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In this paper, we study the limit properties of the ratio for order statistics based on samples from an exponential distribution and obtain the expression of the density functions, the existence of the moments, the strong law of large numbers for    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$R_{nij}$\end{document}    R   n  i  jwith    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$1\leq i< j< m_{n}=m$\end{document}   1  ≤  i  <  j  <   m  n   =  m. We also discuss other limit theorems such as the central limit theorem, the law of iterated logarithm, the moderate deviation principle, the almost sure central limit theorem for self-normalized sums of    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$R_{nij}$\end{document}    R   n  i  jwith    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$2\leq i< j< m_{n}=m$\end{document}   2  ≤  i  <  j  <   m  n   =  m.

Limit properties for ratios of order statistics from exponentials