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Section 1 Review of Limit Theorems for Sequences of Random Variables
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Review Question 2.1.5 (Combine Delta method with Lindeberg-Levy)
Let $$ { z_i } $$ be a sequence of i.i.d. (independently and identically distributed) random variables with $$ \operatorname{E} ( z_i ) = \mu \neq 0 $$ and $$ \operatorname{Var} ( z_i ) = \sigma^2 $$, and let $$ \bar{z}_n $$ be the sample mean. Show that
Lemma 2.5 (the “delta method”): Suppose $$ \mathbf{x}_n $$ is a sequence of $$K$$-dimensional random vectors such that $$ \mathbf{x}_n \to_p \boldsymbol{\beta} $$ and
and suppose $$ \mathbf{a} (\cdot): \mathbb{R}^K \to \mathbb{R}^r $$ has continuous first derivatives with $$ \mathbf{A} ( \boldsymbol{\beta} ) $$ denoting the $$ r \times K $$ matrix of first derivatives evaluated at $$ \boldsymbol{\beta} $$: