Access options Buy single article. Prokhorov's theorem can be extended to consider complex measures or finite signed measures. In mathematics, the Lévy metric is a metric on the space of cumulative distribution functions of one-dimensional random variables.It is a special case of the Lévy–Prokhorov metric, and is named after the French mathematician Paul Lévy. Instant access to the full article PDF.

1 Most important properties of the Lévy metric.

Contents. It has the property that d LP (μ k, μ) → 0 if and only if μ k converges weakly to μ; see [2, Theorem 6.8].

Prokhorov, the general theory of stochastic processes can be regarded as the theory of probability measures in complete separable metric spaces. Then d LP is a metric on M + (S n − 1) called the Lévy–Prokhorov metric. It is named after the French mathematician Paul Lévy and the Soviet mathematician Yuri Vasilyevich Prokhorov; Prokhorov introduced it in 1956 as a generalization of the earlier Lévy metric. By Aleksandrovʼs uniqueness theorem [13, Theorem 3.3.1], a convex body is uniquely determined, up to translation, by its surface area measure.

The definition of the Lévy metric carries over to the set $ M $ of all non-decreasing functions on $ \mathbf R ^ {1} $( infinite values of the metric being allowed). Prokhorov, the general theory of stochastic processes can be regarded as the theory of probability measures in complete separable metric spaces. Theorem: Suppose that is a complete separable metric space and is a family of Borel complex measures on .The following statements are equivalent: is sequentially compact; that is, every sequence has a weakly convergent subsequence. According to the fundamental work of Yu.V. The Lévy metric can be regarded as a special case of the Lévy–Prokhorov metric. Abbreviations for metrics used in Figure 1. Abbreviation Metric D Discrepancy H Hellinger distance I Relative entropy (or Kullback-Leibler divergence) K Kolmogorov (or Uniform) metric L L´evy metric P Prokhorov metric S Separation distance TV Total variation distance W Wasserstein (or Kantorovich) metric χ2 χ2 distance Table 1.

Since stochastic processes depending upon a continuous parameter are basically probability measures on certain subspaces of the space of all functions of a real variable, a particularly important case of … This is a preview of subscription content, log in to check access. According to the fundamental work of Yu.V. 5 In mathematics, the Lévy–Prokhorov metric is a metric on the collection of probability measures on a given metric space. $\begingroup$ The Levy-Prokhorov metric is looks useful from an analytic point of view, but since I'm working with discrete sample distributions I don't think that will be helpful for me. where K ¯ is an explicitly calculated constant, and l π is the Lévy-Prokhorov metric (see Section 2 for definition). The choice of the Levy-Prokhorov metric as a “good” metric that justifies´ the definition of weak convergence above, is connected, first of all, with the fact that the analog of the Skorokhod representation theorem ([16], see also, e.g., [5, Sec.11.7]) holds in the case of merging measures. More precisely, the following is true.

The convergence in l π is equivalent to the weak convergence plus the convergence of first absolute moments (see [8]). An extension of the Levy-Prokhorov metric is considered, and topological conditions of convergence in the sense of this metric are examined. $\endgroup$ – Marc Kjerland Jul 25 '12 at 20:16.