Getting back to the business about asymptotic distribution of the LRT, I was thinking that the convergence ought to depend on how large lambda is, and not the number of sampling volumes.

Doesnt poisson become more "normalish" as lambda gets bigger, or am I mistaken in that regard.

I think increasing lamda and increasing sample size are related cuz

X1 + X2 ~ Poisson(2*lambda) is like doubling lambda.

It suggests that convergence of the lrt depends on both number of samples and the lambda?