In a Bernoulli process with a random variable that is 1 with a probability $p$ and 0 with probability $q = 1 - p$, after $N$ repetitions one uses $\hat{p} = \frac{N_1}{N}$ as estimator of $p$, and then $\langle\hat{p}\rangle = p$ and $\langle\hat{p}^2\rangle = \frac{p(1-p)}{N}$ as $P(N_1)$ is a binomial distribution.
How does the estimator $\hat{z}$ of $z = \frac{p}{q}$ converge as a function of p and N? In other words, what is $\langle\hat{z}^2\rangle(N,p)$?
Thanks a lot for your help.