Delta Method for Asymptotic Variance of the Log Odds Ratio

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Asymptotic Statistics / Delta Method.

Your Task
Let $X_n \sim \frac{1}{n} \text{Binomial}(n, p)$ be the sample proportion of successes. By CLT, $\sqrt{n}(X_n - p) \xrightarrow{d} \mathcal{N}(0, p(1-p))$.
(1) State the univariate Delta Method.
(2) Using the Delta Method, derive the asymptotic distribution of the sample log-odds $g(X_n) = \ln\left(\frac{X_n}{1 - X_n}\right)$.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.