Delta method for Bernoulli odds ($\sqrt{n}(\hat{o}-o)\stackrel{d}{\to }N(0,p/(1-p{)}^3)$)

For independent Bernoulli random variables of fixed success probability $0<p<1$, put $\hat{p}=n^{-1}{\sum }_i{X}_i$ and $o=p/(1-p)$. The central limit theorem gives $\sqrt{n}(\hat{p}-p)\stackrel{d}{\to }N(0,p(1-p))$. Applying the delta method to the odds map $g(p)=p/(1-p)$, whose derivative is $(1-p{)}^{-2}$, givesThe ratio is undefined on the all-success event, whose probability is $p^n$. Any finite extension there has the same limiting probability distribution, since that probability tends to zero. This is an asymptotic statement and does not assert an exact finite-sample variance of the unextended ratio. It is useful for uncertainty in estimated odds when the success probability stays away from one.