Komlós–Major–Tusnády approximation
approximation of the empirical process by a Gaussian process
In probability theory, the Komlós–Major–Tusnády approximation (also known as the KMT approximation, the KMT embedding, or the Hungarian embedding) refers to one of the two strong embedding theorems: 1) approximation of random walk by a standard Brownian motion constructed on the same probability space, and 2) an approximation of the empirical process by a Brownian bridge constructed on the same probability space. It is named after Hungarian mathematicians János Komlós, Gábor Tusnády, and Péter Major, who proved it in 1975.
Theory
Let
U
1
,
U
2
,
…
{\displaystyle U_{1},U_{2},\ldots }
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
]
.
{\displaystyle F_{U,n}(t)={\frac {1}{n}}\sum _{i=1}^{n}\mathbf {1} _{U_{i}\leq t},\quad t\in [0,1].}
Define a uniform empirical process as
α
U
,
n
(
t
)
=
n
(
F
U
,
n
(
t
)
−
t
)
,
t
∈
[
0
,
1
]
.
{\displaystyle \alpha _{U,n}(t)={\sqrt {n}}(F_{U,n}(t)-t),\quad t\in [0,1].}
The Donsker theorem (1952) shows that
α
U
,
n
(
t
)
{\displaystyle \alpha _{U,n}(t)}
converges in law to a Brownian bridge
B
(
t
)
.
{\displaystyle 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
…
{\displaystyle U_{1},U_{2}\ldots }
the empirical process
{
α
U
,
n
(
t
)
,
0
≤
t
≤
1
}
{\displaystyle \{\alpha _{U,n}(t),0\leq t\leq 1\}}
can be approximated by a sequence of Brownian bridges
{
B
n
(
t
)
,
0
≤
t
≤
1
}
{\displaystyle \{B_{n}(t),0\leq t\leq 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
{\displaystyle P\left\{\sup _{0\leq t\leq 1}|\alpha _{U,n}(t)-B_{n}(t)|>{\frac {1}{\sqrt {n}}}(a\log n+x)\right\}\leq be^{-cx}}
for all positive integers n and all
x
>
0
{\displaystyle x>0}
, where a, b, and c are positive constants.
Corollary
A corollary of that theorem is that for any real iid r.v.
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