In mathematics and statistics, Skorokhod's representation theorem is a result that shows that a weakly convergent sequence of probability measures whose limit measure is sufficiently well-behaved can be represented as the distribution/law of a pointwise convergent sequence of random variables defined on a common probability space. It is named for the Ukrainian mathematician A. V. Skorokhod.
Statement
Let ( μ n ) n ∈ N {\displaystyle (\mu _{n})_{n\in \mathbb {N} }} be a sequence of probability measures on a metric space S {\displaystyle S} such that μ n {\displaystyle \mu _{n}} converges weakly to some probability measure μ ∞ {\displaystyle \mu _{\infty }} on S {\displaystyle S} as n → ∞ {\displaystyle n\to \infty } . Suppose also that the support of μ ∞ {\displaystyle \mu _{\infty }} is separable. Then there exist S {\displaystyle S} -valued random variables X n {\displaystyle X_{n}} defined on a common probability space ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},\mathbf {P} )} such that the law of X n {\displaystyle X_{n}} is μ n {\displaystyle \mu _{n}} for all n {\displaystyle n} (including n = ∞ {\displaystyle n=\infty } ) and such that ( X n ) n ∈ N {\displaystyle (X_{n})_{n\in \mathbb {N} }} converges to X ∞ {\displaystyle X_{\infty }} , P {\displaystyle \mathbf {P} } -almost surely.
See also
- Billingsley, Patrick (1999). Convergence of Probability Measures. New York: John Wiley & Sons, Inc. ISBN 0-471-19745-9. (see p. 7 for weak convergence, p. 24 for convergence in distribution and p. 70 for Skorokhod's theorem)