aarjaneiro/planetmath huggingface dataset:
almost surely absolutely bounded random variable

almost surely absolutely bounded random variable

Let

{

Ω

,

E

,

P

}

a

probability space

and let

X

be a

random
variable

; then,

X

is said to be almost surely absolutely

bounded

iff a

M

>

0

exists such that

Pr

⁡

{

|

X

|

≤

M

}

=

1

.

Title

almost surely absolutely bounded random variable

Canonical name

AlmostSurelyAbsolutelyBoundedRandomVariable

Date of creation

2013-03-22 16:14:30

Last modified on

2013-03-22 16:14:30

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

7

Author

Andrea Ambrosio (7332)

Entry type

Definition

Classification

msc 60A10

Generated on Sat Feb 10 11:52:43 2018 by

LaTeXML
Borel measure

Borel measure

Definition 1 -

Let

X

be a

topological space

and

ℬ

be its Borel

σ

-algebra (

http://planetmath.org/BorelSigmaAlgebra

). A

Borel measure

on

X

is a

measure

on the

measurable space

(

X

,

ℬ

)

.

In the literature one can find other different definitions of Borel measure, like the following:

Definition 2 -

Let

X

be a topological space and

ℬ

be its Borel

σ

-algebra. A

Borel measure

on

X

is a measure

μ

on the measurable space

(

X

,

ℬ

)

such that

μ

⁢

(

K

)

<

∞

for all

compact subsets

K

⊂

X

. (ref.

[

1

]

).

Definition 3 -

Let

X

be a topological space and

ℬ

be the

σ

-algebra

generated by

all compact sets of

X

. A

Borel measure

on

X

is a measure

μ

on the measurable space

(

X

,

ℬ

)

such that

μ

⁢

(

K

)

<

∞

for all compact subsets

K

⊂

X

.

Definition 4 -

The restriction (

http://planetmath.org/RestrictionOfAFunction

) of the

Lebesgue measure

to the Borel

σ

-algebra of

ℝ

n

is also sometimes called “the” Borel measure of

ℝ

n

.

Remark -

Definitions

2

and

3

are technically different. For example, when constructing a

Haar measure

on a locally compact group one considers the

σ

-algebra generated by all compact subsets, instead of all closed (or open) sets.

References

1

M.R. Buneci. 2006.,

http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdf

Groupoid

C*-Algebras.,

Surveys in Mathematics and its Applications

, Volume 1: 71–98.

2

A. Connes.1979. Sur la théorie noncommutative de l’ integration,

Lecture Notes in
Math.

, Springer-Verlag, Berlin,

725

: 19-14.

Title

Borel measure

Canonical name

BorelMeasure

Date of creation

2013-03-22 17:34:00

Last modified on

2013-03-22 17:34:00

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

24

Author

asteroid (17536)

Entry type

Definition

Classification

msc 60A10

Classification

msc 28C15

Classification

msc 28A12

Classification

msc 28A10

Related topic

BorelSigmaAlgebra

Related topic

RadonMeasure

Related topic

BorelSpace

Related topic

Measure

Related topic

MeasurableSpace

Related topic

BorelGroupoid

Related topic

BorelMorphism

Related topic

BorelGSpace

Generated on Sat Feb 10 11:52:45 2018 by

LaTeXML
Borel morphism

Borel morphism

Definition 0.1

.

Let

𝔾

B

and

𝔾

B

* be two

groupoids

whose object spaces are Borel. An

algebraic morphism

from

𝔾

B

to

𝔾

B

* is defined as a left action of

𝔾

B

on

𝔾

B

* which commutes with the

multiplication

on

𝔾

B

. Such an algebraic morphism between

Borel groupoids

is said to be a

Borel morphism

if the action of

𝔾

B

on

𝔾

B

* is Borel (viz. ref.

[

1

]

)

References

1

M.R. Buneci. 2006.,

http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdf

Groupoid C*-

Algebras

.,

Surveys in Mathematics and its Applications

, Volume 1: 71–98.

