aarjaneiro/planetmath huggingface dataset:
probability distribution function

probability distribution function

1

Definition

Let

(

Ω

,

𝔅

,

μ

)

be a

measure space

. A

probability distribution function

on

Ω

is a

function

f

:

Ω

⟶

ℝ

such that:

1.

f

is

μ

-measurable

2.

f

is nonnegative

μ

-almost everywhere.

3.

f

satisfies

the

equation

∫

Ω

f

⁢

(

x

)

⁢

𝑑

μ

=

1

The main feature of a probability distribution function is that it induces a

probability measure

P

on the measure space

(

Ω

,

𝔅

)

, given by

P

⁢

(

A

)

:=

∫

A

f

⁢

(

x

)

⁢

𝑑

μ

=

∫

Ω

1

A

⁢

f

⁢

(

x

)

⁢

𝑑

μ

,

for all

A

∈

𝔅

. The

measure

P

is called the

associated probability measure

of

f

. Note that

P

and

μ

are different measures, though they both share the same underlying

measurable space

(

Ω

,

𝔅

)

.

2

Examples

2.1

Discrete case

Let

I

be a

countable set

, and impose the

counting measure

on

I

(

μ

⁢

(

A

)

:=

|

A

|

, the

cardinality

of

A

, for any subset

A

⊂

I

). A probability distribution function on

I

is then a nonnegative function

f

:

I

⟶

ℝ

satisfying the equation

∑

i

∈

I

f

⁢

(

i

)

=

1

.

One example is the

Poisson distribution

P

r

on

ℕ

(for any real number

r

), which is given by

P

r

⁢

(

i

)

:=

e

-

r

⁢

r

i

i

!

for any

i

∈

ℕ

.

Given any

probability space

(

Ω

,

𝔅

,

μ

)

and any

random variable

X

:

Ω

⟶

I

, we can form a

distribution function

on

I

by taking

f

(

i

)

:=

μ

(

{

X

=

i

}

)

. The resulting function is called the

distribution

of

X

on

I

.

2.2

Continuous case

Suppose

(

Ω

,

𝔅

,

μ

)

equals

(

ℝ

,

𝔅

λ

,

λ

)

, the real numbers equipped with

Lebesgue measure

. Then a probability distribution function

f

:

ℝ

⟶

ℝ

is simply a measurable, nonnegative almost everywhere function such that

∫

-

∞

∞

f

⁢

(

x

)

⁢

𝑑

x

=

1

.

The associated measure has Radon–Nikodym

derivative

(

http://planetmath.org/RadonNikodymTheorem

) with respect to

λ

equal to

f

:

d

⁢

P

d

⁢

λ

=

f

.

One defines the

cumulative distribution function

F

of

f

by the

formula

F

(

x

)

:=

P

(

{

X

≤

x

}

)

=

∫

-

∞

x

f

(

t

)

d

t

,

for all

x

∈

ℝ

. A well known example of a probability distribution function on

ℝ

is the

Gaussian distribution

, or

normal distribution

f

⁢

(

x

)

:=

1

σ

⁢

2

⁢

π

⁢

e

-

(

x

-

m

)

2

/

2

⁢

σ

2

.

Title

probability distribution function

Canonical name

ProbabilityDistributionFunction

Date of creation

2013-03-22 12:37:25

Last modified on

2013-03-22 12:37:25

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

11

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 60E99

Synonym

probability density function

Synonym

distribution

Related topic

Measure

Related topic

Stochastic

Related topic

DiscreteDensityFunction

Related topic

DistributionFunction

Related topic

DensityFunction

Related topic

AreaUnderGaussianCurve

Defines

cumulative distribution function

Generated on Sat Feb 10 11:56:44 2018 by

LaTeXML
symmetric random variable

symmetric random variable

Let

(

Ω

,

ℱ

,

P

)

be a

probability space

and

X

a real

random variable

defined on

Ω

.

X

is said to be

symmetric

if

-

X

has the same

distribution function

as

X

. A distribution function

F

:

ℝ

→

[

0

,

1

]

is said to be

symmetric

if it is the distribution function of a

symmetric random variable

.

