Lecture 30 — Feller’s Theorem, Complex Inequalities, and Weak Convergence
1. Feller’s Theorem (restated)
Let ${X_i}{i\ge1}$ be iid with distribution of $X$.
Let ${a_n}{n\ge1}$ be a sequence with $a_n>0$.
Define partial sums:
\(S_n = \sum_{k=1}^n X_k.\)
Theorem (Feller)
(a) If
\(\frac{a_n}{n} \uparrow \infty
\quad\text{and}\quad
\sum_{n=1}^\infty P(\vert X\vert >a_n) < \infty,\)
then
\(\frac{S_n}{a_n} \xrightarrow{\text{a.s.}} 0.\)
(b) If
\(\frac{a_n}{n} \quad\text{is non-decreasing}
\quad\text{and}\quad
\sum_{n=1}^\infty P(\vert X\vert >a_n) = \infty,\)
then
\(\overline{\lim_{n\to\infty}} \frac{\vert S_n\vert }{a_n} = \infty
\qquad\text{a.s.}\)
Remark (page 1)
If (b) holds, then $E\vert X\vert =\infty$.
Reason: if $\frac{a_n}{n}$ is non-decreasing, then
\(a_n \ge n a_1.\)
Thus
\(\sum_n P(\vert X\vert >n a_1)=\infty
\quad\Rightarrow\quad
E\vert X\vert = \infty.\)
2. Two Examples Illustrating (a) and (b)
Example 1 (page 1): St. Petersburg-type distribution
Let
\(P(X = 2^k) = 2^{-k},\qquad k=1,2,\dots\)
Then $E[X]=\infty$.
It is known (Weak Law computation) that \(\frac{S_n}{n\log_2 n} \xrightarrow{P} 1.\)
Thus \(a_n = n\log n.\)
Since $S_n/(n\log n)$ does not converge to zero a.s., Feller(b) applies: \(\sum_n P(\vert X\vert >a_n) = \infty.\)
This checks the divergence of the tail condition.
Example 2 (page 1): Heavy-tailed but finite $p$-moment ( $0<p<1$ )
Assume
\(E\vert X\vert ^p < \infty,\qquad 0 < p < 1.\)
Then \(\sum_{n=1}^\infty P(\vert X\vert >n^{1/p}) = \sum_{n=1}^\infty P(\vert X\vert ^p > n) < \infty.\)
Let \(a_n = n^{1/p}.\)
Since $0<p<1$,
\(\frac{a_n}{n} = n^{1/p - 1} \uparrow \infty.\)
Hence by Feller(a), \(\frac{\vert S_n\vert }{n^{1/p}} \xrightarrow{\text{a.s.}} 0.\)
This is exactly the Marcinkiewicz–Zygmund strong law for $0<p<1$.
=== Starting Durrett Chapter 3 ===
3. Transition to Chapter 3 — Complex Inequalities
Starting on page 2, your notes introduce complex notation for the coming Central Limit Theorem proof.
Let $z_k, w_k \in \mathbb{C}$.
Write $z = a + bi$.
Define \(\theta_i = \max\{1, \vert z_i\vert , \vert w_i\vert \}.\)
A key inequality (derived on page 2):
Inequality A (better than the naïve bound):
\(\vert \sum_{k=1}^n z_k - \prod_{k=1}^n w_k \vert \;\le\; \sum_{k=1}^n \vert z_k - w_k\vert \, \theta_k.\)
This is a flexible inequality used for characteristic function approximations in the CLT.
4. Fundamental Lemma for Characteristic Functions
Given a triangular array $a_{n,m}\in\mathbb{C}$, $1\le m\le n$, suppose:
-
(i)
\(\sum_{m=1}^n a_{n,m} \longrightarrow a \in \mathbb{C}.\) -
(ii)
\(\sup_n \sum_{m=1}^n \vert a_{n,m}\vert < \infty.\) -
(iii)
\(\max_{1\le m\le n} \vert a_{n,m}\vert \xrightarrow{n\to\infty} 0.\)
Then \(\prod_{m=1}^n (1 + a_{n,m}) \;\longrightarrow\; e^a.\)
This is the key analytic tool needed to show that the characteristic function of a normalized sum converges to $e^{-t^2/2}$ in the CLT.
Example
For
\((1 + x/n)^n \to e^x,\)
take $a_{n,m} = x/n$.
Then (i), (ii), and (iii) follow immediately.
5. Inequality for complex exponentials (page 3)
For any complex $z$ with $\vert z\vert \le 1$:
\[\vert e^z - (1+z)\vert \le \vert z\vert ^2.\]This is established by the Taylor series: \(e^z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots,\) and the remainder is bounded by \(\vert z\vert ^2\left(\frac{1}{2!} + \frac{1}{3!} + \cdots\right) < \vert z\vert ^2.\)
This inequality justifies replacing products by exponentials when proving the CLT.
6. Weak Convergence (Distributional Convergence)
(Definition section at end of page 3.)
Definition
We say \(X_n \Rightarrow X\) if the CDFs satisfy \(F_{X_n}(y) \longrightarrow F_X(y) \quad\text{for all }y\text{ at which }F_X \text{ is continuous}.\)
Equivalent characterizations:
-
Convergence of expectations of bounded continuous functions: \(E[g(X_n)] \to E[g(X)], \qquad g:\mathbb{R}\to\mathbb{R}\ \text{bounded, continuous}.\)
-
CDF definition:
$F_X$ is right‐continuous,
\(F(y-) = \lim_{t\uparrow y} F(t),\qquad F(y)=P(X\le y).\)
Weak convergence is the fundamental mode of convergence used in the Central Limit Theorem.
Cheat-Sheet Summary — Lecture 30
- Feller’s theorem gives necessary and sufficient tail conditions for the almost-sure behavior of $\frac{S_n}{a_n}$.
- Examples show how fast or slow a sequence $a_n$ must grow for almost-sure convergence to 0.
- Chapter 3 begins with complex inequalities essential for manipulating characteristic functions in CLT proofs.
- A powerful lemma:
\(\prod_{m=1}^n (1+a_{n,m}) \to e^a\) when the triangular array satisfies summability and uniform smallness. - Weak convergence $X_n\Rightarrow X$ is introduced as the gateway to the Central Limit Theorem.
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