Title

Borel morphism

Canonical name

BorelMorphism

Date of creation

2013-03-22 18:23:36

Last modified on

2013-03-22 18:23:36

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

12

Author

bci1 (20947)

Entry type

Definition

Classification

msc 60A10

Classification

msc 28A12

Classification

msc 28C15

Classification

msc 54H05

Classification

msc 28A05

Related topic

BorelSpace

Related topic

Groupoids

Related topic

CategoryOfBorelSpaces

Related topic

MeasurableFunctions

Related topic

BorelMeasure

Defines

algebraic morphism

Generated on Sat Feb 10 11:52:46 2018 by

LaTeXML
Borel space

Borel space

Definition 0.1

.

A

Borel space

(

X

;

ℬ

⁢

(

X

)

)

is defined as a set

X

, together with
a Borel

σ

-

algebra

(

http://planetmath.org/SigmaAlgebra

)

ℬ

⁢

(

X

)

of subsets of

X

, called

Borel sets

. The Borel algebra on

X

is the smallest

σ

-algebra containing all

open sets

(or, equivalently, all

closed sets

if the

topology

on closed sets is selected).

Remark 0.1

.

Borel sets were named after the French mathematician Emile Borel.

Remark 0.2

.

A

subspace

of a Borel space

(

X

;

ℬ

⁢

(

X

)

)

is a subset

S

⊂

X

endowed with the relative

Borel structure

, that is the

σ

-algebra of all subsets of

S

of the form

S

⁢

⋂

E

, where

E

is a Borel subset of

X

.

Definition 0.2

.

A

rigid Borel space

(

X

r

;

ℬ

⁢

(

X

r

)

)

is defined as a Borel space whose only

automorphism

f

:

X

r

→

X

r

(that is, with

f

being a

bijection

, and also with

f

⁢

(

A

)

=

f

-

1

⁢

(

A

)

for any

A

∈

ℬ

⁢

(

X

r

)

) is the

identity function

1

(

X

r

;

ℬ

⁢

(

X

r

)

)

(ref.

[

2

]

).

Remark 0.3

.

R. M. Shortt and J. Van Mill provided the first construction of a rigid Borel space on a ‘set of large

cardinality

’.

References

1

M.R. Buneci. 2006.,

http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdf

Groupoid

C*-Algebras.,

Surveys in Mathematics and its Applications

, Volume 1: 71–98.

2

B. Aniszczyk. 1991. A rigid Borel space.,

Proceed. AMS.

, 113 (4):1013-1015.,

http://www.jstor.org/pss/2048777

available online.

3

A. Connes.1979. Sur la théorie noncommutative de l’ integration,

Lecture Notes in
Math.

, Springer-Verlag, Berlin,

725

: 19-14.

Title

Borel space

Canonical name

BorelSpace

Date of creation

2013-03-22 18:23:02

Last modified on

2013-03-22 18:23:02

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

22

Author

bci1 (20947)

Entry type

Definition

Classification

msc 60A10

Classification

msc 28C15

Classification

msc 28A12

Classification

msc 54H05

Classification

msc 28A05

Synonym

measurable space

Related topic

BorelSet

Related topic

SigmaAlgebra

Related topic

MeasurableSpace

Related topic

BorelMeasure

Related topic

BorelGroupoid

Related topic

BorelMorphism

Defines

rigid Borel space

Defines

Borel subset space

Generated on Sat Feb 10 11:52:49 2018 by

LaTeXML
Cauchy random variable

Cauchy random variable

X

is a

Cauchy random variable

with

parameters

θ

∈

ℝ

and

β

>

0

∈

ℝ

, commonly denoted

X

∼

C

⁢

a

⁢

u

⁢

c

⁢

h

⁢

y

⁢

(

θ

,

β

)

if

f

X

⁢

(

x

)

=

1

π

⁢

β

⁢

[

1

+

(

x

-

θ

β

)

2

]

.

Cauchy random variables are used primarily for theoretical purposes, the key point being that the values

E

⁢

[

X

]

and

V

⁢

a

⁢

r

⁢

[

X

]

are undefined for Cauchy random variables.

