Lévy-Prokhorov metric

A metric in the space $ \mathfrak M $ of finite Borel measures (cf. Borel measure) on a metric space $ ( U , d ) $, defined by

$$ \pi ( P , Q ) = $$

$$ = \ \inf \{ \epsilon : {P ( A) \leq Q ( A ^ \epsilon ) + \epsilon , Q ( A) \leq P ( A ^ \epsilon ) + \epsilon \textrm{ for all } A \subset \mathfrak B } \} , $$

where $ \mathfrak B $ is the $ \sigma $- algebra of Borel sets of $ ( U , d ) $ and

$$ A ^ \epsilon = \{ {x } : {d ( x , y ) < \epsilon , y \in A } \} . $$

The Lévy–Prokhorov metric was introduced by Yu.V. Prokhorov [1] as a generalization of the Lévy metric. The quantity $ \pi $ changes if in its definition one omits one of the two inequalities and replaces $ \mathfrak B $ by the system of all open or closed sets of $ \mathfrak B $( see [2]).

Most important properties of the Lévy–Prokhorov metric.

1) The metric space $ ( \mathfrak M , \pi ) $ is separable if and only if $ ( U , d ) $ is separable (cf. Separable space).

2) The space $ ( U , d ) $ is complete if the space $ ( \mathfrak M , \pi ) $ is complete (cf. Complete space). The converse is true if the measures of $ \mathfrak M $ have separable supports.

3) In the space $ \mathfrak M $ of probability measures the Lévy–Prokhorov metric has properties analogous to those of the Lévy metric. Namely, the regularity property 3) (cf. Lévy metric) and its corollaries, properties 4) and 5), property 6) (in the case $ U = \mathbf R ^ {1} $), property 7) in part (namely, $ \pi \leq \mathop{\rm var} $), and also an analogue of property 8) if $ ( U , d ) $ is a linear normed space: If $ P _ {a , \sigma } ( A) = P ( \sigma A + a ) $, where $ \sigma > 0 $, $ a \in U $, then for any $ P , Q \in \mathfrak M $,

$$ \pi ( \sigma P , \sigma Q ) \leq \sigma \pi ( P _ {a , \sigma } ,\ Q _ {a , \sigma } ) , $$

$$ \lim\limits _ {\sigma \rightarrow 0 } \pi ( P _ {a , \sigma } , Q _ {a , \sigma } ) = \mathop{\rm var} ( P , Q ) . $$

4) In the case $ U = \mathbf R ^ {k} $ the Lévy–Prokhorov metric in $ \mathfrak M $ can be estimated by means of the characteristic functions $ f $ and $ g $ corresponding to the measures $ P $ and $ Q $( see [3], [4]).

5) The Lévy–Prokhorov metric is a minimal metric with respect to the probability distance

$$ \kappa ( X , Y ) = \inf \{ \epsilon : {P \{ d ( X , Y ) > \epsilon \} < \epsilon } \} , $$

that is, for any random variables $ X $, $ Y $ with fixed marginal distributions $ P _ {X} , P _ {Y} \in \mathfrak M $ one has $ \pi ( P _ {X} , P _ {Y} ) = \inf \kappa ( X , Y ) $ over all joint distributions $ P _ {XY} $.

References

[1]	Yu.V. Prokhorov, "Convergence of random processes and limit theorems in probability theory" Theory Probab. Appl. , 1 (1956) pp. 157–214 Teor. Veroyatnost. i Primenen. , 1 : 2 (1956) pp. 177–238
[2]	R.M. Dudley, "Distances of probability measures and random variables" Ann. Math. Stat. , 39 (1968) pp. 1563–1572
[3]	V.V. Yurinskii, "A smoothing inequality for estimates of the Lévy–Prokhorov distance" Theory Probab. Appl. , 20 (1975) pp. 1–10 Teor. Veroyatnost. i Primenen. , 20 : 1 (1975) pp. 3–12
[4]	V.A. Abramov, "Estimates for the Lévy–Prokhorov distance" Theory Probab. Appl. , 21 (1976) pp. 396–400 Teor. Veroyatnost. i Primenen. , 21 : 2 (1976) pp. 406–410
[5]	V. Strassen, "The existence of probability measures with given marginals" Ann. Math. Stat. , 36 : 2 (1965) pp. 423–439
[6]	P. Billingsley, "Convergence of probability measures" , Wiley (1968)

Comments

See also Distributions, convergence of; Convergence of measures; Weak convergence of probability measures.