Continuous Mapping Theorem

EC484_Econometrics_Analysis

After studying Law of Large Numbers and Central Limit Theorem, we often need to deal with functions of random variables. The Continuous Mapping Theorem helps us understand how functions of converging random variables behave.

The core idea of CMT is mainly: If a sequence of random variables converges (in probability or distribution), then a continuous function of those random variables also converges (in the same mode of convergence).

We have to check if $g(\cdot )$ is still continuous at $x=c$.

Example

An example of CMT is consider $\hat{\mu }=n^{-1}{\sum }_{i=1}^{n}h(y_i){\to }^p\mathbb{E}[h(y)]$ and $g(\cdot )$ is continuous at $\mu$, then

$$g(\hat{\mu }){\to }^{p}g(\mu )$$