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The Skorokhod embedding of a centered random walk states that a random walk with independent identically distributed centered steps of finite variance ${\sigma }^2$ can be realized on an appropriate probability space aswhere $B$ is a standard Brownian motion and the $T_n$ are finite stopping times. More precisely, the stopped positions have the same joint law as the given random walk, and the pairsmay be chosen independent and identically distributed. Their spatial component has the step law, and $\mathbb{E}(T_n-T_{n-1})={\sigma }^2$. Repeating the one-step Skorokhod embedding theorem with the Strong Markov property gives this formulation. In the present normalization, the mean time increment is one, and the strong law of large numbers gives $T_n/n\to 1$ almost surely.

The Donsker invariance principle states that the linearly interpolated diffusively rescaled random walkconverges weakly as a random element of $C[0,1]$, equipped with the uniform norm, to standard Brownian motion restricted to $[0,1]$. At $t=1$ the fractional term is zero. The only step assumptions needed here are zero mean, unit variance, and independent identical distributions; a higher moment or bounded support is not required. This is a functional central limit theorem, concerning the entire interpolated path rather than only its endpoint.

The random-walk form of the Skorokhod embedding theorem says the following. If $S_n=Y_1+\cdots +Y_n$, where the $Y_i$ are independent and identically distributed withthen on a space carrying a Brownian motion $B$ there are stopping timessuch that $(B_{T_n}{)}_{n\ge 0}$ has the same law as $(S_n{)}_{n\ge 0}$, and the increments $T_n-T_{n-1}$ are independent and identically distributed with mean ${\sigma }^2$.

To prove the one-step statement, first note that every centered distribution is a mixture of centered two-point distributions. Indeed, match the equal-mass size-biased measures $x\mathbb{P}(Y\in dx)$ on $(0,\infty )$ and $|x|\mathbb{P}(Y\in dx)$ on $(-\infty ,0)$. This produces a random pair $(L,R)$ of positive numbers such that, conditionally on $(L,R)$, $Y$ has values $-L,R$ with probabilitiesand $\mathbb{E}[LR]=\mathbb{E}{Y}^2={\sigma }^2$. Include the atom at zero by taking the stopping time zero.

Choose $(L,R)$ independently of $B_t$ and stop Brownian motion on first leaving $(-L,R)$. The Brownian exit from an interval formulas give the displayed two-point probabilities and conditional mean stopping time $LR$. Thus $B_T$ has the law of $Y$ and $\mathbb{E}T={\sigma }^2$.

Starting from $T_0=0$, repeat this construction after each $T_{n-1}$. The Strong Markov property makes the new Brownian increments independent copies of the first embedding, proving the Skorokhod embedding of a centered random walk.

Construct the times inductively. Suppose $(B_{T_0},\dots ,B_{T_n})$ has the law of $(S_0,\dots ,S_n)$. Conditional on the past, the martingale increment $S_{n+1}-S_n$ has mean zero and finite second moment. Apply the conditional form of the Skorokhod embedding theorem to this regular conditional law, using the fresh Brownian motion $B_{T_n+t}-B_{T_n}$ supplied by the Strong Markov property. This gives a stopping time increment ${\tau }_{n+1}$ and $T_{n+1}=T_n+{\tau }_{n+1}$ such that the next Brownian increment has the required conditional law. Induction provesfor every $k$.

The Skorokhod embedding theorem states that if $\mu$ is a centered probability law on $\mathbb{R}$ with finite second moment, there is a Brownian stopping time $T$ such that $B_T\sim \mu$, the stopped process $(B_{t\wedge T})$ is uniformly integrable, and $\mathbb{E}T=\int x^2\mu (dx)$.