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( T o p )
1
D e f i n i t i o n
T o g g l e D e f i n i t i o n s u b s e c t i o n
1 . 1
A s a t r a n s i t i o n k e r n e l
1 . 2
A s a r a n d o m e l e m e n t
2
B a s i c r e l a t e d c o n c e p t s
T o g g l e B a s i c r e l a t e d c o n c e p t s s u b s e c t i o n
2 . 1
I n t e n s i t y m e a s u r e
2 . 2
S u p p o r t i n g m e a s u r e
2 . 3
L a p l a c e t r a n s f o r m
3
B a s i c p r o p e r t i e s
T o g g l e B a s i c p r o p e r t i e s s u b s e c t i o n
3 . 1
M e a s u r a b i l i t y o f i n t e g r a l s
3 . 2
U n i q u e n e s s
3 . 3
D e c o m p o s i t i o n
4
R a n d o m c o u n t i n g m e a s u r e
5
S e e a l s o
6
R e f e r e n c e s
T o g g l e t h e t a b l e o f c o n t e n t s
R a n d o m m e a s u r e
4 l a n g u a g e s
● D e u t s c h
● F r a n ç a i s
● N e d e r l a n d s
● Р у с с к и й
E d i t l i n k s
● A r t i c l e
● T a l k
E n g l i s h
● R e a d
● E d i t
● V i e w h i s t o r y
T o o l s
T o o l s
A c t i o n s
● R e a d
● E d i t
● V i e w h i s t o r y
G e n e r a l
● W h a t l i n k s h e r e
● R e l a t e d c h a n g e s
● U p l o a d f i l e
● S p e c i a l p a g e s
● P e r m a n e n t l i n k
● P a g e i n f o r m a t i o n
● C i t e t h i s p a g e
● G e t s h o r t e n e d U R L
● D o w n l o a d Q R c o d e
● W i k i d a t a i t e m
P r i n t / e x p o r t
● D o w n l o a d a s P D F
● P r i n t a b l e v e r s i o n
A p p e a r a n c e
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
Random measures can be defined as transition kernels or as random elements . Both definitions are equivalent. For the definitions, let
E
{\displaystyle E}
be a separable complete metric space and let
E
{\displaystyle {\mathcal {E}}}
be its Borel
σ
{\displaystyle \sigma }
-algebra . (The most common example of a separable complete metric space is
R
n
{\displaystyle \mathbb {R} ^{n}}
)
As a transition kernel [ edit ]
A random measure
ζ
{\displaystyle \zeta }
is a (a.s. ) locally finite transition kernel from an abstract probability space
(
Ω
,
A
,
P
)
{\displaystyle (\Omega ,{\mathcal {A}},P)}
to
(
E
,
E
)
{\displaystyle (E,{\mathcal {E}})}
.[3]
Being a transition kernel means that
For any fixed
B
∈
E
{\displaystyle B\in {\mathcal {\mathcal {E}}}}
, the mapping
ω
↦
ζ
(
ω
,
B
)
{\displaystyle \omega \mapsto \zeta (\omega ,B)}
is measurable from
(
Ω
,
A
)
{\displaystyle (\Omega ,{\mathcal {A}})}
to
(
E
,
E
)
{\displaystyle (E,{\mathcal {E}})}
For every fixed
ω
∈
Ω
{\displaystyle \omega \in \Omega }
, the mapping
B
↦
ζ
(
ω
,
B
)
(
B
∈
E
)
{\displaystyle B\mapsto \zeta (\omega ,B)\quad (B\in {\mathcal {E}})}
is a measure on
(
E
,
E
)
{\displaystyle (E,{\mathcal {E}})}
Being locally finite means that the measures
B
↦
ζ
(
ω
,
B
)
{\displaystyle B\mapsto \zeta (\omega ,B)}
satisfy
ζ
(
ω
,
B
~
)
<
∞
{\displaystyle \zeta (\omega ,{\tilde {B}})<\infty }
for all bounded measurable sets
B
~
∈
E
{\displaystyle {\tilde {B}}\in {\mathcal {E}}}
and for all
ω
∈
Ω
{\displaystyle \omega \in \Omega }
except some
P
{\displaystyle P}
-null set
In the context of stochastic processes there is the related concept of a stochastic kernel, probability kernel, Markov kernel .
