Probability Theory – Klenke’s Proof of Slutzky’s Theorem

convergence-divergencemeasure-theoryprobability theoryprobability-limit-theoremsreal-analysis

In Klenke's book on probability he states Slutzky's theorem as:

Let $X, X_1, X_2, \ldots$ and $Y_1, Y_2\ldots$ be random varaibles with values in $E$. Assume $X_n \xrightarrow{\mathcal{D}} X$ and $d(X_n, Y_n) \xrightarrow{n\rightarrow \infty} 0$ in probability. Then $Y_n \xrightarrow{\mathcal{D}} X$.

Here $E$ is a metric space with metric $d$, and $\xrightarrow{\mathcal{D}}$ means convergence in disribution, i.e. $X_n \xrightarrow{\mathcal{D}} X$ if the distributions $\mu_{X_n}$ of the $X_n$ converge weakly to the distribution $\mu_X$ of $X$. By weak convergence he means
$$\int f \mu_{X_n} \rightarrow \int f \mu_X$$
for all continuous and bounded functions $f$.

His proof is as follows:

Let $f: E \rightarrow \mathbb{R}$ be bounded and Lipschitz continuous with constant $K$. Then $$|f(x) – f(y)| \leq Kd(x,y) \wedge 2\|f\|_{\infty} \quad \text{for all } x, y \in E.$$ Dominated convergence yields $\limsup_{n \rightarrow \infty} \mathbf{E}[|f(X_n) – f(Y_n)|] = 0.$ Hence we have $$\limsup_{n\rightarrow \infty} |\mathbf{E}[f(Y_n)] – \mathbf{E}[f(X)]| \\ \leq \limsup_{n\rightarrow \infty} |\mathbf{E}[f(X)] – \mathbf{E}[f(X_n)]| + \limsup_{n\rightarrow \infty} |\mathbf{E}[f(X_n)] – \mathbf{E}[f(Y_n)]| = 0.$$

I have a few questions about this proof:

  1. Why are we only considering Lipschitz continuous functions? By the definition of weak convergence shouldn't we consider continuous and bounded functions instead?
  2. What is the point of showing $|f(x) – f(y)| \leq Kd(x,y) \wedge 2\|f\|_{\infty}$? It appears we do not use it in the inequalities beneath it.
  3. Why is he using $\limsup$ and not $\lim$ for the dominated convergence theorem? I thought this theorem only applies to $\lim$.
  4. The above proof suggests there is a relationship between $X_n \xrightarrow{\mathcal{D}} X$ and $\limsup_{n\rightarrow \infty} |\mathbf{E}[f(X)] – \mathbf{E}[f(X_n)]| = 0$. What is this relationship?

Best Answer

Let me know if you need additional clarity on the following.

(1) Klenke proves weak convergence is equivalent to $\int f d\mu_n\to \int fd\mu$ for all $f$ Lipschitz and bounded (Portmanteau theorem).

(2) $E[|f(X_n)-f(Y_n)|]\leq E[Kd(X_n,Y_n)\wedge 2\|f\|_\infty]\to 0$ by dominated convergence (version with convergence in probability) since $d(X_n,Y_n)\to^P 0$ and $Kd(X_n,Y_n)\wedge 2\|f\|_\infty$ is bounded.

(3) If $\lim$ exists $\limsup$ exists and is equal to $\lim$. $\limsup$ is used in the next passage since we don't know if $\lim$ exists yet for $|E[f(Y_n)]-E[f(X)]|$.

(4) $\limsup_n|E[f(X_n)]-E[f(X)]|=0\iff \lim_nE[f(X_n)]=E[f(X)]$. For $f$ bounded Lipschitz, this is indeed one of the equivalent definitions of weak convergence in the setting of probability measures (just probability notation).

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