A Weak Convergence Theorem for Order Statistics From Strong-Mixing Processes

Abstract

This paper provides sufficient conditions for the weak convergence in the Skorohod space $D^d\lbrack a, b\rbrack$ of the processes $\{(Y_{1,\lbrack nt\rbrack} - b_n)/a_n, (Y_{2,\lbrack nt\rbrack} - b_n)/a_n, \cdots, (Y_{d,\lbrack_{nt\rbrack}} - b_n)/a_n\}, 0 < a \leqq t \leqq b$, where $Y_{i,n}$ is the $i$th largest among $\{X_1, X_2, \cdots, X_n\}, a_n$ and $b_n$ are normalizing constants, and $\langle X_n: n \geqq 1\rangle$ is a stationary strong-mixing sequence of random variables. Under the conditions given, the weak limits of these processes coincide with those obtained when $\langle X_n: n \geqq 1\rangle$ is a sequence of independent identically distributed random variables.