If $\ X_1,X_2,X_3... $ are iid random variables with Poisson Parameter $λ$,
Does
$(1/{\sqrt n} ) \sum_{1}^{n} $ $(X_{2i-1} - X_{2i})$
Converge in distribution to a normal distribution as $ n \rightarrow \infty $
I've tried to match up with the characteristic function but I keep going wrong and I'm not sure why. Thanks
Define $Z_i=X_{2i-1}-X_{2i}$. Since $X_i$ is iid, $Z_i$ is also iid with $$EZ_i=\lambda-\lambda=0$$ and $$VarZ_i=\lambda+\lambda=2\lambda$$ Therefore for $S_n=\sum_1^nZ_i$ we have $$ES_n=0, VarS_n=2n\lambda$$. Now using the standard (classic) CLT we obtain $$\frac{S_n-ES_n}{\sqrt{VarS_n}}\to N(0,1)$$ or that $$\frac{S_n}{\sqrt{n}}\to N(0,2\lambda)$$