I am stucked in the middle of an exercise:
Let $$X_n,Y_m$$ independent random variables having the Poisson distribution with parameters n and m respectively. Show that $$\frac{(X_n-n)-(Y_m-m)}{\sqrt{X_n+Y_m}} \; \;\underrightarrow{\omega} \;\; N(0,1) \;\mbox{as}\;n,m \rightarrow \infty $$
My problem is the sum in the denominator (for $n+m$ instead, it is an easy exercise because I can split the random variables in several Poisson(1) distributed r.v's and make use of the independence).