39 — Embedding Random Variables into Brownian Motion (Skorokhod Embedding) and CLT via Brownian Motion
1. Thursday’s CB06 Theorem (Review)
Suppose we have a discrete random variable \(P(X = x_k) = p_k,\qquad k=1,\dots,n,\) with \(E[X]=0.\)
Theorem:
There exists a stopping time $\tau$ such that
\(B^0(\tau)\ \overset{d}{=}\ X,
\qquad
E[\tau] = E(X^2).\)
That is, Brownian motion can be stopped so that its stopped value has the same distribution as $X$, and the expected stopping time equals the variance.
2. Why $E[\tau]=E(B_\tau^2)$
We use the martingale \(M_t = B_t^2 - t.\)
Since at the stopping time $\tau$, \(|B^0_{\tau\wedge t}| \le C < \infty\) (bounded because $X$ only takes finitely many values), the stopped process is uniformly integrable, hence a true martingale.
Thus, \(E[B^2_{\tau \wedge t} - (\tau \wedge t)] = 0 \quad \text{for all } t,\) and taking $t\to\infty$, \(E[B_\tau^2] = E(\tau).\)
If $X$ takes values $x_k$, then \(E[B_\tau^2] = E[X^2].\)
3. Induction on the Number of Values of $X$
Base case: $X$ takes only one value
If $X = 0$, take \(\tau = \inf\{t\ge 0 : B_t = 0\} = 0.\)
Inductive step:
Assume the result holds for all random variables taking $n$ distinct values.
Let \(X \in \{x_1 < x_2 < \cdots < x_{n+1}\}.\)
Partition the probability space using \(\mathcal{F}_1 = \sigma\{X = x_k,\ k\le n\ \} \cup \{X = x_{n+1}\}.\)
Define \(Y = E[X \mid \mathcal{F}_1].\)
Then:
- $Y$ takes only $n$ distinct values (by collapsing the last two values of $X$),
- $E[Y]=0$,
- By induction, there is a stopping time $\eta$ such that \(B(\eta) \overset{d}{=} Y.\)
Now refine inside each event ${Y = y_k}$ to reach the desired value of $X$.
Construct \(\tau = \eta + \tau',\) where $\tau’$ is a (conditionally defined) stopping time such that \(B(\eta + \tau') - B(\eta) \overset{d}{=} X - Y.\)
Thus we obtain \(B(\tau) = X.\)
This completes the induction.
4. General Skorokhod-Type Problem (Continuous Case)
Problem:
Let $X$ be any random variable with
\(E[X]=0,\qquad E[X^2]<\infty.\)
Find a stopping time $\tau$ such that \(B(\tau)\ \overset{d}{=} \ X, \qquad E[\tau] = E(X^2).\)
Construction outline
Assume $\Omega=\mathbb R$ with Borel $\sigma$-algebra and $P$ the distribution of $X$.
Let $\mathcal{Q}_k$ be partitions of $\mathbb R$: \(\mathcal{Q}_1 = \{(-\infty,q_1],(q_1,\infty)\},\) \(\mathcal{Q}_2 = \{(-\infty,q_1], (q_1,q_2], (q_2,\infty)\},\) and so on, where $q_1 < q_2 < \cdots < q_n$ are quantiles of $P$.
Define \(X_n = E[X \mid \sigma(\mathcal{Q}_n)].\)
Then:
- $X_n$ has finitely many values,
- $X_n \to X$ a.s. and in $L^2$,
- By the discrete embedding, there exist increasing stopping times \(\tau_1 \le \tau_2 \le \cdots\) such that \(B(\tau_n) \overset{d}{=} X_n.\)
Since conditional expectation contracts $L^2$: \(E[X_n^2] \uparrow E[X^2],\) and hence \(E[\tau_n]=E[X_n^2]\uparrow E[X^2].\)
By monotone convergence, $\tau=\lim \tau_n$ satisfies \(B(\tau) = \lim B(\tau_n) = X, \qquad E[\tau]=E(X^2).\)
This gives the Skorokhod embedding theorem for square-integrable distributions.
5. Application: Deriving the CLT via Brownian Motion
Let ${X_k}_{k=1}^\infty$ be i.i.d. with \(E[X_k]=0,\qquad E[X_k^2]=1.\)
Let \(S_n = X_1 + \cdots + X_n.\)
We want to prove: \(\frac{S_n}{\sqrt{n}} \Rightarrow N(0,1).\)
Step 1 — Embed each $X_k$ into Brownian motion
Find stopping times $\tau_1,\tau_2,\dots$ such that
- $B(\tau_k) - B(\tau_{k-1}) \overset{d}{=} X_k$,
- ${\tau_k - \tau_{k-1}}$ are i.i.d.,
- $E[\tau_k - \tau_{k-1}] = 1$.
Then: \(S_n = B(\tau_n).\)
Step 2 — Show Brownian rescaling appears
From the notes on page 4 (diagram labelled “A”):
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But by the law of large numbers, \(\frac{\tau_n}{n} = \frac{1}{n}\sum_{k=1}^n (\tau_k-\tau_{k-1}) \xrightarrow{a.s.} E[\tau_1] = 1.\)
Thus, \(\frac{S_n}{\sqrt{n}} = B\left(\frac{\tau_n}{n}\right) \xRightarrow[n\to\infty]{} B(1) \sim N(0,1).\)
This yields the Central Limit Theorem via the Skorokhod embedding principle.
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