Remark

. By definition, if a random variable

X

is symmetric, then

E

⁢

[

X

]

exists (

<

∞

). Furthermore,

E

⁢

[

X

]

=

E

⁢

[

-

X

]

=

-

E

⁢

[

X

]

, so that

E

⁢

[

X

]

=

0

. Furthermore, let

F

be the distribution function of

X

. If

F

is continuous at

x

∈

ℝ

, then

F

(

-

x

)

=

P

(

X

≤

-

x

)

=

P

(

-

X

≤

-

x

)

=

P

(

X

≥

x

)

=

1

-

P

(

X

≤

x

)

=

1

-

F

(

x

)

,

so that

F

⁢

(

x

)

+

F

⁢

(

-

x

)

=

1

. This also shows that if

X

has a

density function

f

⁢

(

x

)

, then

f

⁢

(

x

)

=

f

⁢

(

-

x

)

.

There are many examples of symmetric random variables, and the most common one being the

normal random variables

centered at

0

. For any random variable

X

, then the difference

Δ

⁢

X

of two

independent

random variables,

identically distributed

as

X

is symmetric.

Title

symmetric random variable

Canonical name

SymmetricRandomVariable

Date of creation

2013-03-22 16:25:45

Last modified on

2013-03-22 16:25:45

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60E99

Classification

msc 60A99

Defines

symmetric distribution function

Generated on Sat Feb 10 11:56:53 2018 by

LaTeXML
Lindeberg’s central limit theorem

Lindeberg’s central limit theorem

Theorem (Lindeberg’s

central limit theorem

)

Let

X

1

,

X

2

,

…

be

independent

random variables

with

distribution functions

F

1

,

F

2

,

…

, respectively, such that

E

⁢

X

n

=

μ

n

and

Var

⁡

X

n

=

σ

n

2

<

∞

, with at least one

σ

n

>

0

.
Let

S

n

=

X

1

+

⋯

+

X

n

⁢

and

⁢

s

n

=

Var

⁡

(

S

n

)

=

σ

1

2

+

⋯

+

σ

n

2

.

Then the normalized partial sums

S

n

-

E

⁢

S

n

s

n

converge
in

distribution

(

http://planetmath.org/ConvergenceInDistribution

) to a random variable with

normal distribution

N

⁢

(

0

,

1

)

(i.e. the

normal convergence

holds,) if the following

Lindeberg condition

is satisfied:

∀

ε

>

0

,

lim

n

→

∞

⁡

1

s

n

2

⁢

∑

k

=

1

n

∫

|

x

-

μ

k

|

>

ε

⁢

s

n

(

x

-

μ

k

)

2

⁢

𝑑

F

k

⁢

(

x

)

=

0

.

Corollary 1 (Lyapunov’s central limit theorem)

If the

Lyapunov condition

1

s

n

2

+

δ

⁢

∑

k

=

1

n

E

⁢

|

X

k

-

μ

k

|

2

+

δ

→

n

→

∞

0

is satisfied for some

δ

>

0

, the normal convergence holds.

Corollary 2

If

X

1

,

X

2

,

…

are

identically distributed

random variables,

E

⁢

X

n

=

μ

and

Var

⁡

S

n

=

σ

2

, with

0

<

σ

<

∞

, then the normal convergence holds; i.e.

S

n

-

n

⁢

μ

σ

⁢

n

converges in distribution (

http://planetmath.org/ConvergenceInDistribution

) to a random variable with distribution

N

⁢

(

0

,

1

)

.

Reciprocal (Feller)

The reciprocal of Lindeberg’s central limit theorem holds under the following additional assumption:

max

1

≤

k

≤

n

⁡

(

σ

k

2

s

n

2

)

→

n

→

∞

0

.

