Skorokhod theorem

Skorokhod representation theorem

Suppose that $ \{ P _ {n} \} _ {n \geq 1 } $ is a sequence of probability measures on a complete and separable metric space $ ( S, {\mathcal S} ) $ that converges weakly (cf. Weak topology) to a probability measure $ P $( that is, $ {\lim\limits } _ {n} \int _ {S} f {dP _ {n} } = \int _ {S} f {dP } $ for any continuous and bounded function $ f $ on $ S $). Then there exists a probability space $ ( \Omega, {\mathcal F}, {\mathsf P} ) $ and $ S $- valued random elements $ \{ X _ {n} \} $, $ X $ with distributions $ \{ P _ {n} \} $ and $ P $, respectively, such that $ X _ {n} $ converges $ {\mathsf P} $- almost surely to $ X $( cf. Convergence, almost-certain).

If $ S = \mathbf R $, the proof of this result reduces to taking for $ \Omega $ the unit interval $ ( 0,1 ) $ with Lebesgue measure and letting $ X _ {n} ( y ) = \inf \{ z : {P _ {n} ( - \infty,z ] \geq y } \} $, and $ X ( y ) = \inf \{ z : {P ( - \infty,z ] \geq y } \} $, for $ y \in ( 0,1 ) $.

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 $ P $ 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