Convergence in distribution of $\frac{1}{n}max(X_n,Y_n)$, where $X_n\text{ is }U(-n-5,4n-5)$, $Y_n\text{ is }Poiss(10n)$. My idea is to look how $X_n, Y_n$ behave as n tends to infinity, so my first observation is that probability of $X_n$ being infinity is positive as n tends to infinity, however poisson distribution "converges to zero" as n tends to infinity. So I think the only important factor in maximum is the uniform variable, but I might be wrong. Can anyone relate?
The PDF of $\frac{1}{n}max{(X_n,Y_n)}$ is $$\frac{tn+n+5}{5n}e^{-10n}\sum_{k=0}^{\floor{tn}}\frac{10n^{k}}{k!}$$, so we are up to calculating the limit of it when n tends to infinity... $$\text{After a while I think this limit is zero.}$$