Skorokhod's representation theorem
- Skorokhod's representation theorem
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 sequence of random variables defined on a common probability space. It is named for the Ukrainian mathematician A.V. Skorokhod.
tatement of the theorem
Let "μ""n", "n" ∈ N be a sequence of probability measures on a topological space "S"; suppose that "μ""n" converges weakly to some probability measure "μ" on "S" as "n" → ∞. Suppose also that the support of "μ" is separable. Then there exist random variables "X""n", "X" defined on a common probability space (Ω, "F", P) such that
* ("X""n")∗(P) = "μ""n" (i.e. "μ""n" is the distribution/law of "X""n");
* "X"∗(P) = "μ" (i.e. "μ" is the distribution/law of "X"); and
* "X""n"("ω") → "X"("ω") as "n" → ∞ for every "ω" ∈ Ω.
References
* (see theorem 29.6)
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