Historical remark

Title

Lindeberg’s central limit theorem

Canonical name

LindebergsCentralLimitTheorem

Date of creation

2013-03-22 13:14:25

Last modified on

2013-03-22 13:14:25

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

19

Author

Koro (127)

Entry type

Theorem

Classification

msc 60F05

Synonym

Lyapunov’s central limit theorem

Synonym

central limit theorem

Synonym

lyapunov condition

Synonym

lindeberg condition

Related topic

TightAndRelativelyCompactMeasures

Defines

normal convergence

Defines

liapunov’s central limit theorem

Defines

liapunov condition

Generated on Sat Feb 10 11:56:56 2018 by

LaTeXML
tight and relatively compact measures

tight and relatively compact measures

Tight and relatively compact measuresFernando Sanz

Definition 1

.

Let

ℳ

=

{

μ

i

,

i

∈

I

}

be a family of finite

measures

on the
Borel subsets of a metric space

Ω

. We say that

ℳ

is tight
iff for each

ϵ

>

0

there is a compact set

K

such that

μ

i

⁢

(

Ω

-

K

)

<

ϵ

for all

i

. We say that

ℳ

is
relatively compact iff each sequence in

ℳ

has a

subsequence

converging weakly to a finite measure on

ℬ

⁢

(

Ω

)

.

If

{

F

i

,

i

∈

I

}

is a family of

distribution functions

,
relative compactness or tightness of

{

F

i

}

refers to relative
compactness or tightness of the corresponding measures.

Theorem

.

Let

{

F

i

,

i

∈

I

}

be a family of distribution functions with

F

i

⁢

(

∞

)

-

F

i

⁢

(

-

∞

)

<

M

<

∞

for all

i

. The family is
tight iff it is relatively compact.

Proof.

Coming soon…(needs other

theorems

before)
∎

Title

tight and relatively compact measures

Canonical

name

TightAndRelativelyCompactMeasures

Date of creation

2013-03-22 17:19:37

Last modified on

2013-03-22 17:19:37

Owner

fernsanz (8869)

Last modified by

fernsanz (8869)

Numerical id

6

Author

fernsanz (8869)

Entry type

Definition

Classification

msc 60F05

Related topic

LindebergsCentralLimitTheorem

Generated on Sat Feb 10 11:56:57 2018 by

LaTeXML
Kolmogorov’s strong law of large numbers

Kolmogorov’s strong law of large numbers

Let

X

1

,

X

2

,

…

be a

sequence

of

independent

random variables

, with finite

expectations

. The

strong law of large numbers

holds if one of the following conditions is satisfied:

1.

The random variables are

identically distributed

;

2.

For each

n

, the

variance

of

X

n

is finite, and

∑

n

=

1

∞

Var

⁡

[

X

n

]

n

2

<

∞

.

Title

Kolmogorov’s strong law of large numbers

Canonical name

KolmogorovsStrongLawOfLargeNumbers

Date of creation

2013-03-22 13:13:12

Last modified on

2013-03-22 13:13:12

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Theorem

Classification

msc 60F15

Synonym

Kolmogorov’s criterion

Related topic

MartingaleProofOfKolmogorovsStrongLawForSquareIntegrableVariables

Related topic

ProofOfKolmogorovsStrongLawForIIDRandomVariables

Generated on Sat Feb 10 11:56:59 2018 by

LaTeXML
martingale proof of Kolmogorov’s strong law for square integrable variables

martingale proof of Kolmogorov’s strong law for square integrable variables

We apply the

martingale convergence theorem

to prove the following result.

Theorem

.

Let

X

1

,

X

2

,

…

be

independent

random variables

such that

∑

n

Var

⁡

[

X

n

]

/

n

2

<

∞

. Then, setting

S

n

=

1

n

⁢

∑

k

=

1

n

(

X

k

-

𝔼

⁢

[

X

k

]

)

we have

S

n

→

0

as

n

→

∞

, with

probability

one.

To prove this, we start by constructing a

martingale

,

M

n

=

∑

k

=

1

n

X

k

-

𝔼

⁢

[

X

k

]

k

.

