Skorohod representation theorem via disintegrations
Tóm tắt
Let (µ
n
: n ≥ 0) be Borel probabilities on a metric space S such that µ
n
→ µ0 weakly. Say that Skorohod representation holds if, on some probability space, there are S-valued random variables X
n
satisfying X
n
∼ µ
n
for all n and X
n
→ X
0 in probability. By Skorohod’s theorem, Skorohod representation holds (with X
n
→ X
0 almost uniformly) if µ0 is separable. Two results are proved in this paper. First, Skorohod representation may fail if µ0 is not separable (provided, of course, non separable probabilities exist). Second, independently of µ0 separable or not, Skorohod representation holds if W(µ
n
, µ0) → 0 where W is Wasserstein distance (suitably adapted). The converse is essentially true as well. Such a W is a version of Wasserstein distance which can be defined for any metric space S satisfying a mild condition. To prove the quoted results (and to define W), disintegrable probability measures are fundamental.
Tài liệu tham khảo
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