Title

Cauchy random variable

Canonical

name

CauchyRandomVariable

Date of creation

2013-03-22 11:54:32

Last modified on

2013-03-22 11:54:32

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

11

Author

mathcam (2727)

Entry type

Definition

Classification

msc 60A10

Synonym

Cauchy distribution

Generated on Sat Feb 10 11:52:51 2018 by

LaTeXML
Dirac measure

Dirac measure

Let

X

be a

nonempty set

. Let

𝒫

⁢

(

X

)

denote the

power set

of

X

. Then

(

X

,

𝒫

⁢

(

X

)

)

is a

measurable space

.

Let

x

∈

X

. The

Dirac measure

concentrated at

x

is

δ

x

:

𝒫

⁢

(

X

)

→

{

0

,

1

}

defined by

δ

x

⁢

(

E

)

=

{

1

if

⁢

x

∈

E

0

if

⁢

x

∉

E

.

Note that the Dirac measure

δ

x

is indeed a

measure

:

1.

Since

x

∉

∅

, we have

δ

x

⁢

(

∅

)

=

0

.

2.

If

{

A

n

}

n

∈

ℕ

is a

sequence

of

pairwise disjoint

subsets of

X

, then one of the following must happen:

–

x

∉

⋃

n

∈

ℕ

A

n

, in which case

δ

x

⁢

(

⋃

n

∈

ℕ

A

n

)

=

0

and

δ

x

⁢

(

A

n

)

=

0

for every

n

∈

ℕ

;

–

x

∈

⋃

n

∈

ℕ

A

n

, in which case

x

∈

A

n

0

for exactly one

n

0

∈

ℕ

, causing

δ

x

⁢

(

⋃

n

∈

ℕ

A

n

)

=

1

,

δ

x

⁢

(

A

n

0

)

=

1

, and

δ

x

⁢

(

A

n

)

=

0

for every

n

∈

ℕ

with

n

≠

n

0

.

Also note that

(

X

,

𝒫

⁢

(

X

)

,

δ

x

)

is a

probability space

.

Let

ℝ

¯

denote the

extended real numbers

. Then for any

function

f

:

X

→

ℝ

¯

, the

integral

of

f

with respect to the Dirac measure

δ

x

is

∫

X

f

⁢

𝑑

δ

x

=

f

⁢

(

x

)

.

In other words, integration with respect to the Dirac measure

δ

x

amounts to evaluating the function at

x

.

If

X

=

ℝ

,

m

denotes

Lebesgue measure

,

A

is a Lebesgue measurable subset of

ℝ

, and

δ

(no ) denotes the Dirac delta function, then for any

measurable function

f

:

ℝ

→

ℝ

, we have

∫

A

δ

⁢

(

t

-

x

)

⁢

f

⁢

(

t

)

⁢

𝑑

m

⁢

(

t

)

=

∫

A

f

⁢

𝑑

δ

x

=

f

⁢

(

x

)

⁢

δ

x

⁢

(

A

)

.

Moreover, if

f

is defined so that

f

⁢

(

t

)

=

1

for all

t

∈

A

, the above becomes

∫

A

δ

⁢

(

t

-

x

)

⁢

𝑑

m

⁢

(

t

)

=

∫

A

𝑑

δ

x

=

δ

x

⁢

(

A

)

.

In other words, the function

δ

⁢

(

t

-

x

)

(with

x

∈

ℝ

fixed and

t

a real variable) behaves like a

Radon-Nikodym derivative

of

δ

x

with respect to

m

.

Note that, just as the Dirac delta function is a misnomer (it is not really a function), there is not really a Radon-Nikodym derivative of

δ

x

with respect to

m

, since

δ

x

is not

absolutely continuous

with respect to

m

.

Title

Dirac measure

Canonical

name

DiracMeasure

Date of creation

2013-03-22 17:19:40

Last modified on

2013-03-22 17:19:40

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

18

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 60A10

Classification

msc 26A42

Classification

msc 28A25

Classification

msc 28A12

Classification

msc 28A10

Related topic

Measure

Related topic

Integral2

Related topic

DiracDeltaFunction

Generated on Sat Feb 10 11:53:00 2018 by

LaTeXML
isoperimetric inequality

isoperimetric inequality

The classical

isoperimetric inequality

says that if a planar
figure has

perimeter

P

and area

A

, then

4

⁢

π

⁢

A

≤

P

2

,

where the equality holds if and only if the figure is a circle.
That is, the circle is the figure that encloses the largest area
among all figures of same perimeter.