As a random element [ edit ]
Define
M
~
:=
{
μ
∣
μ
is measure on
(
E
,
E
)
}
{\displaystyle {\tilde {\mathcal {M}}}:=\{\mu \mid \mu {\text{ is measure on }}(E,{\mathcal {E}})\}}
and the subset of locally finite measures by
M
:=
{
μ
∈
M
~
∣
μ
(
B
~
)
<
∞
for all bounded measurable
B
~
∈
E
}
{\displaystyle {\mathcal {M}}:=\{\mu \in {\tilde {\mathcal {M}}}\mid \mu ({\tilde {B}})<\infty {\text{ for all bounded measurable }}{\tilde {B}}\in {\mathcal {E}}\}}
For all bounded measurable
B
~
{\displaystyle {\tilde {B}}}
, define the mappings
I
B
~
:
μ
↦
μ
(
B
~
)
{\displaystyle I_{\tilde {B}}\colon \mu \mapsto \mu ({\tilde {B}})}
from
M
~
{\displaystyle {\tilde {\mathcal {M}}}}
to
R
{\displaystyle \mathbb {R} }
. Let
M
~
{\displaystyle {\tilde {\mathbb {M} }}}
be the
σ
{\displaystyle \sigma }
-algebra induced by the mappings
I
B
~
{\displaystyle I_{\tilde {B}}}
on
M
~
{\displaystyle {\tilde {\mathcal {M}}}}
and
M
{\displaystyle \mathbb {M} }
the
σ
{\displaystyle \sigma }
-algebra induced by the mappings
I
B
~
{\displaystyle I_{\tilde {B}}}
on
M
{\displaystyle {\mathcal {M}}}
. Note that
M
~
|
M
=
M
{\displaystyle {\tilde {\mathbb {M} }}|_{\mathcal {M}}=\mathbb {M} }
.
A random measure is a random element from
(
Ω
,
A
,
P
)
{\displaystyle (\Omega ,{\mathcal {A}},P)}
to
(
M
~
,
M
~
)
{\displaystyle ({\tilde {\mathcal {M}}},{\tilde {\mathbb {M} }})}
that almost surely takes values in
(
M
,
M
)
{\displaystyle ({\mathcal {M}},\mathbb {M} )}
[3] [4] [5]
Basic related concepts [ edit ]
Intensity measure [ edit ]
For a random measure
ζ
{\displaystyle \zeta }
, the measure
E
ζ
{\displaystyle \operatorname {E} \zeta }
satisfying
E
[
∫
f
(
x
)
ζ
(
d
x
)
]
=
∫
f
(
x
)
E
ζ
(
d
x
)
{\displaystyle \operatorname {E} \left[\int f(x )\;\zeta (\mathrm {d} x)\right]=\int f(x )\;\operatorname {E} \zeta (\mathrm {d} x)}
for every positive measurable function
f
{\displaystyle f}
is called the intensity measure of
ζ
{\displaystyle \zeta }
. The intensity measure exists for every random measure and is a s-finite measure .
Supporting measure [ edit ]
For a random measure
ζ
{\displaystyle \zeta }
, the measure
ν
{\displaystyle \nu }
satisfying
∫
f
(
x
)
ζ
(
d
x
)
=
0
a.s.
iff
∫
f
(
x
)
ν
(
d
x
)
=
0
{\displaystyle \int f(x )\;\zeta (\mathrm {d} x)=0{\text{ a.s. }}{\text{ iff }}\int f(x )\;\nu (\mathrm {d} x)=0}
for all positive measurable functions is called the supporting measure of
ζ
{\displaystyle \zeta }
. The supporting measure exists for all random measures and can be chosen to be finite.
Laplace transform [ edit ]
For a random measure
ζ
{\displaystyle \zeta }
, the Laplace transform is defined as
L
ζ
(
f
)
=
E
[
exp
(
−
∫
f
(
x
)
ζ
(
d
x
)
)
]
{\displaystyle {\mathcal {L}}_{\zeta }(f )=\operatorname {E} \left[\exp \left(-\int f(x )\;\zeta (\mathrm {d} x)\right)\right]}
for every positive measurable function
f
{\displaystyle f}
.
Basic properties [ edit ]
Measurability of integrals [ edit ]
For a random measure
ζ
{\displaystyle \zeta }
, the integrals
∫
f
(
x
)
ζ
(
d
x
)
{\displaystyle \int f(x )\zeta (\mathrm {d} x)}
and
ζ
(
A
)
:=
∫
1
A
(
x
)
ζ
(
d
x
)
{\displaystyle \zeta (A ):=\int \mathbf {1} _{A}(x )\zeta (\mathrm {d} x)}
for positive
E
{\displaystyle {\mathcal {E}}}
-measurable
f
{\displaystyle f}
are measurable, so they are random variables .
