Suppose $X_1, X_2, \ldots$ are independent random variables with distribution: $$ \mathbb{P}(X_n = 0) = \frac{1}{n}, \, \mathbb{P}(X_n = 2n) = 1 - \frac{1}{n} $$ Let $Y_n = \frac{X_1 + X_2 + \ldots + X_n}{n} - n$. Does $Y_n$ converge weakly? If so, identify the limit.
At first, taking $X_{n,k} = \frac{X_k - (2k - 1)}{n}$ we easily see that $\sum_{k=1}^{n}X_{n,k} = Y_n$. My attempts were to find out that $\textbf{Var} Y_n = 2 - \frac{2}{n}$ and $\mathbb{E}Y_n \equiv 1$, so one would expect that $Y_n$ would converge in distribution to $\mathcal{N}(-1, 2)$ due to the central limit theorem. However, checking the Lindeberg condition is really bad, I do not really see how to tackle it. In addition, the Lyapunov condition is not satisfied, so it looks really bad in my opinion.
EDIT: $\mathbb{E}X_{n,k} = -\frac{1}{n}$, $\mathbf{Var}X_{n,k} = \frac{4k-4}{n^2}$