The Skorokhod embedding of a centered random walk states that a random walk with independent identically distributed centered steps of finite variance

σ^{2}

can be realized on an appropriate probability space as

S_{n} = B_{T_{n}}, 0 = T_{0} \leq T_{1} \leq T_{2} \leq \dots,

(1)

where

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 pairs

(T_{n} - T_{n - 1}, B_{T_{n}} - B_{T_{n - 1}}), n \geq 1,

(2)

may be chosen independent and identically distributed. Their spatial component has the step law, and

E (T_{n} - T_{n - 1}) = σ^{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 walk

W_{n} (t) = \frac{S _{⌊ n t ⌋} + ( n t - ⌊ n t ⌋) X _{⌊ n t ⌋ + 1}}{n}, 0 \leq t \leq 1, S_{0} = 0,

(3)

converges 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.