Uniqueness [ edit ]
The distribution of a random measure is uniquely determined by the distributions of
∫
f
(
x
)
ζ
(
d
x
)
{\displaystyle \int f(x )\zeta (\mathrm {d} x)}
for all continuous functions with compact support
f
{\displaystyle f}
on
E
{\displaystyle E}
. For a fixed semiring
I
⊂
E
{\displaystyle {\mathcal {I}}\subset {\mathcal {E}}}
that generates
E
{\displaystyle {\mathcal {E}}}
in the sense that
σ
(
I
)
=
E
{\displaystyle \sigma ({\mathcal {I}})={\mathcal {E}}}
, the distribution of a random measure is also uniquely determined by the integral over all positive simple
I
{\displaystyle {\mathcal {I}}}
-measurable functions
f
{\displaystyle f}
.[6]
Decomposition [ edit ]
A measure generally might be decomposed as:
μ
=
μ
d
+
μ
a
=
μ
d
+
∑
n
=
1
N
κ
n
δ
X
n
,
{\displaystyle \mu =\mu _{d}+\mu _{a}=\mu _{d}+\sum _{n=1}^{N}\kappa _{n}\delta _{X_{n}},}
Here
μ
d
{\displaystyle \mu _{d}}
is a diffuse measure without atoms, while
μ
a
{\displaystyle \mu _{a}}
is a purely atomic measure.
Random counting measure [ edit ]
A random measure of the form:
μ
=
∑
n
=
1
N
δ
X
n
,
{\displaystyle \mu =\sum _{n=1}^{N}\delta _{X_{n}},}
where
δ
{\displaystyle \delta }
is the Dirac measure , and
X
n
{\displaystyle X_{n}}
are random variables, is called a point process [1] [2] or random counting measure . This random measure describes the set of N particles, whose locations are given by the (generally vector valued) random variables
X
n
{\displaystyle X_{n}}
. The diffuse component
μ
d
{\displaystyle \mu _{d}}
is null for a counting measure.
In the formal notation of above a random counting measure is a map from a probability space to the measurable space (
N
X
{\displaystyle N_{X}}
,
B
(
N
X
)
{\displaystyle {\mathfrak {B}}(N_{X})}
) a measurable space . Here
N
X
{\displaystyle N_{X}}
is the space of all boundedly finite integer-valued measures
N
∈
M
X
{\displaystyle N\in M_{X}}
(called counting measures).
The definitions of expectation measure, Laplace functional, moment measures and stationarity for random measures follow those of point processes . Random measures are useful in the description and analysis of Monte Carlo methods , such as Monte Carlo numerical quadrature and particle filters .[7]
See also [ edit ]
References [ edit ]
^ a b Jan Grandell, Point processes and random measures, Advances in Applied Probability 9 (1977) 502-526. MR 0478331 JSTOR A nice and clear introduction.
^ a b Kallenberg, Olav (2017). Random Measures, Theory and Applications . Probability Theory and Stochastic Modelling. Vol. 77. Switzerland: Springer. p. 1. doi :10.1007/978-3-319-41598-7 . ISBN 978-3-319-41596-3 .
^ Klenke, Achim (2008). Probability Theory . Berlin: Springer. p. 526. doi :10.1007/978-1-84800-048-3 . ISBN 978-1-84800-047-6 .
^ Daley, D. J.; Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes . Probability and its Applications. doi :10.1007/b97277 . ISBN 0-387-95541-0 .
^ Kallenberg, Olav (2017). Random Measures, Theory and Applications . Probability Theory and Stochastic Modelling. Vol. 77. Switzerland: Springer. p. 52. doi :10.1007/978-3-319-41598-7 . ISBN 978-3-319-41596-3 .
^ "Crisan, D., Particle Filters: A Theoretical Perspective , in Sequential Monte Carlo in Practice, Doucet, A., de Freitas, N. and Gordon, N. (Eds), Springer, 2001, ISBN 0-387-95146-6
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Random_measure&oldid=1225711216 "
C a t e g o r i e s :
● M e a s u r e s ( m e a s u r e t h e o r y )
● S t o c h a s t i c p r o c e s s e s
● T h i s p a g e w a s l a s t e d i t e d o n 2 6 M a y 2 0 2 4 , a t 0 6 : 3 9 ( 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 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
● P r i v a c y p o l i c y
● A b o u t W i k i p e d i a
● D i s c l a i m e r s
● C o n t a c t W i k i p e d i a
● C o d e o f C o n d u c t
● D e v e l o p e r s
● S t a t i s t i c s
● C o o k i e s t a t e m e n t
● M o b i l e v i e w