If

ℱ

n

is the

σ

-algebra (

http://planetmath.org/SigmaAlgebra

) generated by

X

1

,

…

⁢

X

n

then

𝔼

[

M

n

+

1

∣

ℱ

n

]

=

M

n

+

𝔼

[

X

n

+

1

∣

ℱ

n

]

-

𝔼

[

X

n

+

1

]

n

+

1

=

M

n

.

Here, the independence of

X

n

+

1

and

ℱ

n

has been used to imply that

𝔼

[

X

n

+

1

∣

ℱ

n

]

=

𝔼

[

X

n

+

1

]

. So,

M

is a martingale with respect to the

filtration

(

ℱ

n

)

n

∈

ℕ

.

Also, by the independence of the

X

n

, the

variance

of

M

n

is

Var

⁡

[

M

n

]

=

∑

k

=

1

n

Var

⁡

[

X

k

/

k

]

≤

∑

k

=

1

∞

Var

⁡

[

X

k

]

k

2

<

∞

.

So, the

inequality

𝔼

⁢

[

|

M

n

|

]

≤

𝔼

⁢

[

M

n

2

]

=

Var

⁡

[

M

n

]

shows that

M

is an

L

1

-

bounded

martingale, and the martingale convergence theorem says that the limit

M

∞

=

lim

n

→

∞

⁡

M

n

exists and is finite, with probability one.

The strong law now follows from Kronecker’s lemma, which states that for

sequences

of

real numbers

x

1

,

x

2

,

…

and

0

<

b

1

,

b

2

,

…

such that

b

n

strictly increases to

infinity

and

∑

n

x

n

/

b

n

converges

to a finite limit, then

b

n

-

1

⁢

∑

k

=

1

n

x

k

tends to

0

as

n

→

∞

. In our case, we take

x

n

=

X

n

-

𝔼

⁢

[

X

n

]

and

b

n

=

n

to deduce that

n

-

1

⁢

∑

k

=

1

n

(

X

k

-

𝔼

⁢

[

X

k

]

)

converges to zero with probability one.

References

1

David Williams,

Probability with martingales

,
Cambridge Mathematical Textbooks, Cambridge University Press, 1991.

2

Olav Kallenberg,

Foundations

of modern probability

, Second edition. Probability and its Applications. Springer-Verlag, 2002.

Title

martingale proof of Kolmogorov’s strong law for

square integrable

variables

Canonical name

MartingaleProofOfKolmogorovsStrongLawForSquareIntegrableVariables

Date of creation

2013-03-22 18:33:51

Last modified on

2013-03-22 18:33:51

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

4

Author

gel (22282)

Entry type

Proof

Classification

msc 60F15

Classification

msc 60G42

Related topic

MartingaleConvergenceTheorem

Related topic

KolmogorovsStrongLawOfLargeNumbers

Related topic

StrongLawOfLargeNumbers

Related topic

ProofOfKolmogorovsStrongLawForIIDRandomVariables

Generated on Sat Feb 10 11:57:02 2018 by

LaTeXML
proof of martingale convergence theorem

proof of martingale convergence theorem

Let

(

X

n

)

n

∈

ℕ

be a

supermartingale

such that

𝔼

⁢

|

X

n

|

≤

M

, and let

a

<

b

. We define a

random variable

counting how many times the process crosses the stripe between

a

and

b

:

U

n

:=

max

⁡

{

0

,

r

∈

{

1

,

…

,

n

}

∣

∃

0

≤

s

1

<

t

1

<

s

2

<

t

2

<

…

<

s

r

<

t

r

≤

n

⁢

∀

i

∈

{

1

,

…

,

r

}

:

X

s

i

≤

a

∧

X

t

i

≥

b

}

.

