A pitcher contains $X + 1$ blue balls and $Y + 1$ red balls. It is known that $X, Y$ are independent random variables, and it is given that $X \sim \mathrm{Poisson}(n), Y \sim\mathrm{Poisson}(2n)$. Calculate the limit: $\displaystyle\lim_{n\to \infty}\mathbb P (X + n = Y) $
I tried to condition on the value of $Y$ and to use the Law of total probability but it did not lead to a solution. ( I tried to sum: $ \sum_{i=0}^\infty\mathbb P(X+n=Y|Y=i)\mathbb P(Y=i) $ and get an expression that I can calculate its limit as n goes to infinity)
Let $\{V_i\}_{i=1}^n$ and $\{W_i\}_{i=1}^n$ be independent random variables s.t. $V_i\sim \text{Poisson}(1)$ and $W_i\sim \text{Poisson}(2)$. Also let $X_n:=\sum_{i=1}^n V_i$ and $Y_n:=\sum_{i=1}^n W_i$. Then
$$ p_n:=\mathsf{P}(X_n+n=Y_n)=\mathsf{P}\!\left(n^{-1/2}(X_n-n)=n^{-1/2}(Y_n-2n)\right). $$ Using the CLT, $$ n^{-1/2}\begin{bmatrix} X_n-n \\ Y_n-2n \end{bmatrix}\xrightarrow{d}N\!\left(0, \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix} \right). $$ Therefore, for any $\epsilon>0$, \begin{align} \limsup_{n\to\infty}p_n&\le \limsup_{n\to\infty}\mathsf{P}\!\left(n^{-1/2}|X_n-Y_n+n|<\epsilon\right) \\ &= 2\Phi(\epsilon;0,3)-1\le \sqrt{\frac{2}{3\pi}}\,\epsilon. \end{align}