Skorokhod's representation theorem

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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 of the theorem

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 \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

References

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