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 Soviet mathematician A. V. Skorokhod.
Statement
Let \( \mu _{n} \), \( n\in \mathbb {N} \) be a sequence of probability measures on a metric space S such that \( \mu _{n} \) converges weakly to some probability measure \( \mu _{\infty } \) on S as n → ∞ {\displaystyle n\to \infty } n\to \infty . Suppose also that the support of \( \mu _{\infty } \) is separable. Then there exist random variables \( X_{n} \) defined on a common probability space \( (\Omega ,{\mathcal {F}},\mathbf {P} ) \) such that the law of \( X_{n} \) is \( \mu _{n} \) for all n (including \( n=\infty \) ) and such that \( X_{n} \) converges to \( X_{\infty } \) , \( \mathbf {P} \) -almost surely.
See also
Convergence in distribution
References
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)
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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