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7 — H–S 0–1 Law, Applications, and Return-Time Results

1. Behavior of Boundary-Hitting Times

Let ${X_k}_{k\ge 1}$ be iid with

  • $E(X_1)=0$,
  • $E(X_1^2)=1$.

Define partial sums
\(S_n=\sum_{k=1}^n X_k.\)

For $c>0$, define the hitting time \(T_c=\inf\{n\ge 1 : S_n > c\sqrt{n}\}.\)

Theorem

\(E[T_c] = \begin{cases} \infty, & c\ge 1,\\[6pt] <\infty, & c<1. \end{cases}\)

First Step: Show $P(T_c<\infty)=1$ for all $c>0$

Using the CLT, \(\frac{S_n}{\sqrt{n}} \Rightarrow N(0,1).\)

Hence for large $n$, \(P\left(\left|\frac{S_n}{\sqrt{n}}\right| > c\right) \approx P(|Z|>c) > 0.\)

Thus, \(P\left(\left|\frac{S_n}{\sqrt{n}}\right| > c \ \text{i.o.}\right) > 0.\)

Since
\(\bigcup_{k\ge 1}\left\{\left|\frac{S_k}{\sqrt{k}}\right| > c\right\}\) is increasing in $k$,
\(P\left(\left|\frac{S_k}{\sqrt{k}}\right| > c\right) > 0 \quad\Rightarrow\quad P(T_c<\infty)=1.\)

2. Proving $E(T_c)=\infty$ for $c\ge 1$

Assume by contradiction that $E(T_c)<\infty$.

By Wald’s Second Equation, \(E(S_T^2)=E(X_1^2)\,E(T)=E(T).\)

But at the stopping time $T_c$, \(S_{T_c} > c\sqrt{T_c} \quad\Rightarrow\quad S_{T_c}^2 > c^2 T_c.\)

Then \(E(S_T^2) > E(c^2 T_c) = c^2 E(T_c) \ge E(T_c).\)

This contradicts Wald’s equation, which gave
\(E(S_T^2) = E(T_c).\)

Thus: \(E(T_c)=\infty \quad \text{for } c\ge 1.\)

3. Hewitt–Savage 0–1 Law

Let ${X_k}$ be iid.
Let $A$ be a permutable event, meaning:

$\pi(A)=A$ for every finite permutation $\pi$ of the coordinates.

Then: \(P(A)\in\{0,1\}.\)

Durrett Probability 4e - Theorem 4.1.1. - Hewitt-Savage 0-1 law.

If $X_1, X_2,…$ are i.i.d. and $A\in\mathcal{E}$ then $P(A)\in{0,1}$.

where

  • finite permutation $\pi$ - a bijection $\pi : \mathbb{N}\to\mathbb{N}$ s.t. ${i\in\mathbb{N} : \pi(i)\ne i}$ is finite.
  • permutable event A - $\pi^{-1}A = A$ where $\pi^{-1}A\equiv {\omega : \pi omega \in A}$
  • exchangeable $\sigma$-field $\mathcal{E}$ - Is a set of permutable events.

My own words: HS 0–1: For i.i.d. sequences, any event invariant under all finite permutations has probability 0 or 1.

Examples

(1) Tail events

Do not change if you permute a finite number of coordinates.
Thus tail events satisfy the 0–1 law.

(2) Events of the form

\(\{ S_n \in B \ \text{i.o.}\}.\)
This describes behavior “infinitely often,” which is also permutation-invariant.

4. Applications of Hewitt–Savage 0–1 Law

Let $S_n=\sum_{k=1}^n X_k$.

There are four possible long-run behaviors for a random walk:

  1. $X_1\equiv 0$ a.s.
    Then $S_n=0$ for all $n$.

  2. $S_n \to +\infty$ a.s.
    Happens when $E(X_1)>0$ (by SLLN).

  3. $S_n \to -\infty$ a.s.
    Happens when $E(X_1)<0$.

  4. Oscillation:
    \(-\infty = \liminf_{n\to\infty} S_n < \limsup_{n\to\infty} S_n = \infty.\)

Example for (4):

  • If $X\stackrel{d}{=}-X$ (symmetric),
  • and $E(X)=0$,
  • and $E(X^2)<\infty$,
  • and $X\not\equiv 0$,

then \(P(S_n>\sqrt{n} \ \text{i.o.}) > 0.\)

Since the event is permutable, H–S 0–1 law gives probability 1: \(P(S_n>\sqrt{n} \ \text{i.o.})=1.\)

5. Consequences and Tail-Limit Argument

By H–S, \(\lim_{n\to\infty} S_n = C \quad\text{a.s., where } C\in[-\infty,\infty].\)

Let
\(S_n' = S_{n+1} - X_1 = \sum_{k=2}^{n+1} X_k.\)

Since ${S_n’}$ and ${S_n}$ have the same distribution, \(\limsup_{n\to\infty} S_n' = \limsup_{n\to\infty} S_n = C.\)

Thus: \(C - X_1 = C.\)

If $|C|<\infty$, this forces
\(X_1 = 0 \ \text{a.s.}\)

Thus unless $X_1\equiv 0$, we must have:
\(C\in\{-\infty,\infty\},\)
and similarly for $\liminf$.

6. Proof Components for H–S 0–1 Law

Symmetric difference metric: \(A\Delta C\subseteq (A\Delta B)\cup (B\Delta C).\)

If
\(P(A_n\Delta A)\to 0,\) then:

  1. $P(A_n)\to P(A)$
  2. $P(A_n\cap A)\to P(A)$

If also
\(P(B_n\Delta A)\to 0,\) then \(P(A_n\cap B_n)\to P(A).\)

This is used to show invariance under finite permutations forces probability $0$ or $1$.

End of Lecture 8

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