$X_1, X_2, \dots, X_n$, are $n$ mutually independent r.v.s. $Y_1,\dots,Y_n$ are another set of mutually independent r.v.s. $X_k\mid Y_k=y_k\sim N(y_k,y_k^2)$ and $Y_k\sim\text{uniform}(-k,k)$ for $k=1,\dots,n$. Define $\bar{X}_n=\frac{\sum_{i=1}^nX_i}{n}$ the sample mean of $X_i$'s.
I've calculated E$[\bar{X}_n]=0$ and Var$(\bar{X}_n)=\frac{n(n+1)(2n+1)}{9n^2}=\frac{(n+1)(2n+1)}{9n}$.
How can we prove that $\bar{X}_n$ doesn't converge to E$[\bar{X}_n]$ in probability?
Thanks!