Lévy–Prokhorov metric

Lévy–Prokhorov metric is a measure of the difference between two probability measures on a given metric space. It is named after the French mathematician Paul Lévy and the Russian mathematician Yuri V. Prokhorov, who independently contributed to its development. The Lévy–Prokhorov metric is an important concept in probability theory and has applications in various fields such as statistics, mathematical analysis, and quantitative finance.

Definition

Given a metric space \((X, d)\), and two probability measures \(\mu\) and \(\nu\) on \(X\), the Lévy–Prokhorov metric \(d_{LP}(\mu, \nu)\) is defined as the infimum of all \(\varepsilon > 0\) such that for every Borel set \(A \subseteq X\),

\[ \mu(A) \leq \nu(A^\varepsilon) + \varepsilon \quad \text{and} \quad \nu(A) \leq \mu(A^\varepsilon) + \varepsilon, \]

where \(A^\varepsilon = \{x \in X : \exists y \in A \text{ such that } d(x, y) < \varepsilon\}\) denotes the \(\varepsilon\)-expansion of \(A\).

Properties

The Lévy–Prokhorov metric has several important properties that make it useful in the study of probability measures:

• It is a metric, meaning it satisfies the properties of non-negativity, identity of indiscernibles, symmetry, and the triangle inequality.
• It metrizes the weak convergence of probability measures. That is, a sequence of probability measures \(\{\mu_n\}\) converges weakly to a probability measure \(\mu\) if and only if \(d_{LP}(\mu_n, \mu) \to 0\) as \(n \to \infty\).
• It is invariant under isometries of the underlying metric space \(X\), making it a useful tool for comparing probability measures on different spaces.

Applications

The Lévy–Prokhorov metric finds applications in various areas of mathematics and applied sciences:

• In probability theory, it is used to study the convergence of random variables and the stability of stochastic processes.
• In statistics, it provides a basis for comparing empirical distributions and for goodness-of-fit tests.
• In quantitative finance, it is used to assess the risk and performance of financial models by comparing the predicted and actual distributions of returns.

See also

• Prokhorov's theorem
• Weak convergence (probability theory)
• Total variation distance of probability measures
• Kolmogorov–Smirnov test