Consider a stationary sequence $X_1,X_2,...X_n$ such that $X_i \sim N(0,1)$ and their correlation sequence is given by $r_n=E(X_iX_{i+n})$. It is well known that if $r_n \log n \rightarrow 0$ then the maxima $M_n=\max_{1 \leq i \leq n} X_i$ converges in distribution to a Gumbel distribution and if $r_n \log n \rightarrow \lambda>0$ then $M_n$ converges to a mixture of Gumbel and standard normal distribution.

My question is what would happen when the dependence is large, i.e when $r_n \log n \rightarrow \infty$? I can see some related results in literature which show that $M_n \xrightarrow{d} N(0,1)$ in this case, under some extra assumptions about the smoothness of $r_n$. Would it be true if we assume nothing other than $r_n \log n \rightarrow \infty$?