# Continuous Mapping Theorem (CMT) for a sequence of random vectors

I need help proving the Continuous Mapping Theorem (CMT) for random vectors. I'm currently reading Econometric Analysis for Cross Section and Panel Data by Jeffrey M. Wooldridge (Chapter 3, pp. 40 - 41, 2nd edition). Unfortunately, he leaves it to the reader to prove most asymptotic results. Additionally, almost every other econometrics textbook I read simply states the result.

Definition 1: A sequence of random variables $x_n$ converges in distribution to a continuous random variable $x$ if and only if $\forall s \in \mathbb{R} \ \forall \epsilon >0 \ \exists N \ s.t. \ \forall n>N \; |Prob(x_n \leq s) - Prob(x \leq s)|<\epsilon$. We write $x_n \to^d x.$ [Note: A continuous random variable is one for which the cumulative distribution function is continuous.]

Definition 2: A sequence of K $\times$ 1 random vectors $\mathbf{x}_n$ converges in distribution to the continuous random $K \times 1$ vector $\mathbf{x}$ if and only if $\forall \mathbf{c} \in \mathbb{R}^{K}$ such that $\mathbf{c}^T\mathbf{c} = 1$, $\mathbf{c}^T\mathbf{x}_n \to^d \mathbf{c}^T\mathbf{x}$, and we write $\mathbf{x}_n \to^d \mathbf{x}.$

Theorem 1: Let $\mathbf{x}_n$ be a sequence of $K \times 1$ random vectors such that $\mathbf{x}_n \to^d \mathbf{x}$. If $\mathbf{g}:\mathbb{R}^k\to\mathbb{R}^{\ell}$ is a continuous function, then $\mathbf{g}(\mathbf{x}_n)$ $\to^d$ $\mathbf{g}(\mathbf{x}).$

Definition 3: A sequence of random variables $x_n$ is bounded in probability if and only if $\forall \epsilon>0 \ \exists b_{\epsilon}>0 \ \exists N \ s.t. \forall n>N \ Prob(|x_n|>b_{\epsilon})$. A vector $\mathbf{x}_n$ is bounded in probability if and only if the random variables which constitute the vector of random variables are themselves bounded in probability.

Theorem 2: If $\mathbf{x}_n \to^d \mathbf{x}$, where $\mathbf{x}$ is a $K \times 1$ vector, then $\mathbf{x}_n = O_p(1)$.

I need rigorous proofs for Theorems 1 and 2. This problem has been frustrating me for a couple days now, so any help would go a long way.

Thanks.

CS

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 There are several equivalent definitions of convergence in distribution; which one(s) are you using? – Nate Eldredge Oct 30 '12 at 13:57 I made some adjustments. Hope that helps. – Christian Oct 30 '12 at 16:18