Question
Suppose we have an ergodic positive stochastic process $(X_n)_{n \in \mathbb{N}}$ (in particular I'm mainly interested in the case where $(X_n)_n$ is an aperiodic, irreducible, positive recurrent Markov Chain or an $AR(1)$ process but general results are always nice to have). Assume that we are far enough in time s.t. we can assume that each $X_n$ seperately is distributed according to the steady state of the chain, say $X$. I would like to say as much as possible about $\mbox{Var}(\sum_{n=1}^N \delta\{X_n > i_n\} )$ i.f.o. correlation of $(X_n)_n$, i.e. $\mbox{Corr}(X_n,X_{n-1})$. Here we use the noteation $\delta\{X_n > i_n\}$ is one if $X_n > i_n$ and zero otherwise.
Start of general solution
We have: $$ \mbox{Var}(\sum_{n=1}^N \delta\{X_n > i_n\} ) = \sum_{n=1}^N \mbox{Var}(\delta\{X_n > i_n\}) + 2 \sum_{1 \leq n < m \leq N} \mbox{Cov}(\delta\{X_n > i_n\}, \delta\{X_m > i_m\}) $$ The first part of this expression isn't influenced by the correlation between subsequent $X_n$ thus it remains to look at $\mbox{Cov}(\delta\{X_n > i_n\}, \delta\{X_m > i_m\})$. It seems somewhat logical that as the correlation $\mbox{Corr}(X_n,X_{n-1})$ increases the value of $$ \sum_{1 \leq n < m \leq N} \mbox{Cov}(\delta\{X_n > i_n\}, \delta\{X_m > i_m\}) $$ will also increase. For the two dimensional case I have been able to show this to be true in case we are working with a Markov Chain and $i_n = N - n$.
In the general case we can easily show $\mbox{Cov}(\delta\{X_n > i_n\}, \delta\{X_m > i_m\})$ to be equal to: $$ \mathbb{P}\{X > i_n\} \cdot (\mathbb{P}\{X_m > i_m \mid X_n > i_n\} - \mathbb{P}\{ X_m > i_m \}) $$ again it seems logical that $(\mathbb{P}\{X_m > i_m \mid X_n > i_n\} - \mathbb{P}\{ X_m > i_m \})$ will be greater than $0$ if the correlation is positive and the higher the correlation increases.
I have tried showing this for a geneal Markov chain and an $AR(1)$ process but without success.