Obviously

U

n

+

1

≥

U

n

therefore

U

∞

=

lim

n

→

∞

⁡

U

n

exists almost surely. Next we will construct a new process that mirrors the movement of

X

n

but only if the original process is underway of going from below

a

to over

b

, and is

constant

otherwise. To do this let

C

1

:=

χ

[

X

0

<

a

]

,

C

k

:=

χ

[

C

k

-

1

=

0

∧

X

k

-

1

<

a

]

+

χ

[

C

k

-

1

=

1

∧

X

k

-

1

≤

b

]

for

k

≥

2

, and define

Y

0

:=

0

,

Y

n

:=

∑

k

=

1

n

C

k

⁢

(

X

k

-

X

k

-

1

)

. Then

Y

n

is also a supermartingale, and the

inequality

Y

n

≥

(

b

-

a

)

⁢

U

n

-

|

X

n

-

a

|

holds, which gives

0

≥

𝔼

⁢

(

Y

n

)

≥

(

b

-

a

)

⁢

𝔼

⁢

(

U

n

)

-

𝔼

⁢

|

X

n

-

a

|

. After rearrangement we get

𝔼

⁢

(

U

n

)

≤

𝔼

⁢

|

X

n

-

a

|

b

-

a

≤

𝔼

⁢

|

X

n

|

+

|

a

|

b

-

a

≤

M

+

|

a

|

b

-

a

.

Therefore by the

monotone convergence theorem

𝔼

⁢

(

U

∞

)

=

lim

n

→

∞

⁡

𝔼

⁢

(

U

n

)

≤

M

+

|

a

|

b

-

a

<

∞

,

which means

ℙ

(

U

∞

=

∞

)

=

0

. Since

a

and

b

were arbitrary

X

=

lim

n

→

∞

⁡

X

n

exists almost surely. Now the

Fatou lemma

gives

𝔼

⁢

|

X

|

=

𝔼

⁢

(

lim

n

→

∞

⁡

|

X

n

|

)

≤

lim inf

n

→

∞

⁡

𝔼

⁢

|

X

n

|

≤

M

<

∞

.

Thus

X

n

is in fact

convergent

almost surely, and

X

∈

L

1

.

∎

Title

proof of martingale convergence theorem

Canonical name

ProofOfMartingaleConvergenceTheorem

Date of creation

2013-03-22 18:34:33

Last modified on

2013-03-22 18:34:33

Owner

scineram (4030)

Last modified by

scineram (4030)

Numerical id

4

Author

scineram (4030)

Entry type

Proof

Classification

msc 60F15

Classification

msc 60G44

Classification

msc 60G46

Classification

msc 60G42

Generated on Sat Feb 10 11:57:09 2018 by

LaTeXML
strong law of large numbers

strong law of large numbers

A

sequence

of

random variables

X

1

,

X

2

,

…

with finite

expectations

in a

probability space

is said to satisfiy the

strong law of large numbers

if

1

n

⁢

∑

k

=

1

n

(

X

k

-

E

⁡

[

X

k

]

)

→

a

.

s

.

0

,

where

a

.

s

.

stands for convergence almost surely.

When the random variables are

identically distributed

, with expectation

μ

,
the law becomes:

1

n

⁢

∑

k

=

1

n

X

k

→

a

.

s

.

μ

.

Kolmogorov’s strong law of large numbers

theorems

give conditions on the random variables under which the law is satisfied.

Title

strong law of large numbers

Canonical name

StrongLawOfLargeNumbers

Date of creation

2013-03-22 13:13:10

Last modified on

2013-03-22 13:13:10

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

11

Author

Koro (127)

Entry type

Definition

Classification

msc 60F15

Related topic

MartingaleProofOfKolmogorovsStrongLawForSquareIntegrableVariables

Generated on Sat Feb 10 11:57:11 2018 by

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filtered probability space

filtered probability space

A

filtered probability space

, or

stochastic basis

,

(

Ω

,

ℱ

,

(

ℱ

t

)

t

∈

T

,

ℙ

)

consists of a

probability space

(

Ω

,

ℱ

,

ℙ

)

and a

filtration

(

http://planetmath.org/FiltrationOfSigmaAlgebras

)

(

ℱ

t

)

t

∈

T

contained

in

ℱ

. Here,

T

is the time

index set

, and is an ordered set — usually a subset of the

real numbers

— such that

ℱ

s

⊆

ℱ

t

for all

s

<

t

in

T

.

