Skorokhod theorem

Skorokhod representation theorem

Suppose that is a sequence of probability measures on a complete and separable metric space that converges weakly (cf. Weak topology) to a probability measure (that is, for any continuous and bounded function on ). Then there exists a probability space and -valued random elements , with distributions and , respectively, such that converges -almost surely to (cf. Convergence, almost-certain).

If , the proof of this result reduces to taking for the unit interval with Lebesgue measure and letting , and , for .

In [a1] the theorem has been extended to general separable metric spaces, while in [a4] the result is proved for an arbitrary metric space, assuming that the limit probability is concentrated on a separable set. Extensions of this theorem to non-metrizable topological spaces are discussed in [a2].

References

[a1]	R.M. Dudley, "Distance of probability measures and random variables" Ann. Math. Stat. , 39 (1968) pp. 1563–1572
[a2]	A. Schief, "Almost surely convergent random variables with given laws" Probab. Th. Rel. Fields , 81 (1989) pp. 559–567
[a3]	A.V. Skorokhod, "Limit theorems for stochastic processes" Th. Probab. Appl. , 1 (1956) pp. 261–290
[a4]	M.J. Wichura, "On the construction of almost uniformly convergent random variables with given weakly convergent image laws" Ann. Math. Stat. , 41 (1970) pp. 284–291