A Skorohod representation theorem without separability

作者：Berti Patrizia^*; Pratelli Luca; Rigo Pietro

来源：Electronic Communications in Probability, 2013, 18(0).

DOI：10.1214/ECP.v18-2793

• Convergence of probability measures
• Perfect probability measure
• Separable probability measure
• Skorohod representation theorem
• Uniform distance

Let (S, d) be a metric space, G a sigma-field on S and (mu(n) : n %26gt;= 0) a sequence of probabilities on G. Suppose G countably generated, the map (x, y) bar right arrow d (x, y) measurable with respect to G circle times G, and mu(n) perfect for n %26gt; 0. Say that (mu(n)) has a Skorohod representation if, on some probability space, there are random variables X-n such that %26lt;br%26gt;X-n similar to mu(n) for all n %26gt;= 0 and d(X-n, X-0) -%26gt;(P) 0. %26lt;br%26gt;It is shown that (mu(n)) has a Skorohod representation if and only if %26lt;br%26gt;(lim)(sup)(n)(f)vertical bar mu(n)(f)-mu(0)(f)vertical bar=0, %26lt;br%26gt;where sup is over those f : S -%26gt; [-1, 1] which are G-universally measurable and satisfy vertical bar f(x) - f(y)vertical bar %26lt;= 1 boolean AND d (x, y). An useful consequence is that Skorohod representations are preserved under mixtures. The result applies even if mu(0) fails to be d-separable. Some possible applications are given as well.

• 出版日期2013-10-18