ON THE STOCHASTIC CONVERGENCE OF REPRESENTATIONS BASED ON WASSERSTEIN METRICS

• 成果类型: Article
• 署名作者: TUERO, A
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989394
• 发表日期: 1993
• 页码: 72-85
• 关键词: random-variables; distance; LAWS

摘要

Suppose that P and P(n), n is-an-element-of N, are probabilities on a real, separable Hilbert space, V. It is known that if P satisfies some regularity conditions and X is such that P(X) = P, then there exist mappings H(n): V --> V, such that P(Hn(X)) = P(n) and the Wasserstein distance between P(n) and P coincides with (integral\\x - H(n)(x)\\2 dp)1/2, n is-an-element-of N. In this paper we prove that the weak convergence of (P(n)) to P is enough to ensure that {H(n)(X)} converges to X in measure, and that, if V = R(p), then the convergence is also a.e. This property seems to be characteristic of finite-dimensional spaces, because we include an example, with V infinite-dimensional and P Gaussian, where a.e. convergence does not hold.