In theory of probability, the Komlós–Major–Tusnády approximation (also known as the KMT approximation, the KMT embedding, or the Hungarian embedding) is an approximation of the empirical process by a Gaussian process constructed on the same probability space. It is named after Hungarian mathematicians János Komlós, Gábor Tusnády, and Péter Major.
Let
U
1
,
U
2
,
…
be independent uniform (0,1) random variables. Define a uniform empirical distribution function as
F
U
,
n
(
t
)
=
1
n
∑
i
=
1
n
1
U
i
≤
t
,
t
∈
[
0
,
1
]
.
Define a uniform empirical process as
α
U
,
n
(
t
)
=
n
(
F
U
,
n
(
t
)
−
t
)
,
t
∈
[
0
,
1
]
.
The Donsker theorem (1952) shows that
α
U
,
n
(
t
)
converges in law to a Brownian bridge
B
(
t
)
.
Komlós, Major and Tusnády established a sharp bound for the speed of this weak convergence.
Theorem (KMT, 1975) On a suitable probability space for independent uniform (0,1) r.v.
U
1
,
U
2
…
the empirical process
{
α
U
,
n
(
t
)
,
0
≤
t
≤
1
}
can be approximated by a sequence of Brownian bridges
{
B
n
(
t
)
,
0
≤
t
≤
1
}
such that
P
{
sup
0
≤
t
≤
1
|
α
U
,
n
(
t
)
−
B
n
(
t
)
|
>
1
n
(
a
log
n
+
x
)
}
≤
b
e
−
c
x
for all positive integers
n and all
x
>
0
, where
a,
b, and
c are positive constants.
A corollary of that theorem is that for any real iid r.v.
X
1
,
X
2
,
…
,
with cdf
F
(
t
)
,
it is possible to construct a probability space where independent sequences of empirical processes
α
X
,
n
(
t
)
=
n
(
F
X
,
n
(
t
)
−
F
(
t
)
)
and Gaussian processes
G
F
,
n
(
t
)
=
B
n
(
F
(
t
)
)
exist such that
lim sup
n
→
∞
n
ln
n
∥
α
X
,
n
−
G
F
,
n
∥
∞
<
∞
,
almost surely.
