Extending the one series theorem to $k$-dependent sequences

The problem goes: For some $k \ge 1$, let $\{X_i\}$ be a sequence of $k$-dependent random variables. That is, for each $n\ge 1$, $\sigma(X_1,\ldots,X_n)$ and $\sigma(X_{n+k+1},\ldots)$ are independent. In addition, $EX_i=0, \forall i$. If $\sum_n \text{var} X_n<\infty$, must $\sum_n X_n$ converge almost surely?

I know from the proof of the one series theorem that was given us, I only need to show $T_n \to_p 0$, where $T_n=\sup_{m\ge n}|S_m-S_n|$, and $S_n=\sum_{i=1}^n X_i$. However, in the proof of the one series theorem, this was done by Kolmogorov's maximal inequality, which hinges on $\{X_i\}$ being independent, which is not the case here. Thank you for any help.

We have for each $i_0\in\{0,\dots,k-1\}$ that the series $$\sum_{j=1}^\infty X_{kj+i_0}$$ converges almost surely. Hence there is a set $\Omega'\subset\Omega$ of probability one for which if $\omega\in \Omega'$, then for each $i_0\in\{0,\dots,k-1\}$, the series $\sum_{j=1}^\infty X_{kj+i_0}$ is convergent.
Assume we have a sequence $(c_n)_{n\geqslant 1}$ of real numbers such that $S':=\sum_n c_{2n}$ and $S'':=\sum_n c_{2n+1}$ are convergent. Define $S'_N:=\sum_{j=1}^Nc_{2j}$ and $S''_N:=\sum_{j=1}^Nc_{2j-1}$ and $S_N:=\sum_{j=1}^Nc_j$. Then $$S_{2N}=S'_N+S''_N\to S'+S''$$ and similarly, $S_{2N+1}\to S'+S''$. The same idea applies for $k$ not necessarily equal to $2$.
• Thank you! I should have thought about breaking the series into $k$ subseries. – Fang Jing Dec 2 '13 at 18:40