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?



Variance of sum of $m$ dependent random variables
First, the random variable (r.v.) $Y$ plays no role here, since $Y/\sqrt n\to0$. 


Second, $\sigma^2$ may be zero. However, in the abstract of [Janson](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=8&ved=2ahUKEwjzvpeOs_DgAhUh4IMKHcsUDvUQFjAHegQIABAC&url=https%3A%2F%2Farxiv.org%2Fpdf%2F1312.1563&usg=AOvVaw1ilXV3k-sugaFEmxz4u71y) we find this complete answer to your question: 



> 
> It is well-known that the central limit theorem holds for partial sums
>  of a stationary sequence $(X\_i)$ of $m$-dependent random variables
>  with finite variance; however, the limit may be degenerate with
>  variance $0$ even if $\text{Var}\,(X\_i)\ne0$. We show that this happens only in
>  the case when $X\_i − \text{E}\,X\_i = Y\_i − Y\_{i−1}$ for an $(m − 1)$-dependent
>  stationary sequence $(Y\_i)$ with finite variance (a result implicit in
>  earlier results)
> 
> 
> 


