# fixedpoint.jp

## 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.