binomial normal with dependent success probability

Question

Suppose $Z$ is a random variable distributed according to binomial distribution $B(n,p)$. For constant $p$ it is known that the distribution can be well approximated by the normal distribution. If $pn$ is constant, then it converge to a Poisson distribution. What in the intermidiate case? For example to what is the limit distribution for $p=\Omega(\log(n)/n)$? Is there any known theorem, which specifies this case?

Answer 1

The Poisson approximation still works in the $p=\Omega(\log( n)/n)$ case. According to Le Cam's inequality, the total variation distance between the actual binomial distribution and its approximating Poisson distribution is of order $np^2$, which in your case is $O((\log n)^2/n)$. Also note that the Poisson distribution with large expectation (in your case, $O(\log n)$) is itself approximated by the CLT, but not in total variation.

The specifics of your application will dictate both the order of growth of $pn$ and the desired degree of approximation of distributions, so I hesitate to give one recipe that fits all occasions. If, for instance, $np\to\infty$ and $np^2\to 0$, the Binomial is well approximated by the Poisson and the Poisson, after rescaling, by the Gaussian. If $np^2\to0$ but $np$ remains bounded, you lose the Gaussian approximation but retain the Poisson. But if $p=\Omega(1/\log n)$ Le Cam's inequality looses grip, at least when used this way.