Difference between revisions of "Lévy-Prokhorov metric"
Ulf Rehmann (talk | contribs) m (moved Lévy–Prokhorov metric to Levy-Prokhorov metric: ascii title) |
Ulf Rehmann (talk | contribs) m (moved Levy-Prokhorov metric to Lévy-Prokhorov metric: accented title) |
(No difference)
|
Revision as of 07:54, 26 March 2012
A metric in the space of finite Borel measures (cf. Borel measure) on a metric space
, defined by
![]() |
![]() |
where is the
-algebra of Borel sets of
and
![]() |
The Lévy–Prokhorov metric was introduced by Yu.V. Prokhorov [1] as a generalization of the Lévy metric. The quantity changes if in its definition one omits one of the two inequalities and replaces
by the system of all open or closed sets of
(see [2]).
Most important properties of the Lévy–Prokhorov metric.
1) The metric space is separable if and only if
is separable (cf. Separable space).
2) The space is complete if the space
is complete (cf. Complete space). The converse is true if the measures of
have separable supports.
3) In the space 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
), property 7) in part (namely,
), and also an analogue of property 8) if
is a linear normed space: If
, where
,
, then for any
,
![]() |
![]() |
4) In the case the Lévy–Prokhorov metric in
can be estimated by means of the characteristic functions
and
corresponding to the measures
and
(see [3], [4]).
5) The Lévy–Prokhorov metric is a minimal metric with respect to the probability distance
![]() |
that is, for any random variables ,
with fixed marginal distributions
one has
over all joint distributions
.
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.
Lévy-Prokhorov metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L%C3%A9vy-Prokhorov_metric&oldid=23380