Further research on limit theorems for self-normalized sums

Abstract

In this paper, we show that self-normalized versions of central limit theorem and almost sure central limit theorem hold with more general weight sequence both for i.i.d. random sequence and $\varphi -$mixing sequence. Our conclusions generalize and improve the known results from the logarithmic averages which used traditionally in almost sure central limit theorem to some general averages.

Keywords:

• Almost sure central limit theorem
• domain of attraction of the normal law
• $\varphi -$mixing sequence
• self-normalized (weighted) sums

Mathematics Subject Classification:

• 60F15
• 60f05

Additional information

Funding

This work was supported by the National Natural Science Foundation of China under Grant number 11771178; the Science and Technology Development Program of Jilin Province under Grant number 20170101152JC; Science and Technology Program of Jilin Educational Department during the 13th Five-Year Plan Period under Grant number JJKH20180110KJ.