The Continuous Mapping Theorem

A huge amount of statistical practice relies on transforming an estimator — squaring it, taking a log, plugging it into a ratio — and then wanting to know what happens to that transformed quantity as the sample size grows. The continuous mapping theorem is the tool that justifies this: it says that if a sequence converges (in probability, in distribution, or almost surely) to some limit, and you apply a continuous function to every term in the sequence, the transformed sequence converges too, to the function applied to that same limit — you don't have to redo the convergence analysis from scratch for the transformed quantity.

Formally, if $X_n \to X$ (in any of the standard modes of convergence) and $g$ is a continuous function (or continuous at every point $X$ can take, for the almost-sure and in-probability cases), then:

$$g(X_n) \to g(X).$$

In plain English: convergence passes through continuous functions unchanged — if a sequence is settling toward a limit, applying a continuous transformation to every term in the sequence gives you a new sequence settling toward the transformed limit, no extra assumptions needed beyond the function's continuity.

Worked example. Suppose $\bar X_n \xrightarrow{p} \mu$ (a sample mean converging in probability to the true mean, by the law of large numbers), and you want to know what $\bar X_n^2$ converges to. Since $g(x) = x^2$ is continuous, the continuous mapping theorem immediately gives $\bar X_n^2 \xrightarrow{p} \mu^2$ — no separate proof needed. The same logic justifies that $1/\bar X_n \to 1/\mu$ (for $\mu \neq 0$, where $g(x)=1/x$ is continuous), which is exactly the kind of step used constantly when deriving the asymptotic behavior of ratio estimators like the Sharpe ratio or a regression coefficient.