Lecture 33 — Uniformly Continuous Functions, Helly Selection, and Tightness
This lecture connects several ideas:
- Reducing weak convergence tests to uniformly continuous bounded functions,
- Approximating bounded uniformly continuous functions by smooth functions via Gaussian averaging,
- Helly’s Selection Theorem for subsequence extraction of distribution functions,
- Tightness of a family of distributions.
1. Weak Convergence: Restriction to Uniformly Continuous Bounded Functions
We have already seen:
\[X_n \Rightarrow X \quad\Longleftrightarrow\quad E f(X_n) \to E f(X) \quad\text{for all bounded continuous } f. \tag{1}\]Claim
It suffices to check (1) for uniformly continuous bounded functions.
Definition: Uniformly Continuous
$f:\mathbb R\to\mathbb R$ is uniformly continuous if
\[\forall\varepsilon>0\ \exists\delta>0\ \text{ such that }\ \vert x_1-x_2\vert <\delta \ \Rightarrow\ \vert f(x_1)-f(x_2)\vert <\varepsilon \quad\forall x_1,x_2.\]The diagram on page 1 (vertical bound on jump size) emphasizes that the continuity modulus is uniform in $x$.
2. Approximation of Bounded Uniformly Continuous $f$ by Smooth Functions
The key idea (illustrated in the “moving average / bump smoothing’’ picture on page 1) uses Gaussian convolution.
Let
\(T_x = x + \sigma Y, \qquad Y\sim N(0,1),\) and define the Gaussian smoothing of $f$:
\[f_\sigma(x) = E[f(T_x)] = E[f(x+\sigma Y)].\]Properties of $f_\sigma$
-
$f_\sigma \in C_B^\infty(\mathbb R)$ (all derivatives exist and are bounded).
This is the smooth bell‐curve averaging in the diagram. -
$f_\sigma \to f$ uniformly as $\sigma\to 0$ (page 1–2 argument):
\[\vert f(x+\sigma Y) - f(x)\vert \le \varepsilon\cdot\mathbf{1}_{\{\vert \sigma Y\vert \le\delta\}} + 2\vert f\vert _\infty\mathbf{1}_{\{\vert \sigma Y\vert >\delta\}}.\]
Since $f$ is uniformly continuous and bounded,
for small $\sigma$,The tail probability $P(\vert Y\vert >\delta/\sigma)\to 0$ as $\sigma\to 0$.
\[\sup_x \vert f_\sigma(x) - f(x)\vert \to 0.\]
Thus:
Therefore:
Any bounded uniformly continuous $f$ can be approximated uniformly by smooth bounded functions.
3. Lifting Weak Convergence Through Gaussian Smoothing
Suppose for all $\sigma>0$,
\[X_n + \sigma Y \Rightarrow X + \sigma Y, \tag{2}\]with the same independent $Y\sim N(0,1)$.
Then for all $f\in C_B^\infty$,
\[E[f(X_n+\sigma Y)] \to E[f(X+\sigma Y)].\]Because $f_\sigma\to f$ uniformly and the error bound does not depend on $n$, you get (page 2):
\[\limsup_{n\to\infty} \vert E[f(X_n)] - E[f(X)]\vert \le 3\varepsilon \quad\text{for small enough }\sigma.\]Thus, assuming (2) for all $\sigma>0$, we conclude:
\[E[f(X_n)] \to E[f(X)] \qquad\forall f\in C_B(\mathbb R).\]This gives another route to establishing weak convergence.
4. Helly’s Selection Theorem
This is the major theorem in the second half of the notes (yellow highlight on page 1–2).
Let ${F_n}$ be a sequence of distribution functions (i.e. CDFs).
Theorem (Helly Selection)
There exists a subsequence $F_{n_k}$ and a function $F\in C^+$ (i.e. non-decreasing, right-continuous) such that:
\[F_{n_k}(x) \to F(x) \quad\text{at every continuity point of }F.\]Durrett Probability 4.1e - Theorem 3.2.6 Helly’s Selection Theorem (p. 88)
For every sequence $F_n$ of distribution functions, there exists a subsequence $F_{n_k}$ and a right continuous nondecreasing function $F$ so that $\lim_{n\to\infty}F_{n_k}(y)=F(y)$ at all continuity points $y$ of $F$.
Sketch of Proof (as given in the notes)
Use a diagonal argument on the rationals $\mathbb Q={q_1,q_2,\dots}$:
-
For $q_1$: the sequence $F_n(q_1)\in [0,1]$ is bounded, hence has a convergent subsequence.
Call this subsequence $(1,k)$. -
For $q_2$: look at ${F_{(1,k)}(q_2)}$.
Select a further subsequence $(2,k)$ that converges at $q_2$. -
Continue: for $q_m$, extract a subsequence $(m,k)\subseteq(m-1,k)$ with convergence at $q_m$.
-
Diagonal selection:
Define $n_k = (k,k)$.
Then $F_{n_k}(q_m)\to \tilde F(q_m)$ for all $m$.
Define:
\[F(x) = \inf\{\tilde F(q_m) : q_m \ge x\}.\]This gives a nondecreasing, RCLL function $F$.
Then $F_{n_k}(x) \to F(x)$ at continuity points.
Important Note (page 2 diagram)
$F$ may not be a proper CDF (it may not satisfy $F(-\infty)=0$ or $F(\infty)=1$).
See the example on page 2 of notes where limit is a non-probability staircase.
Thus we must add a criterion ensuring tightness to guarantee proper limits.
5. Tightness
To ensure the Helly limit is a valid CDF, we impose tightness.
Definition (Tight Family)
A family ${F_n}$ of CDFs is tight if:
\[\forall \varepsilon>0\,\exists M>0\text{ such that } \sup_n P(\vert X_n\vert \ge M) = \sup_n\bigl(1 - F_n(M) + F_n(-M)\bigr) <\varepsilon.\]This was written in the notes as:
“Uniform tightness: same $M$ for all $n$.”
6. Tightness Results (Theorems 3.2.7 and 3.2.8)
Theorem 3.2.7
If $F_n\Rightarrow F$ and all are CDFs, then ${F_n}\cup{F}$ is tight.
Theorem 3.2.8
If there exists a measurable $\varphi\ge0$ such that
- $\varphi(x)\to\infty$ as $\vert x\vert \to\infty$,
- \[C = \sup_n \int \varphi(x)\,dF_n(x) < \infty,\]
then ${F_n}$ is tight.
Your notes sketch the proof (page 2):
- Choose $M$ so that $\varphi(x)\ge 1/\varepsilon$ for $\vert x\vert \ge M$.
- Then: \(P(\vert X_n\vert >M) \le \varepsilon \int\varphi(x)\,dF_n(x) \le \varepsilon C.\)
7. Concluding Comments
The lecture ends with a preview:
- Next time: Central Limit Theorem.
- Need only:
- finite second moment,
- Lindeberg–Feller condition.
Cheat-Sheet Summary — Lecture 33
- Weak convergence can be checked using bounded uniformly continuous $f$.
- Any such $f$ can be approximated uniformly by smooth $f_\sigma\in C_B^\infty$ via Gaussian smoothing.
- If $X_n+\sigma Y\Rightarrow X+\sigma Y$ for all $\sigma>0$, then $X_n\Rightarrow X$.
- Helly Selection Theorem: from any sequence of CDFs extract a subsequence converging at continuity points.
- Helly limits may fail to be proper CDFs; tightness fixes this.
- Tightness criteria:
- $\varphi$-moment bound,
- or already known weak convergence.
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