Let’s assume that we have samples of two independent Bernoulli random variables, Ber(θ1) and Ber(θ2).
How do we prove that (ˉX1−ˉX2)−(θ1−θ2)√θ1(1−θ1)n1+θ2(1−θ2)n2d→N(0,1)?
Assume that n1≠n2.
B=(ˉX2−θ2)/b. We have
In terms of characteristic functions it means
We want to prove that
Since A and B are independent,
as we wish it to be.
This proof is incomplete. Here we need some estimates for uniform convergence of characteristic functions. However in the case under consideration we can do explicit calculations. Put p=θ1, m=n1.
as t3m−3/2→0. Thus, for a fixed t,
(even if a→0 or b→0), since |e−y−(1−y/m)m|≤y2/2m when y/m<1/2 (see https://math.stackexchange.com/questions/2566469/uniform-bounds-for-1-y-nn-exp-y/ ).
Note that similar calculations may be done for arbitrary (not necessarily Bernoulli) distributions with finite second moments, using the expansion of characteristic function in terms of the first two moments.