convergence in probability is preserved under continuous transformations

Theorem 1.

Let $g\colon\mathbb{R}^{k}\to\mathbb{R}^{l}$ be a continuous function. If $\{X_{n}\}$ are $\mathbb{R}^{k}$ -valued random variables converging to $X$ in probability, then $\{g(X_{n})\}$ converge in probability to $g(X)$ also.

Proof.

Suppose first that $g$ is uniformly continuous. Given $\epsilon>0$ , there is $\delta>0$ such that $\lVert g(X_{n})-g(X)\rVert<\epsilon$ whenever $\lVert X_{n}-X\rVert<\delta$ . Therefore,

\mathbb{P}\bigl{(}\lVert g(X_{n})-g(X)\rVert\geq\epsilon\bigr{)}\leq\mathbb{P}% \bigl{(}\lVert X_{n}-X\rVert\geq\delta\bigr{)}\to 0

as $n\to\infty$ .

Now suppose $g$ is not necessarily uniformly continuous on $\mathbb{R}^{k}$ . But it will be uniformly continuous on any compact set $\{x\in\mathbb{R}^{k}\colon\lVert x\rVert\leq m\}$ for $m\geq 0$ . Consequently, if $X_{n}$ and $X$ are bounded (by $m$ ), then the proof just given is applicable. Thus we attempt to reduce the general case to the case that $X_{n}$ and $X$ are bounded.

Let

f_{m}(x)=\begin{cases}x\,,&\lVert x\rVert\leq m\\ mx/\lVert x\rVert\,,&\lVert x\rVert\geq m\end{cases}

Clearly, $f_{m}\colon\mathbb{R}^{k}\to\mathbb{R}^{k}$ is continuous; in fact, it can be verified that $f_{m}$ is uniformly continuous on $\mathbb{R}^{k}$ . (This is geometrically obvious in the one-dimensional case.)

Set $X_{n}^{m}=f_{m}(X_{n})$ and $X^{m}=f_{m}(X)$ , so that $X_{n}^{m}$ converge to $X^{m}$ in probability for each $m\geq 0$ .

We now show that $g(X_{n})$ converge to $g(X)$ in probability by a four-step estimate. Let $\epsilon>0$ and $\delta>0$ be given. For any $m\geq 0$ (which we will later),

\mathbb{P}\bigl{(}\lVert g(X_{n})-g(X)\rVert\geq\delta\bigr{)}\leq\mathbb{P}% \bigl{(}\lVert g(X_{n}^{m})-g(X^{m})\rVert\geq\delta\bigr{)}+\mathbb{P}\bigl{(% }\lVert X_{n}\rVert\geq m\bigr{)}+\mathbb{P}\bigl{(}\lVert X\rVert\geq m\bigr{% )}\,.

Choose $M$ such that for $m\geq M$ ,

\mathbb{P}\bigl{(}\lVert X\rVert\geq m\bigr{)}\leq\mathbb{P}\bigl{(}\lVert X% \rVert\geq M\bigr{)}<\frac{\epsilon}{4}\,.

(This is possible since $\lim_{m\to\infty}\mathbb{P}\bigl{(}\lVert X\rVert\geq m\bigr{)}=\mathbb{P}% \bigl{(}\bigcap_{m=0}^{\infty}\{\lVert X\rVert\geq m\}\bigr{)}=\mathbb{P}(% \emptyset)=0$ .)

In particular, let $m=M+1$ . Since $X_{n}^{m}$ converge in probability to $X^{m}$ and $X_{n}^{m}$ , $X^{m}$ are bounded, $g(X_{n}^{m})$ converge in probability to $g(X^{m})$ . That means for $n$ large enough,

\mathbb{P}\bigl{(}\lVert g(X_{n}^{m})-g(X^{m})\rVert\geq\delta\bigr{)}<\frac{% \epsilon}{4}\,.

Finally, since $\lVert X_{n}\rVert\leq\lVert X_{n}-X\rVert+\lVert X\rVert$ , and $X_{n}$ converge to $X$ in probability, we have

\mathbb{P}\bigl{(}\lVert X_{n}\rVert\geq m=M+1\bigr{)}\leq\mathbb{P}\bigl{(}% \lVert X_{n}-X\rVert\geq 1\bigr{)}+\mathbb{P}\bigl{(}\lVert X\rVert\geq M\bigr% {)}<\frac{\epsilon}{4}+\frac{\epsilon}{4}

for large enough $n$ .

Collecting the previous inequalities together, we have

\mathbb{P}\bigl{(}\lVert g(X_{n})-g(X)\rVert\geq\delta\bigr{)}<\epsilon

for large enough $n$ . ∎

Title	convergence in probability is preserved under continuous transformations
Canonical name	ConvergenceInProbabilityIsPreservedUnderContinuousTransformations
Date of creation	2013-03-22 16:15:05
Last modified on	2013-03-22 16:15:05
Owner	stevecheng (10074)
Last modified by	stevecheng (10074)
Numerical id	10
Author	stevecheng (10074)
Entry type	Theorem
Classification	msc 60A10