Limit theorems for ratios of order statistics from uniform distributions

Abstract

AbstractLet $\{X_{ni}, 1 \leq i \leq m_{n}, n\geq 1\}${Xni,1≤i≤mn,n≥1} be an array of independent random variables with uniform distribution on $[0, \theta _{n}]$[0,θn], and $\{X_{n(k)}, k=1, 2, \ldots , m_{n}\}${Xn(k),k=1,2,…,mn} be the kth order statistics of the random variables $\{X_{ni}, 1 \leq i \leq m_{n}\}${Xni,1≤i≤mn}. We study the limit properties of ratios $\{R_{nij}=X_{n(j)}/X_{n(i)}, 1\leq i < j \leq m_{n}\}${Rnij=Xn(j)/Xn(i),1≤i<j≤mn} for fixed sample size $m_{n}=m$mn=m based on their moment conditions. For $1=i < j \leq m$1=i<j≤m, we establish the weighted law of large numbers, the complete convergence, and the large deviation principle, and for $2=i < j \leq m$2=i<j≤m, we obtain some classical limit theorems and self-normalized limit theorems.