A Strong Invariance Principle Concerning the $J$-Upper Order Statistics for Stationary Gaussian Sequences

Abstract

It is shown that in the case of stationary Gaussian processes, the $J$th $(J \geq 1)$ record times $\{T_n, n \geq 1\}$ and the corresponding $J$-upper order statistics $\{X_{T_n - J + 1, T_n}, \ldots, X_{T_n, T_n}\}$ can almost surely be identified via a translation of the time index $n$ to the corresponding elements defined on a sequence of independent and identically distributed random variables. A construction method for approximating sequences of record times and the corresponding upper order statistics introduced by Haiman (1987a, b) for the case $J = 1$ is extended and applied under weaker conditions concerning the covariance function, and also under different sets of new hypotheses.