Filtered probability spaces form the setting for defining and studying

stochastic processes

. A process

X

t

with time index

t

ranging over

T

is said to be

adapted

if

X

t

is an

ℱ

t

-measurable

random variable

for every

t

.

When the index set

T

is an

interval

(

http://planetmath.org/Interval

) of the real numbers (i.e., continuous-time), it is often convenient to impose further conditions. In this case, the filtered probability space is said to

satisfy

the

usual conditions

or

usual hypotheses

if the following conditions are met.

•

The probability space

(

Ω

,

ℱ

,

ℙ

)

is

complete

(

http://planetmath.org/CompleteMeasure

).

•

The

σ

-algebras

ℱ

t

contain all the sets in

ℱ

of zero

probability

.

•

The filtration

ℱ

t

is right-continuous. That is, for every non-maximal

t

∈

T

, the

σ

-algebra

ℱ

t

+

≡

⋂

s

>

t

ℱ

s

is equal to

ℱ

t

.

Given any filtered probability space, it can always be enlarged by passing to the

completion

of the probability space, adding zero probability sets to

ℱ

t

, and by replacing

ℱ

t

by

ℱ

t

+

. This will then satisfy the usual conditions.
In fact, for many types of processes defined on a complete probability space, their

natural filtration

will already be right-continuous and the usual conditions met.
However, the process of completing the probability space depends on the specific

probability measure

ℙ

and in many situations, such as the study of

Markov processes

, it is necessary to study many different

measures

on the same space. A much weaker condition which can be used is that the

σ

-algebras

ℱ

t

are

universally complete

, which is still strong enough to apply much of the ‘heavy machinery’ of stochastic processes, such as the Doob-Meyer decomposition,

section

theorems

, etc.

Title

filtered probability space

Canonical name

FilteredProbabilitySpace

Date of creation

2013-03-22 18:36:51

Last modified on

2013-03-22 18:36:51

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Definition

Classification

msc 60G05

Related topic

FiltrationOfSigmaAlgebras

Defines

stochastic basis

Defines

usual conditions

Defines

usual hypotheses

Generated on Sat Feb 10 11:57:13 2018 by

LaTeXML
filtration of σ-algebras

filtration of

σ

-algebras

For an ordered set

T

, a

filtration

of

σ

-algebras (

http://planetmath.org/SigmaAlgebra

)

(

ℱ

t

)

t

∈

T

is a

collection

of

σ

-algebras on an underlying set

Ω

, satisfying

ℱ

s

⊆

ℱ

t

for all

s

<

t

in

T

. Here,

t

is understood as the time variable, taking values in the

index set

T

, and

ℱ

t

represents the collection of all events observable up until time

t

. The index set is usually a subset of the

real numbers

, with common examples being

T

=

ℤ

+

for discrete-time and

T

=

ℝ

+

for continuous-time scenarios.
The collection

(

ℱ

t

)

t

∈

T

is a filtration on a

measurable space

(

Ω

,

ℱ

)

if

ℱ

t

⊆

ℱ

for every

t

. If, furthermore, there is a

probability measure

defined on the underlying measurable space then this gives a

filtered probability space

.
The alternative notation

(

ℱ

t

,

t

∈

T

)

is often used for the filtration or, when the index set

T

is clear from the

context

, simply

(

ℱ

t

)

or

𝐅

.

Filtrations are widely used for studying

stochastic processes

, where a process

X

t

with time ranging over the set

T

is said to be

adapted

to the filtration if

X

t

is an

ℱ

t

-

measurable

random variable

for each time

t

.

Conversely

, any stochastic process

(

X

t

)

t

∈

T

generates a filtration. Let

ℱ

t

be the smallest

σ

-algebra with respect to which

X

s

is measurable for all

s

≤

t

,

ℱ

t

=

σ

(

X

s

:

s

≤

t

)

.

This defines the smallest filtration to which

X

is adapted, known as the

natural filtration

of

X

.

