In mathematics, the Lévy–Prokhorov metric (sometimes known just as the Prokhorov metric) is a metric (i.e., a definition of distance) on the collection of probability measures on a given metric space. It is named after the French mathematician Paul Lévy and the Soviet mathematician Yuri Vasilyevich Prokhorov; Prokhorov introduced it in 1956 as a generalization of the earlier Lévy metric.
Let
(
M
,
d
)
be a metric space with its Borel sigma algebra
B
(
M
)
. Let
P
(
M
)
denote the collection of all probability measures on the measurable space
(
M
,
B
(
M
)
)
.
For a subset
A
⊆
M
, define the ε-neighborhood of
A
by
A
ε
:=
{
p
∈
M
|
∃
q
∈
A
,
d
(
p
,
q
)
<
ε
}
=
⋃
p
∈
A
B
ε
(
p
)
.
where
B
ε
(
p
)
is the open ball of radius
ε
centered at
p
.
The Lévy–Prokhorov metric
π
:
P
(
M
)
2
→
[
0
,
+
∞
)
is defined by setting the distance between two probability measures
μ
and
ν
to be
π
(
μ
,
ν
)
:=
inf
{
ε
>
0
|
μ
(
A
)
≤
ν
(
A
ε
)
+
ε
and
ν
(
A
)
≤
μ
(
A
ε
)
+
ε
for all
A
∈
B
(
M
)
}
.
For probability measures clearly
π
(
μ
,
ν
)
≤
1
.
Some authors omit one of the two inequalities or choose only open or closed
A
; either inequality implies the other, and
(
A
¯
)
ε
=
A
ε
, but restricting to open sets may change the metric so defined (if
M
is not Polish).
If
(
M
,
d
)
is separable, convergence of measures in the Lévy–Prokhorov metric is equivalent to weak convergence of measures. Thus,
π
is a metrization of the topology of weak convergence on
P
(
M
)
.
The metric space
(
P
(
M
)
,
π
)
is separable if and only if
(
M
,
d
)
is separable.
If
(
P
(
M
)
,
π
)
is complete then
(
M
,
d
)
is complete. If all the measures in
P
(
M
)
have separable support, then the converse implication also holds: if
(
M
,
d
)
is complete then
(
P
(
M
)
,
π
)
is complete.
If
(
M
,
d
)
is separable and complete, a subset
K
⊆
P
(
M
)
is relatively compact if and only if its
π
-closure is
π
-compact.
