I originally posted this question in Mathstackexchange, but since I got no answer I'm posting it also here.

Let $X_1,X_2,...$ be a sequence of identically distributed and $m$-dependent random variables with $\mathbb{E}[X_i]=0$, $0<\operatorname{Var}(X_i)<\infty$ ($m$-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 $\operatorname{Var}(Y)<\infty$.

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?