The analogous statement is true in arbitrary dimension. The

d

-dimensional ball has the largest volume among all figures of
equal

surface area

.

The isoperimetric inequality can alternatively be stated using the

ϵ

-

neighborhoods

. An

ϵ

-neighborhood of a set

S

,
denoted here by

S

ϵ

, is the set of all points whose

distance

to

S

is at most

ϵ

. The isoperimetric
inequality in terms of

ϵ

-neighborhoods states that

vol

⁡

(

S

ϵ

)

≥

vol

⁡

(

B

ϵ

)

where

B

is the ball of
the same volume as

S

. The classical isoperimetric inequality can
be recovered by taking the limit

ϵ

→

0

.
The advantage of this formulation is that it
does not depend on the notion of surface area, and so can be
generalized to arbitrary

measure spaces

with a metric.

An example when this general formulation proves useful is the
Talagrand’s isoperimetric theory dealing with Hamming (

http://planetmath.org/HammingDistance

)-like
distances in product spaces. The theory has proven to be very
useful in many applications of

probability

to

combinatorics

.

References

1

Noga Alon and Joel H. Spencer.

The

probabilistic method

.

John Wiley & Sons, Inc., second edition, 2000.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0996.05001

Zbl 0996.05001.

2

Jiří Matoušek.

Lectures on Discrete Geometry

, volume 212 of

GTM

.

Springer, 2002.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0999.52006

Zbl 0999.52006.

Title

isoperimetric inequality

Canonical name

IsoperimetricInequality

Date of creation

2013-03-22 14:02:47

Last modified on

2013-03-22 14:02:47

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

12

Author

bbukh (348)

Entry type

Theorem

Classification

msc 60A10

Classification

msc 51M25

Classification

msc 51M16

Generated on Sat Feb 10 11:53:06 2018 by

LaTeXML
joint continuous density function

joint continuous density function

Let

X

1

,

X

2

,

…

,

X

n

be

n

random variables

all defined on the same

probability space

. The

joint continuous density function

of

X

1

,

X

2

,

…

,

X

n

, denoted by

f

X

1

,

X

2

,

…

,

X

n

⁢

(

x

1

,

x

2

,

…

,

x

n

)

, is the

function

f

X

1

,

X

2

,

…

,

X

n

:

ℝ

n

→

ℝ

such that for any domain

D

⊂

ℝ

n

, we have

∫

D

f

X

1

,

X

2

,

…

,

X

n

(

u

1

,

u

2

,

…

,

u

n

)

d

u

1

d

u

2

…

d

u

n

=

Prob

(

X

1

,

X

2

,

…

,

X

n

∈

D

)

As in the case where

n

=

1

, this function

satisfies

:

1.

f

X

1

,

X

2

,

…

,

X

n

⁢

(

x

1

,

…

,

x

n

)

≥

0

∀

(

x

1

,

…

,

x

n

)

2.

∫

x

1

,

…

,

x

n

f

X

1

,

X

2

,

…

,

X

n

⁢

(

u

1

,

u

2

,

…

,

u

n

)

⁢

𝑑

u

1

⁢

𝑑

u

2

⁢

…

⁢

𝑑

u

n

=

1

As in the single variable case,

f

X

1

,

X

2

,

…

,

X

n

does not represent the

probability

that each of the random variables takes on each of the values.

