View source for Lévy-Prokhorov metric

A metric in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l0583201.png" /> of finite Borel measures (cf. [[Borel measure|Borel measure]]) on a [[Metric space|metric space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l0583202.png" />, defined by

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l0583205.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l0583206.png" />-algebra of Borel sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l0583207.png" /> and

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The Lévy–Prokhorov metric was introduced by Yu.V. Prokhorov [[#References|[1]]] as a generalization of the [[Lévy metric|Lévy metric]]. The quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l0583209.png" /> changes if in its definition one omits one of the two inequalities and replaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832010.png" /> by the system of all open or closed sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832011.png" /> (see [[#References|[2]]]).

==Most important properties of the Lévy–Prokhorov metric.==


1) The metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832012.png" /> is separable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832013.png" /> is separable (cf. [[Separable space|Separable space]]).

2) The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832014.png" /> is complete if the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832015.png" /> is complete (cf. [[Complete space|Complete space]]). The converse is true if the measures of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832016.png" /> have separable supports.

3) In the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832017.png" /> of probability measures the Lévy–Prokhorov metric has properties analogous to those of the Lévy metric. Namely, the regularity property 3) (cf. [[Lévy metric|Lévy metric]]) and its corollaries, properties 4) and 5), property 6) (in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832018.png" />), property 7) in part (namely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832019.png" />), and also an analogue of property 8) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832020.png" /> is a linear normed space: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832023.png" />, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832024.png" />,

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4) In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832027.png" /> the Lévy–Prokhorov metric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832028.png" /> can be estimated by means of the characteristic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832030.png" /> corresponding to the measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832032.png" /> (see [[#References|[3]]], [[#References|[4]]]).

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

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that is, for any random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832035.png" /> with fixed marginal distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832036.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832037.png" /> over all joint distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832038.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.M. Dudley,   "Distances of probability measures and random variables"  ''Ann. Math. Stat.'' , '''39'''  (1968)  pp. 1563–1572</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V. Strassen,   "The existence of probability measures with given marginals"  ''Ann. Math. Stat.'' , '''36''' :  2  (1965)  pp. 423–439</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P. Billingsley,   "Convergence of probability measures" , Wiley  (1968)</TD></TR></table>



====Comments====
See also [[Distributions, convergence of|Distributions, convergence of]]; [[Convergence of measures|Convergence of measures]]; [[Weak convergence of probability measures|Weak convergence of probability measures]].