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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Statement in probability theory
Donsker's invariance principle for simple random walk on
Z
{\displaystyle \mathbb {Z} }
.
In probability theory , Donsker's theorem (also known as Donsker's invariance principle , or the functional central limit theorem ), named after Monroe D. Donsker , is a functional extension of the central limit theorem for empirical distribution functions. Specifically, the theorem states that an appropriately centered and scaled version of the empirical distribution function converges to a Gaussian process .
Let
X
1
,
X
2
,
X
3
,
…
{\displaystyle X_{1},X_{2},X_{3},\ldots }
be a sequence of independent and identically distributed (i.i.d.) random variables with mean 0 and variance 1. Let
S
n
:=
∑
i
=
1
n
X
i
{\displaystyle S_{n}:=\sum _{i=1}^{n}X_{i}}
. The stochastic process
S
:=
(
S
n
)
n
∈
N
{\displaystyle S:=(S_{n})_{n\in \mathbb {N} }}
is known as a random walk . Define the diffusively rescaled random walk (partial-sum process) by
W
(
n
)
(
t
)
:=
S
⌊
n
t
⌋
n
,
t
∈
[
0
,
1
]
.
{\displaystyle W^{(n )}(t ):={\frac {S_{\lfloor nt\rfloor }}{\sqrt {n}}},\qquad t\in [0,1].}
The central limit theorem asserts that
W
(
n
)
(
1
)
{\displaystyle W^{(n )}(1 )}
converges in distribution to a standard Gaussian random variable
W
(
1
)
{\displaystyle W(1 )}
as
n
→
∞
{\displaystyle n\to \infty }
. Donsker's invariance principle[1] [2] extends this convergence to the whole function
W
(
n
)
:=
(
W
(
n
)
(
t
)
)
t
∈
[
0
,
1
]
{\displaystyle W^{(n )}:=(W^{(n )}(t ))_{t\in [0,1]}}
. More precisely, in its modern form, Donsker's invariance principle states that: As random variables taking values in the Skorokhod space
D
[
0
,
1
]
{\displaystyle {\mathcal {D}}[0,1]}
, the random function
W
(
n
)
{\displaystyle W^{(n )}}
converges in distribution to a standard Brownian motion
W
:=
(
W
(
t
)
)
t
∈
[
0
,
1
]
{\displaystyle W:=(W(t ))_{t\in [0,1]}}
as
n
→
∞
.
{\displaystyle n\to \infty .}
Donsker-Skorokhod-Kolmogorov theorem for uniform distributions.
Donsker-Skorokhod-Kolmogorov theorem for normal distributions
Formal statement [ edit ]
Let F n be the empirical distribution function of the sequence of i.i.d. random variables
X
1
,
X
2
,
X
3
,
…
{\displaystyle X_{1},X_{2},X_{3},\ldots }
with distribution function F. Define the centered and scaled version of F n by
G
n
(
x
)
=
n
(
F
n
(
x
)
−
F
(
x
)
)
{\displaystyle G_{n}(x )={\sqrt {n}}(F_{n}(x )-F(x ))}
indexed by x ∈ R . By the classical central limit theorem , for fixed x , the random variable G n (x ) converges in distribution to a Gaussian (normal) random variable G (x ) with zero mean and variance F (x )(1 − F (x )) as the sample size n grows.
Theorem (Donsker, Skorokhod, Kolmogorov) The sequence of G n (x ), as random elements of the Skorokhod space
D
(
−
∞
,
∞
)
{\displaystyle {\mathcal {D}}(-\infty ,\infty )}
, converges in distribution to a Gaussian process G with zero mean and covariance given by
cov
[
G
(
s
)
,
G
(
t
)
]
=
E
[
G
(
s
)
G
(
t
)
]
=
min
{
F
(
s
)
,
F
(
t
)
}
−
F
(
s
)
{\displaystyle \operatorname {cov} [G(s ),G(t )]=E[G(s )G(t )]=\min\{F(s ),F(t )\}-F(s )}
F
(
t
)
.
{\displaystyle {F}(t ).}
The process G (x ) can be written as B (F (x )) where B is a standard Brownian bridge on the unit interval.
History and related results [ edit ]
Kolmogorov (1933) showed that when F is continuous , the supremum
sup
t
G
n
(
t
)
{\displaystyle \scriptstyle \sup _{t}G_{n}(t )}
and supremum of absolute value,
sup
t
|
G
n
(
t
)
|
{\displaystyle \scriptstyle \sup _{t}|G_{n}(t )|}
converges in distribution to the laws of the same functionals of the Brownian bridge B (t ), see the Kolmogorov–Smirnov test . In 1949 Doob asked whether the convergence in distribution held for more general functionals, thus formulating a problem of weak convergence of random functions in a suitable function space .[3]
In 1952 Donsker stated and proved (not quite correctly)[4] a general extension for the Doob–Kolmogorov heuristic approach. In the original paper, Donsker proved that the convergence in law of G n to the Brownian bridge holds for Uniform[0,1] distributions with respect to uniform convergence in t over the interval [0,1].[2]
However Donsker's formulation was not quite correct because of the problem of measurability of the functionals of discontinuous processes. In 1956 Skorokhod and Kolmogorov defined a separable metric d , called the Skorokhod metric , on the space of càdlàg functions on [0,1], such that convergence for d to a continuous function is equivalent to convergence for the sup norm, and showed that G n converges in law in
D
[
0
,
1
]
{\displaystyle {\mathcal {D}}[0,1]}
to the Brownian bridge.
Later Dudley reformulated Donsker's result to avoid the problem of measurability and the need of the Skorokhod metric. One can prove[4] that there exist X i , iid uniform in [0,1] and a sequence of sample-continuous Brownian bridges B n , such that
‖
G
n
−
B
n
‖
∞
{\displaystyle \|G_{n}-B_{n}\|_{\infty }}
is measurable and converges in probability to 0. An improved version of this result, providing more detail on the rate of convergence, is the Komlós–Major–Tusnády approximation .
See also [ edit ]
References [ edit ]
^ Doob, Joseph L. (1949). "Heuristic approach to the Kolmogorov–Smirnov theorems" . Annals of Mathematical Statistics . 20 (3 ): 393–403. doi :10.1214/aoms/1177729991 . MR 0030732 . Zbl 0035.08901 .
^ a b Dudley, R.M. (1999). Uniform Central Limit Theorems . Cambridge University Press. ISBN 978-0-521-46102-3 .
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Donsker%27s_theorem&oldid=1222813291 "
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● T h i s p a g e w a s l a s t e d i t e d o n 8 M a y 2 0 2 4 , a t 0 2 : 0 6 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
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