Let us have a stationary time series $X=(X_t, t\in\mathbb{Z})$ following e.g. AR(p) model (i.e. there are $a_1, \dots, a_p$ such that $X_t=a_1X_{t-1}+\dots + a_pX_{t-p} + N_t$ where $N_t$ are some iid noise variables with possibly heavy tailes).
Let the stationary distribution function be $F$ (i.e. df of $X_0$). Notation $X_{(k)}$ represents the $k-$th order statistic from $0$ to $n$, e.g. $X_{(1)}=\min_{i=0, \dots, n} X_i$ and $X_{(n-k+1)}$ is the $k$-th largest value out $X_0, \dots, X_n$. Let $k_n\in\mathbb{N}$ fulfill $$k_n\to\infty, \frac{k_n}{n}\to 0 \text{, as } n\to\infty.$$
Is the following true $$F(X_{(n-k_n+1)})\overset{n\to\infty}{\to} 1^{-}?$$
If not, what "reasonable" condition should we assume so that it will be true?