Title

joint continuous density function

Canonical name

JointContinuousDensityFunction

Date of creation

2013-03-22 11:54:58

Last modified on

2013-03-22 11:54:58

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

11

Author

mathcam (2727)

Entry type

Definition

Classification

msc 60A10

Synonym

joint mass function

Synonym

joint density function

Synonym

joint distribution

Generated on Sat Feb 10 11:53:08 2018 by

LaTeXML
joint cumulative distribution function

joint cumulative distribution function

Let

X

1

,

X

2

,

…

,

X

n

be

n

random variables

all defined on the same

probability space

. The

joint cumulative distribution function

of

X

1

,

X

2

,

…

,

X

n

, denoted by

F

X

1

,

X

2

,

…

,

X

n

⁢

(

x

1

,

x

2

,

…

,

x

n

)

, is the following function:

F

X

1

,

X

2

,

…

,

X

n

:

R

n

→

R

F

X

1

,

X

2

,

…

,

X

n

(

x

1

,

x

2

,

…

,

x

n

)

=

P

[

X

1

≤

x

1

,

X

2

≤

x

2

,

…

,

X

n

≤

x

n

]

As in the unidimensional case, this function satisfies:

1.

lim

(

x

1

,

…

,

x

n

)

→

(

-

∞

,

…

,

-

∞

)

⁡

F

X

1

,

X

2

,

…

,

X

n

⁢

(

x

1

,

…

,

x

n

)

=

0

and

lim

(

x

1

,

…

,

x

n

)

→

(

∞

,

…

,

∞

)

⁡

F

X

1

,

X

2

,

…

,

X

n

⁢

(

x

1

,

…

,

x

n

)

=

1

2.

F

X

1

,

X

2

,

…

,

X

n

⁢

(

x

1

,

…

,

x

n

)

is a

monotone

,

nondecreasing function

.

3.

F

X

1

,

X

2

,

…

,

X

n

⁢

(

x

1

,

…

,

x

n

)

is

continuous

from the right in each variable.

The way to evaluate

F

X

1

,

X

2

,

…

,

X

n

⁢

(

x

1

,

…

,

x

n

)

is the following:

F

X

1

,

X

2

,

…

,

X

n

⁢

(

x

1

,

…

,

x

n

)

=

∫

-

∞

x

1

∫

-

∞

x

2

⋯

⁢

∫

-

∞

x

n

f

X

1

,

X

2

,

…

,

X

n

⁢

(

u

1

,

…

,

u

n

)

⁢

𝑑

u

1

⁢

𝑑

u

2

⁢

⋯

⁢

𝑑

u

n

(if

F

is continuous) or

F

X

1

,

X

2

,

…

,

X

n

⁢

(

x

1

,

…

,

x

n

)

=

∑

i

1

≤

x

1

,

…

,

i

n

≤

x

n

f

X

1

,

X

2

,

…

,

X

n

⁢

(

i

1

,

…

,

i

n

)

(if

F

is discrete),

where

f

X

1

,

X

2

,

…

,

X

n

is the

joint density function

of

X

1

,

…

,

X

n

.

Title

joint cumulative distribution function

Canonical name

JointCumulativeDistributionFunction

Date of creation

2013-03-22 11:54:52

Last modified on

2013-03-22 11:54:52

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 60A10

Synonym

joint cumulative distribution

Generated on Sat Feb 10 11:53:11 2018 by

LaTeXML
measure

measure

Let

(

E

,

ℬ

⁢

(

E

)

)

be a

measurable space

. A

measure

on

(

E

,

ℬ

⁢

(

E

)

)

is a

function

μ

:

ℬ

⁢

(

E

)

→

ℝ

∪

{

∞

}

with values in the

extended real numbers

such that:

1.

μ

⁢

(

A

)

≥

0

for

A

∈

ℬ

⁢

(

E

)

, with equality if

A

=

∅

2.

μ

⁢

(

⋃

i

=

0

∞

A

i

)

=

∑

i

=

0

∞

μ

⁢

(

A

i

)

for any

sequence

of

pairwise disjoint

sets

A

i

∈

ℬ

⁢

(

E

)

.

Occasionally, the term

positive measure

is used to distinguish measures as defined here from more general notions of measure which are not necessarily restricted to the non-negative extended reals.

The second property above is called

countable additivity

, or

σ

-additivity. A

finitely additive measure

μ

has the same definition except that

ℬ

⁢

(

E

)

is only required to be an algebra and the second property above is only required to hold for finite unions. Note the slight abuse of terminology: a finitely additive measure is not necessarily a measure.

