# Variance of sum of $m$ dependent random variables

Let $$X_1,X_2,...$$ be a sequence of identically distributed and $$m$$-dependent random variables with $$\mathbb{E}[X_i]=0$$, $$0 ($$m$$-dependent means that each $$X_i$$ is independent of $$X_{i+j}$$ for $$|i-j|\ge m$$.

Suppose $$Y$$ is a random variable with $$\mathbb{E}[Y]=0$$ and $$Var(Y)<\infty$$.

Assume also that $$Y$$ is independent of $$X_m,X_{m+1},...$$

We know that $$\frac{Y+\sum_{i=1}^{n}X_i}{\sqrt{n}}\overset{d}{\longrightarrow} N(0,\sigma^2)$$

from the Hoeffding-Robbins theorem, but I am struggling to show that $$\sigma^2>0$$ even though intuitively it seems true.

Do you have any ideas?