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My textbook has this theorem:

Suppose that $X_1,X_2,\cdots$ converges in probability to a random variable $X$ and that $h$ is a continuous function. Then $h(X_1), h(X_2),\cdots$ converges in probability to $h(X)$

Failed to come up with a proof immediately, I looked at the proof provided in the solution:

If $h$ is continuous, given $\epsilon > 0$ there exits $\delta>0$ such that $|h(x_n)−h(x)| < \epsilon$  for $|x_n−x| < \delta$. Since $X_1,\cdots, X_n$ converges in probability to the random variable $X$, then $\lim_{n\to\infty} P(|X_n − X| < \delta) = 1$. Thus $\lim_{n\to\infty} P(|h(X_n) − h(X)| < \epsilon) = 1$.

The part follows "since" does not really make sense to me. I think the subtly lies in the fact that $X_n$ is a random variable (that is, a function defined on some sigma-algebra $\Omega$). So for a given $\omega \in \Omega$, for a given $n$, and for a given $\epsilon >0$, we indeed can find $\delta$ such that $$|X_n( \omega) - X (\omega)| < \delta \text{ implies that } |h(X_n( \omega) - h(X (\omega))|<\epsilon$$

, but the choice of such $\delta$ seems to depend on $\omega$ and $n$? My expectation was that, in the end, we want something like $$\{\omega\in \Omega: |X_n(\omega) - X(\omega)|<\delta\} \subset \{ \omega\in\Omega: |h(X_n(\omega) - h(X(\omega))| < \epsilon\}$$

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  • $\begingroup$ What were the exact hypotheses on $h$. In particular, do they assume $h$ is uniformly continuous? $\endgroup$ – kimchi lover Sep 24 '17 at 2:11
  • $\begingroup$ @kimchilover Well, it doesn't say uniformly continuous in the theorem. $\endgroup$ – 3x89g2 Sep 24 '17 at 2:12
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    $\begingroup$ Look at the wikipedia article on continuous mapping theorem. It's clear that your textbook's "proof" is an abridgment of the argument given there. I'm not 100% sure of the details of the wikipedia argument, but it is a step in the right direction. $\endgroup$ – kimchi lover Sep 24 '17 at 2:32
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    $\begingroup$ The claimed proof is wrong because the statement "given $\epsilon > 0$ there exists $\delta>0$ such that $|h(x_n)−h(x)| < \epsilon$ for $|x_n - x| < \delta$" is not generally true for continuous functions; it's the definition of uniformly continuous. But if $h$ were uniformly continuous the proof would be fine. The $\delta$ really does only depend on $\epsilon$ and not on anything else in sight. The choice to call the second point $x_n$ is perhaps part of the confusion; they should have written "$|h(y) - h(x)| < \epsilon$ for $|x-y| < \delta$." $\endgroup$ – Nate Eldredge Sep 24 '17 at 5:15
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    $\begingroup$ @Misakov: Yes, correct. If $h$ is merely continuous then $\delta$ also depends on $x$. $\endgroup$ – Nate Eldredge Sep 24 '17 at 15:00
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The OP is right: the "proof" in the book is incomplete. It assumes $h$ is uniformly continuous, that is, that for every $\epsilon>0$ there is a single $\delta>0$ such that for all $x$ the usual good stuff happens. The proof sketched in the wikipedia article is more serious, if hard to follow. My library does not have a copy of the book the wikipedia's proof seems taken from, so I came up with a separate argument, which is (I think) a bit more conceptual than the wikipedia's argument, or at least relies on standard facts in a more self-evident way.

These facts are: in a separable metric space $(S,d)$, convergence in probability is metrizable by the metric $d_P(X,Y)=E\min(1,d(X,Y))$, and in a separable metric space, convergence in probability $X_n\xrightarrow P X$ implies convergence in distribution $X_n\xrightarrow D X$. See the wikipedia article or the discussion in Billingsley's Convergence of Probability Measures, (first, 1968 edition) p.26.

Here is a sketch of how to use them to prove the result. We have $X_n\xrightarrow{P}X,$ where the rvs are $S$-valued, and $h:S\to S'$ is continuous. Then we also have $(X_n,X)\xrightarrow{P}(X,X)$ and $(X_n,X)\xrightarrow{D}(X,X)$. (The first follows from use of $d_P$; the second is standard.) Define $g:S\times S\to \mathbb R$ by $g(u,v) = \max(1,d(h(u),h(v)))$, where $d(\cdot,\cdot)$ is the metric on $S'$. It is clear that $g$ is continuous and bounded, so $(X_n,X)\xrightarrow{D}(X,X)$ implies $\lim_{n\to\infty} Eg(X_n,X) = Eg(X,X)$. But $Eg(X,X) = 0$, so we know $\lim_{n>0} d_P(Y_n,Y)=0,$ where $Y_n=h(X_n)$ and $Y=h(X).$ This implies $Y_n\xrightarrow P Y$.

ADDED NEXT DAY: Grimmett's textbook has a clean and simple proof. We want to show that for arbitrary $\delta>0$ we have $P(|Y_n-Y|>\delta)\to0$. It suffices to show that for each $\eta>0$ for all $n$ sufficiently large, $P(|Y_n-Y|>\delta)<\eta$. So pick $K$ so large that $P(|X|>K)<\eta/2$, and observe that the function $h$ is uniformly continuous on $[-K,K]$, and hence there exists an $\epsilon>0$ such that $|u|\le K$ and $|u-v|\le\epsilon$ together imply $|h(u)-h(v)|<\delta$. Thus, $P(|Y_n-Y|>\delta)<P(|X|>K) + P(|X_n-X|>\epsilon)<\eta/2 + P(|X_n-X|>\epsilon)<\eta$ for all $n$ sufficiently large.

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