The triple

(

E

,

ℬ

⁢

(

E

)

,

μ

)

is called a

measure space

. If

μ

⁢

(

E

)

=

1

, then it is called a

probability space

, and the measure

μ

is called a

probability measure

.

Lebesgue measure

on

ℝ

n

is one important example of a measure.

Title

measure

Canonical name

Measure

Date of creation

2013-03-22 11:57:33

Last modified on

2013-03-22 11:57:33

Owner

djao (24)

Last modified by

djao (24)

Numerical id

19

Author

djao (24)

Entry type

Definition

Classification

msc 60A10

Classification

msc 28A10

Related topic

LpSpace

Related topic

SigmaFinite

Related topic

Integral2

Related topic

Distribution

Related topic

LebesgueMeasure

Defines

measure space

Defines

probability space

Defines

probability measure

Defines

countably additive

Defines

finitely additive

Defines

σ

-additive

Defines

positive measure

Generated on Sat Feb 10 11:53:13 2018 by

LaTeXML
outer measure

outer measure

Definition

[

1

,

2

,

3

]

Let

X

be a set, and let

𝒫

⁢

(

X

)

be the

power set

of

X

. An

outer measure

on

X

is a

function

μ

∗

:

𝒫

⁢

(

X

)

→

[

0

,

∞

]

satisfying the

properties

1.

μ

∗

⁢

(

∅

)

=

0

.

2.

If

A

⊂

B

are subsets in

X

, then

μ

∗

⁢

(

A

)

≤

μ

∗

⁢

(

B

)

.

3.

If

{

A

i

}

is a

countable

collection

of subsets of

X

,
then

μ

∗

⁢

(

⋃

i

A

i

)

≤

∑

i

μ

∗

⁢

(

A

i

)

.

Here, we can make two remarks. First, from (1) and (2), it follows
that

μ

∗

is a

positive

function on

𝒫

⁢

(

X

)

. Second,
property (3) also holds for any finite collection of subsets since
we can always append an

infinite

sequence

of

empty sets

to
such a collection.

References

1

A. Mukherjea, K. Pothoven,

Real and

Functional analysis

,
Plenum press, 1978.

2

A. Friedman,

Foundations

of Modern

Analysis

,
Dover publications, 1982.

3

G.B. Folland,

Real Analysis

: Modern Techniques and Their Applications

, 2nd ed, John Wiley & Sons, Inc., 1999.

Title

outer measure

Canonical

name

OuterMeasure

Date of creation

2013-03-22 13:45:20

Last modified on

2013-03-22 13:45:20

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Definition

Classification

msc 60A10

Classification

msc 28A10

Related topic

CaratheodorysExtensionTheorem

Related topic

CaratheodorysLemma

Related topic

ProofOfCaratheodorysExtensionTheorem

Generated on Sat Feb 10 11:53:18 2018 by

LaTeXML
relation between almost surely absolutely bounded random variables and their absolute moments

relation between almost surely absolutely bounded random variables and their absolute moments

Let

{

Ω

,

E

,

P

}

a

probability space

and let

X

be a

random
variable

; then, the following are

equivalent

:

1)

Pr

⁡

{

|

X

|

≤

M

}

=

1

i.e.

X

is
absolutely

bounded

almost surely;

2)

E

⁢

[

|

X

|

k

]

≤

M

k

∀

k

≥

1

,

k

∈

N

Proof.

1)

⟹

2)

Let’s define

F

=

{

ω

∈

Ω

:

|

X

⁢

(

ω

)

|

>

M

}

;

Then by

hypothesis

Pr

⁡

{

Ω

\

F

}

=

1

and

Pr

⁡

{

F

}

=

0

.

We have:

E

⁢

[

|

X

|

k

]

=

∫

Ω

|

X

|

k

⁢

𝑑

P

=

∫

Ω

\

F

|

X

|

k

⁢

𝑑

P

+

∫

F

|

X

|

k

⁢

𝑑

P

=

∫

Ω

\

F

|

X

|

k

⁢

𝑑

P

≤

∫

Ω

\

F

M

k

⁢

𝑑

P

=

M

k

⁢

Pr

⁡

{

Ω

\

F

}

=

M

k

.

