z)for allx,y,z,a∈X(rectangle inequality).

©2012 Ding and Karapınar; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.