# Sampling distribution from two independent Bernoulli populations

Let’s assume that we have samples of two independent Bernoulli random variables, $\mathrm{Ber}(\theta_1)$ and $\mathrm{Ber}(\theta_2)$.

How do we prove that ?

Assume that $n_1\neq n_2$.

Put
$a=\frac{\sqrt{\theta_1(1-\theta_1)}}{\sqrt{n_1}}$, $b=\frac{\sqrt{\theta_2(1-\theta_2)}}{\sqrt{n_2}}$,
$A=(\bar{X}_1-\theta_1)/a$,
$B=(\bar{X}_2-\theta_2)/b$. We have
$A\to_d N(0,1),\ B\to_d N(0,1)$.
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 $p=\theta_1,\ m=n_1$.

as $t^3m^{-3/2}\to 0$. Thus, for a fixed $t$,

(even if $a\to 0$ or $b\to 0$), since $\left|e^{-y}-(1-y/m)^m\right|\le {y^2}/{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.