2)

⟹

1)

Let’s define

F

=

{

ω

∈

Ω

:

|

X

⁢

(

ω

)

|

>

M

}

F

n

=

{

ω

∈

Ω

:

|

X

⁢

(

ω

)

|

>

M

+

1

n

}

⁢

⁢

∀

n

≥

1

.

Then we have obviously

F

n

⊆

F

n

+

1

(in fact, if

ω

∈

F

n

⟹

|

X

⁢

(

ω

)

|

>

M

+

1

n

>

M

+

1

n

+

1

⟹

ω

∈

F

n

+

1

) and

F

=

⋃

n

=

1

∞

F

n

(in fact, let

ω

∈

F

; let

N

=

⌈

1

|

X

⁢

(

ω

)

|

-

M

⌉

; then

|

X

⁢

(

ω

)

|

>

M

+

1

N

, that is

ω

∈

F

N

); this means that

F

=

lim

n

→

∞

⁡

F

n

in the meaning of sets

sequences

convergence (

http://planetmath.org/SequenceOfSetsConvergence

).

So the continuity from below property (

http://planetmath.org/PropertiesForMeasure

) of

probability

can be applied:

Pr

⁡

{

F

}

=

Pr

⁡

{

lim

n

→

∞

⁡

F

n

}

=

lim

n

→

∞

⁡

Pr

⁡

{

F

n

}

.

Now, for any

k

≥

1

,

M

k

≥

E

⁢

[

|

X

|

k

]

=

∫

Ω

|

X

⁢

(

ω

)

|

k

⁢

𝑑

P

=

∫

Ω

\

F

n

|

X

⁢

(

ω

)

|

k

⁢

𝑑

P

+

∫

F

n

|

X

⁢

(

ω

)

|

k

⁢

𝑑

P

≥

∫

F

n

|

X

⁢

(

ω

)

|

k

⁢

𝑑

P

≥

∫

F

n

(

M

+

1

n

)

k

⁢

𝑑

P

=

(

M

+

1

n

)

k

⁢

Pr

⁡

{

F

n

}

.

that is

Pr

⁡

{

F

n

}

≤

(

M

M

+

1

n

)

k

⁢

for any

⁢

k

≥

1

so that the only acceptable value for

Pr

⁡

{

F

n

}

is

Pr

⁡

{

F

n

}

=

0

whence the thesis.
∎

Acknowledgements: due to helpful discussions with Mathprof.

Title

relation between almost surely absolutely bounded random variables and their absolute moments

Canonical name

RelationBetweenAlmostSurelyAbsolutelyBoundedRandomVariablesAndTheirAbsoluteMoments

Date of creation

2013-03-22 16:14:33

Last modified on

2013-03-22 16:14:33

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

8

Author

Andrea Ambrosio (7332)

Entry type

Theorem

Classification

msc 60A10

Generated on Sat Feb 10 11:53:25 2018 by

LaTeXML
adapted process

adapted process

Let

{

X

t

∣

t

∈

T

}

be a

stochastic process

defined on a

probability space

(

Ω

,

ℱ

,

P

)

and

{

ℱ

t

∣

t

∈

T

}

a

filtration

(an increasing sequence of sigma

subalgebras

of

ℱ

), where

T

is a

linearly ordered

subset of

ℝ

with a minimum

t

0

. Then the process

{

X

t

}

is said to be

adapted

to

the filtration

{

ℱ

t

}

if for each

t

≥

t

0

,

X

t

is

ℱ

t

-measurable (

http://planetmath.org/MathcalFMeasurableFunction

):

X

t

-

1

⁢

(

B

)

∈

ℱ

t

⁢

for each Borel set

⁢

B

∈

ℝ

.

A stochastic process is an

adapted process

if it is adapted to some filtration.

Title

adapted process

Canonical name

AdaptedProcess

Date of creation

2013-03-22 16:16:43

Last modified on

2013-03-22 16:16:43

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

19

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 60A99

Classification

msc 60G07

Synonym

adapted

Generated on Sat Feb 10 11:53:27 2018 by

LaTeXML