Lecture 37 — Continuity Theorem
This lecture has two major parts:
- Completing the Inversion Formula for characteristic functions (CFs), including boundary cases.
- Proving the Continuity Theorem:
convergence of characteristic functions implies convergence in distribution.
1. Inversion Formula Revisited
From Lecture 36, let $X$ be a real-valued r.v. with characteristic function $\varphi_X(t)$, and let
$U\sim \mathrm{Unif}(a,b)$, independent of $X$.
Set $Y = X - U$.
Then:
\[\varphi_Y(t) = \varphi_X(t)\,\varphi_{-U}(t), \qquad \varphi_{-U}(t) = \frac{e^{-ita}-e^{-itb}}{it(b-a)}.\]Main inversion identity (page 1):
\(\frac{b-a}{2\pi} \int_{-T}^{T} \varphi_X(t)\,\varphi_{-U}(t)\,dt \;\xrightarrow[T\to\infty]{ } P(a<X<b) +\frac{P(X=a)+P(X=b)}{2}. \tag{1}\)
This recovers the mass in $(a,b)$ plus half–mass at the endpoints.
1.1 Boundary case: recovering the mass at a point
Page 1 shows the special case $a=b=x$.
Using the formula with $a=b=x$ (interpreting via limiting uniform variables):
This is the Dirichlet kernel limit: \(\frac{\sin(Ty)}{\pi y} \to \delta_0(y),\) as indicated by the sketch on page 1 (“hope $x-a=0$”).
Thus (2) recovers point mass at any $x\in\mathbb{R}$.
1.2 When $X$ has a density
Page 1–2:
If $\int_{-\infty}^\infty \vert \varphi_X(t)\vert \,dt <\infty$, then $X$ has a PDF $f_X$, with:
Moreover, the truncated integral approximates the density:
\[\lim_{T\to\infty} \frac{1}{2\pi} \vert \int_{-T}^{T} e^{-itx}\varphi_X(t)\,dt\vert = f_X(x).\](Your notes show the inequality bounding the tail by $(1/2T)\int \vert \varphi_X\vert \to0$.)
2. Continuity Theorem (Durrett 3.3.6)
This is the heart of Lecture 37, written across both pages.
Continuity Theorem: Part (i)
If $X_n\Rightarrow X$, then for every $t\in\mathbb{R}$:
\[\varphi_{X_n}(t) \to \varphi_X(t). \tag{4}\]This follows from boundedness of $e^{itX_n}$ and dominated convergence.
Continuity Theorem: Part (ii)
Let $\varphi_n(t)$ be the CFs of $X_n$.
Assume:
- $\varphi_n(t)\to g(t)$ for every $t\in\mathbb R$,
- $g$ is continuous at 0.
Then:
- ${X_n}$ is tight,
- $g(t)$ is the CF of some $X$,
- and
\(X_n \Rightarrow X. \tag{5}\)
This is the most powerful implication:
Pointwise convergence of CFs + continuity at 0 ⇒ weak convergence.
Durrett Probability 4.1e - Theorem 3.3.6 Continuity Theorem
Let $\mu_n,1\ge n\ge\infty$ be probability measures with characteristic functions $\varphi_n$. (i) If $\mu_n\Rightarrow \mu_\infty$ then $\varphi_n(t)\to \varphi_\infty(t)$ for all $t$. (ii) If $\varphi_n(t)$ converges pointwise to a limit $\varphi(t)$ that is continuous at $0$, then the associated sequence of distributions $\mu_n$ is tight and converges weakly to the measure $\mu$ with characteristic function $\varphi$.
3. Proof Sketch of the Continuity Theorem
The handwritten proof on pages 1–2 gives the core ideas:
(1) show tightness; (2) identify the limit law using inversion;
(3) show convergence on a subsequence gives same CF, therefore whole sequence converges.
3.1 Key Lemma on Page 2
For any $u>0$,
\[\frac{1}{u}\int_{-u}^{u} \big(1-\varphi(t)\big)\,dt = 2 - \frac{1}{u}\int_{-u}^{u}\varphi(t)\,dt. \tag{6}\]Since $\vert \varphi(t)\vert \le 1$, the right-hand side is bounded by 2.
Further in the notes (page 2):
\[\vert X\vert \ge a \ \Rightarrow\ 1 - \varphi(t) \ge c\,P(\vert X\vert \ge a) \quad\text{for suitable }c>0. \tag{7}\]The intuition is that if $\vert X\vert $ is large, the oscillation in $e^{itX}$ forces the average in (6) up.
Combining (6)–(7), for each $\varepsilon>0$ one finds an $M$ such that:
\[\frac{1}{u}\int_{-u}^u \big(1-g(t)\big)\,dt <\varepsilon \quad\Rightarrow\quad P(\vert X\vert >M) < \varepsilon. \tag{8}\]This step (page 2 boxed conclusion):
\[P(\vert X_n\vert \ge M) < \varepsilon \quad\text{uniformly in }n,\]is exactly tightness.
Thus the sequence of distributions is tight.
3.2 Extracting a subsequence
By tightness + Helly–Prokhorov (Lecture 33),
we may extract $X_{n_k}\Rightarrow X$ for some subsequence.
Then by part (i):
\[\varphi_{n_k}(t) \to \varphi_X(t). \tag{9}\]But the hypothesis says $\varphi_{n_k}(t)\to g(t)$.
Thus:
So $g$ is a characteristic function.
Since every convergent subsequence has the same limit law, the whole sequence converges:
\[X_n \Rightarrow X.\]4. Summary of the Logic
The Continuity Theorem ties everything together:
\[X_n\Rightarrow X \quad\Longleftrightarrow\quad \varphi_{X_n}(t)\to\varphi_X(t)\;\forall t.\]More generally:
-
Given just CF convergence,
\[X_n \Rightarrow X.\]
if the limit $g(t)$ is continuous at 0, then it is a valid CF of some $X$, and
This is the foundation of all the CF-based limit theorems, especially the CLT, stable laws, and the Lévy–Khintchine formula.
Cheat-Sheet Summary — Lecture 37
- Inversion formula recovers probabilities: \(\frac{b-a}{2\pi}\int_{-T}^{T}\varphi_X(t)\varphi_{-U}(t)\,dt \to P(a<X<b)+\tfrac12(P(X=a)+P(X=b)).\)
-
If $\int \vert \varphi_X\vert !<\infty$, then $X$ has a density and
\(f_X(x)=\frac{1}{2\pi}\int e^{-itx}\varphi_X(t)\,dt.\) - Continuity Theorem:
- $X_n\Rightarrow X \implies \varphi_{X_n}\to\varphi_X$.
- If $\varphi_n\to g$ pointwise and $g$ is continuous at $0$,
then ${X_n}$ is tight, $g$ is a CF, and $X_n\Rightarrow X$.
-
The key lemma:
\(\frac{1}{u}\!\int_{-u}^{u}(1-\varphi(t))dt = 2 - \frac{1}{u}\!\int_{-u}^{u}\varphi(t)dt.\)Used to force tightness.
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