Let $X = (X_1, \dots, X_n, \dots)$ be a sequence of random variables. We assume that the process X is stationary i.e. for any integer $k$, any set of indices $i_1 < \dots < i_k$ and any integer $s$ the joint law of the $k$-uples $(X_{i_1}, \dots X_{i_k})$ and $(X_{i_1 + s}, \dots X_{i_k + s})$ are equal. Of course this implies that the random variables are identically distributed. We assume that $X_1$ is centered and has a finite variance $V_1$. We let $S_n = \sum_{k=1}^n X_k$ be the partial sum of order $n$.

What are the possible behaviours for $V_n := \mathbb{E}(S_n^2)$ as $n$ goes to $+ \infty$ ?

If the $X_i$ are mutually independent then of course $V_n = n V_1$. If the $X_i$ are only $m$-dependent (i.e. $Cov(X_i, X_j) = 0$ as soon as $|i-j| > m$) then if I am not mistaking the following alternative holds : either $V_n$ diverges as a constant times $n$ or it is bounded (and periodic).

We may always write $$ V_n = \mathbb{E}((\sum_{i=1}^n X_i)^2) = \sum_{i,j =1}^n Cov(X_i, X_j) = n V_1 + 2\sum_{i=1}^{n-1} Cov(X_1, X_i) (n-i) $$ where we have used the hypothesis of stationarity to write that $Cov(X_i, X_j)$ only depends on $|i-j|$.

Let us consider the particular case where $\sum_{i=1}^{+\infty} |Cov(X_1, X_i)|$ is finite. Then we have $V_n \sim n ( V_1 + 2 \sum_{i=1}^{+\infty} Cov(X_1, X_i))$ except when $V_1 + 2 \sum_{i=1}^{+\infty} Cov(X_1, X_i)$ is not $0$. In this case, is there anything that can be said for the behaviour of $V_n$ ?

If $\{Cov(X_1, X_j)\}_j$ was simply a bounded sequence $\{a_j\}_j$ it seems that there could be any type of asymptotic behavior for the sum $na_0 + 2\sum_{i=1}^{n-1} (n-i) a_i$ (e.g. it could be equivalent to $n^{\alpha}$ for any $\alpha < 1$) but there is some weak relations between the various correlations, do they hinder such arbitrary behavior ?