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An upper bound of the Lévy–Prokhorov metric between probability distributions (2019-03-06)

The Lévy-Prokhorov metric is a metric on the set of probability distributions on the Borel \(\sigma\)-algebra \(\mathfrak{B}\) of metric space \((X, d)\). Call the Lévy-Prokhorov metric \(\pi\): \[\pi(P, Q) := \inf \{\varepsilon > 0 \mid P(A) \leq Q(A^\varepsilon) + \varepsilon \text{ and } Q(A) \leq P(A^\varepsilon) + \varepsilon \text{ for all } A \in \mathfrak{B} \}\] where \(A^\varepsilon := \{x \in X \mid d(a, x) < \varepsilon \text{ for some } a \in A \}\).

Also, there is another metric on the same set of probability measures on \(\mathfrak{B}\), called the total variation distance \(\delta\). It is defined by \[\delta(P, Q) := \sup_{A \in \mathfrak{B}} \lvert P(A) - Q(A) \rvert.\]

It is derived from the above definitions that the total variation distance between \(P\) and \(Q\) is an upper bound of the Lévy–Prokhorov metric between them, i.e., \[\pi(P, Q) \leq \delta(P, Q).\]Proof. If \(P(A) = Q(A)\) for each \(A \in \mathfrak{B}\), then \(\pi(P, Q) = \delta(P, Q) = 0\). Otherwise \(\delta(P, Q) > 0\), and for any \(A \in \mathfrak{B}\), \(P(A) \leq Q(A) + \delta(P, Q) \leq Q(A^{\delta(P, Q)}) + \delta(P, Q)\) and \(Q(A) \leq P(A) + \delta(P, Q) \leq P(A^{\delta(P, Q)}) + \delta(P, Q)\). QED.


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