Lecture 32 — Scheffé’s Theorem, Total Variation, and Portmanteau Properties

1. Scheffé’s Theorem

Let $f_n$ be the pdf of $X_n$ and let $f$ be the pdf of $X$.
Assume:

\[f_n(x) \to f(x) \quad\text{pointwise for all }x\in\mathbb R.\]

(Sketched on page 1.)

Goal

Show that $\sup_{B\in\mathcal B(\mathbb R)} \vert P(X_n\in B) - P(X\in B) \vert \;\longrightarrow\; 0,$ i.e. convergence in total variation.

Key inequality

For any Borel set $B$, $\vert P(X_n\in B) - P(X\in B)\vert = \vert \int_B f_n(x)\,dx - \int_B f(x)\,dx\vert \le \int_B \vert f_n(x) - f(x)\vert \,dx.$

Thus, $\sup_B \vert P(X_n\in B) - P(X\in B)\vert \le \int_{\mathbb R} \vert f_n(x)-f(x)\vert \,dx. \tag{1}$

Hence it suffices to show: $\int_{\mathbb R}\vert f_n-f\vert \to 0.$

2. Positivity decomposition

Using $a = a^+ - a^-, \qquad \vert a\vert = a^+ + a^-,$ write: $\int_{\mathbb R}(f - f_n) = \int (f - f_n)^+ - \int (f - f_n)^-.$

But $\int (f - f_n) = 0 \quad\text{since both integrate to 1}.$

Thus, $\int \vert f - f_n\vert = 2\int (f - f_n)^+. \tag{2}$

Dominating function

On page 1 your notes check the DCT bound:

If $f_n(x) > f(x)$ then
$(f(x)-f_n(x))^+ = 0 \le f(x)$.
If $f_n(x) < f(x)$ then
$0 < f(x)-f_n(x)\le f(x)$.

So: $(f(x)-f_n(x))^+ \le f(x).$

Since $f_n\to f$ pointwise and the dominating function $f$ is integrable,
DCT gives:

\[\int (f-f_n)^+ \to 0.\]

Plugging into (2):

\[\int \vert f_n-f\vert \to 0.\]

Then (1) gives total variation convergence.

3. Consequence for Distribution Functions

In our setting, $P(X=x)=0$ for all $x\in\mathbb R$.
Thus the CDFs are continuous.

If $\vert M_n - M\vert _{TV}\to 0$, then:

\[F_{X_n}(x) = P(X_n \le x) \to P(X\le x) = F_X(x) \quad\text{uniformly in }x.\]

So CDF convergence is uniform.

This comment appears in your notes as:

“Convergence should be uniformly.”

4. Total Variation Norm

For probability measures $M_n(B)=P(X_n\in B)$:

\[\vert M_n - M\vert = \sup_{B\in \mathcal B(\mathbb R)} \vert M_n(B) - M(B)\vert \to 0.\]

This is stronger than convergence in distribution.

5. Example — The Empirical Median of Uniform[0,1]

(Derived on pages 1–2; reference to Durrett 3.2.6.)

Let ${U_k}$ be i.i.d. uniform on $[0,1]$.
Let $U_{(1)}\le U_{(2)}\le \dots \le U_{(2n+1)}$ be the order statistics.
The sample median is $V_{n+1} = U_{(n+1)}.$

Density (page 2 diagram)

$f_{V_{n+1}}(x) = \binom{2n+1}{n,n} x^n (1-x)^n, \qquad 0<x<1.$

This comes from the multinomial probability that exactly $n$ observations lie left of $x$ and $n$ right of $x$.

Mean

$E[V_{n+1}] = \frac12.$

Variance

Using the Beta integral identity (page 2): $\int_0^1 x^k(1-x)^l\,dx = \frac{k!\,l!}{(k+l+1)!},$ your notes compute: $\operatorname{Var}(V_{n+1}) = \frac{n}{4(2n+1)(2n+2)} \sim \frac{1}{8n}.$

Normalization

Define $Y_n = 2\sqrt{2n}\left(V_{n+1} - \frac12\right).$

Then (page 2): $f_{Y_n}(y) = \phi_n(y) = \left(1 - \frac{y^2}{2n}\right)^n \quad\to\quad e^{-y^2/2},$

so $Y_n \Rightarrow N(0,1).$

This is a CLT for the sample median under uniform sampling.

6. Portmanteau Theorem (Durrett Thm 3.2.5)

Your notes (page 2–3) list the four equivalent conditions for weak convergence.

Let $X_n\Rightarrow X$.

The following are equivalent:

(CDF convergence)
$X_n \Rightarrow X$.
(Open sets)
$\liminf_{n\to\infty} P(X_n\in O) \ge P(X\in O),\qquad \forall\ O\ \text{open}.$
(Closed sets)
$\limsup_{n\to\infty} P(X_n\in C) \le P(X\in C),\qquad \forall\ C\ \text{closed}.$
(Boundary condition)
For all Borel sets $A$ with boundary $\partial A$ satisfying $P(X \in \partial A)=0,$ we have $P(X_n\in A)\to P(X\in A).$

Durrett Probability 3.2.5 - Portmanteau Theorem

The following are equivalent: $\begin{aligned} &\quad(i)\quad X_n\Rightarrow X_\infty, \\ &\quad(ii)\quad \text{ for all open sets } G, \liminf_{n\to\infty} P(X_n\in G)\ge P(X_\infty\in G), \\ &\quad(iii)\quad \text{ for all closed sets } F, \limsup_{n\to\infty} P(X_n\in F)\le P(X_\infty\in F), \\ &\quad(iv)\quad \text{ for all Borel sets }A\text{ with } P(X_\infty\in \partial A)=0, \lim_{n\to\infty} P(X_n\in A)= P(X_\infty\in A). \end{aligned}$

Sketch: (4) ⇒ (1)

(Page 3 diagram.)

Take the interval $A=(-\infty,x]$.
Its boundary is ${x}$.
If $P(X=x)=0$, then (4) gives:

\[P(X_n\le x) \to P(X\le x),\]

which is exactly the definition of convergence in distribution.

Sketch: (1) ⇒ (2)

If $X_n\to X$ almost surely (the stronger assumption used in notes for intuition): $\mathbb{1}_{\{X_n\in O\}} \to \mathbb{1}_{\{X\in O\}}$ pointwise except on a null set.

Then by Fatou’s lemma:

\[\liminf_n P(X_n\in O) = \liminf_n E[\mathbb{1}_{\{X_n \in O\}}] \ge E[\liminf_n \mathbb{1}_{\{X_n\in O\}}] = P(X\in O).\]

(2) ⇒ (3)

Apply (2) to $O = C^c$.
Then take complements.

(3) ⇒ (4)

For any Borel set $A$:

\[A^\circ \subseteq A \subseteq \overline A.\]

Since $\partial A = \overline A \cap (\overline{A^\circ})^c,$ the condition $P(X\in\partial A)=0$ ensures:

\[P(X\in \overline A) = P(X\in A) = P(X\in A^\circ).\]

Apply (2) and (3) to the interior and closure.

Cheat-Sheet Summary — Lecture 32

Scheffé’s theorem: pointwise convergence of pdfs ⇒ convergence in total variation.
Uses positivity decomposition and DCT.
Total variation convergence implies uniform CDF convergence.
Empirical median CLT for uniform samples: $2\sqrt{2n}\left(V_{n+1}-1/2\right) \Rightarrow N(0,1).$
Portmanteau Theorem gives 4 equivalent formulations of weak convergence:
1. CDF convergence.
2. Open-set inequality.
3. Closed-set inequality.
4. Convergence on sets with zero boundary mass.

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