On the joint limiting distribution of sums and maxima of stationary normal sequences

Let $X_1,X_2,\dots$ be a stationary sequence of standard normal random variables. Let $\rho_n=\mathbf{E}(X_1 X_{n+1})$. Ho and Hsing derived the asymptotic joint distribution of $\sum_{i=1}^n X_i$ and $\max_{1\le i\le n}X_i$ for the case $\rho_n\log n\to\gamma\in[0,\infty)$. In this paper we extend this result for the case where $\rho_n$ is convex with $\rho_n=o(1)$, and $(\rho_n\log n)^{-1}$ is monotone with $(\rho_n\log n)^{-1}=o(1)$.