Delta Method

Definition

Suppose we know that \(c_{n}(X_{n} - a)\) converges in distribution to some \(X\) and \(c_{n}\) is strictly increasing and nonnegative. That is, \(c_{n}(X_{n} -a) \xrightarrow d X\). Assume we have a function \(f(z)\) that is continuously differentiable at point \(a\). Then

\[ c_{n}(f(X_{n})-f(a)) \xrightarrow d f'(a)X. \]