Almost sure central limit theorem for the location and height of extreme order statistics and high values

Abstract

In this article, the almost sure central limit theorem for the joint version of location and height of extreme order statistics for independent and identically distributed random variables is proved. The almost sure central limit theorem for the joint version of the location of the maximum and the location of the first exceedances of high levels is also proved.

Keywords:

• Almost sure central limit theorem
• order statistics
• location
• height
• exceedances

AMS CLASSIFICATION:

• Primary 62F15
• Secondary 60G70

Acknowledgment

The author would like to thank the referee and the editor for their valuable suggestions, which greatly improve the original results of the paper.

Additional information

Funding

Research supported by Natural Science Foundation of Zhejiang Province of China (No. LY18A010020), National Science Foundation of China (No. 11501250), Xinmiao Talents Project of Zhejiang Province of China (No. 2019R417013) and SRT of Jiaxing University.