Given a filtration, there are various limiting

σ

-algebras which can be defined. The values at plus and minus

infinity

are

ℱ

∞

=

σ

⁢

(

⋃

t

ℱ

t

)

,

ℱ

-

∞

=

⋂

t

ℱ

t

,

which

satisfy

ℱ

-

∞

⊆

ℱ

t

⊆

ℱ

∞

.
In continuous-time, when the index set is an

interval

of the real numbers, the left and right limits can be defined at any time. They are,

ℱ

t

+

=

⋂

s

>

t

ℱ

s

,

ℱ

t

-

=

σ

⁢

(

⋃

s

<

t

ℱ

s

)

,

except if

t

is the

maximum

of

T

it is often convenient to set

ℱ

t

+

=

ℱ

t

or, if

t

is the

minimum

,

ℱ

t

-

=

ℱ

t

. It is easily verified that

ℱ

s

⊆

ℱ

s

+

⊆

ℱ

t

-

⊆

ℱ

t

for all times

s

<

t

.
Furthermore,

(

ℱ

t

+

)

and

(

ℱ

t

-

)

are themselves filtrations.

A filtration is said to be right-continuous if

ℱ

t

=

ℱ

t

+

for every

t

so, in particular,

(

ℱ

t

+

)

is always the smallest right-continuous filtration larger than

(

ℱ

t

)

.

Title

filtration of

σ

-algebras

Canonical name

FiltrationOfsigmaalgebras

Date of creation

2013-03-22 18:37:13

Last modified on

2013-03-22 18:37:13

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Definition

Classification

msc 60G05

Synonym

filtration of sigma-algebras

Related topic

FilteredProbabilitySpace

Related topic

Filtration

Defines

natural filtration

Generated on Sat Feb 10 11:57:15 2018 by

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measurability of stochastic processes

measurability of stochastic processes

For a continuous-time stochastic process

adapted

to a given filtration (

http://planetmath.org/FiltrationOfSigmaAlgebras

)

(

ℱ

t

)

t

∈

ℝ

+

on a

measurable space

(

Ω

,

ℱ

)

, there are various conditions which can be placed either on its

sample paths

or on its measurability when considered as a function from

ℝ

+

×

Ω

to

ℝ

. The following theorem lists the dependencies between these properties.

Theorem

.

Let

(

X

t

)

t

∈

R

+

be a real valued

stochastic process

.
Then,

X

is

optional

if it is adapted and right-continuous, it is

predictable

if it is adapted and left-continuous. Furthermore, each of the following properties implies the next.

1.

X

is predictable.

2.

X

is optional.

3.

X

is

progressive

.

4.

X

is adapted and jointly measurable.

Title

measurability of stochastic processes

Canonical name

MeasurabilityOfStochasticProcesses

Date of creation

2013-03-22 18:37:29

Last modified on

2013-03-22 18:37:29

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Theorem

Classification

msc 60G05

Related topic

MeasurabilityOfStoppedProcesses

Generated on Sat Feb 10 11:57:20 2018 by

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measurability of stopped processes

measurability of stopped processes

Let

X

be a real-valued

stochastic process

and

τ

be a

stopping time

. If

X

satisfies any of the following properties then so does the

stopped process

X

τ

.

1.

X

is jointly measurable.

2.

X

is

progressively measurable

.

3.

X

is

optional

.

4.

X

is

predictable

.

In particular, if

X

is a right-continuous and

adapted process

then it is progressive (alternatively, it is optional). Then, the stopped process

X

τ

will also be progressive and is therefore right-continuous and adapted.

Also, for any progressive process

X

and bounded stopping time

τ

≤

t

, the above result shows that

X

τ

=

X

t

τ

will be

ℱ

t

-measurable.

Title

measurability of stopped processes

Canonical name

MeasurabilityOfStoppedProcesses

Date of creation

2013-03-22 18:39:00

Last modified on

2013-03-22 18:39:00

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

4

Author

gel (22282)

Entry type

Theorem

Classification

msc 60G05

Related topic

MeasurabilityOfStochasticProcesses

Generated on Sat Feb 10 11:57:22 2018 by

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