aarjaneiro/planetmath huggingface dataset:
Zipf’s law

Zipf’s law

Zipf’s law (named for Harvard linguistic professor George Kingsley Zipf) models the occurrence of distinct objects in particular sorts of collections. Zipf’s law says that the

i

th most frequent object will appear

1

/

i

θ

times the frequency of the most frequent object, or that the

i

th most frequent object from an object “vocabulary” of size

V

occurs

O

⁢

(

i

)

=

n

i

θ

⁢

H

θ

⁢

(

V

)

times in a collection of

n

objects, where

H

θ

⁢

(

V

)

is the harmonic number of order

θ

of

V

.

(generated by GNU Octave and gnuplot)

Figure 1:

A typical Zipf-law rank

distribution

. The y-axis represents occurrence frequency, and the x-axis represents rank (highest at the left)

Zipf’s law typically holds when the “objects” themselves have a

property

(such as length or size) which is modelled by an

exponential distribution

or other skewed distribution that places

restrictions

on how often “larger” objects can occur.

An example of where Zipf’s law applies is in English texts, to frequency of word occurrence. The commonality of English words follows an exponential distribution, and the nature of communication is such that it is more efficient to place emphasis on using shorter words. Hence the most common words tend to be short and appear often, following Zipf’s law.

The value of

θ

typically ranges between 1 and 2, and is between 1.5 and 2 for the English text case.

Another example is the

populations

of cities. These follow Zipf’s law, with a few very populous cities, falling off to very numerous cities with a small population. In this case, there are societal forces which supply the same type of “restrictions” that limited which length of English words are used most often.

A final example is the income of companies. Once again the ranked incomes follow Zipf’s law, with competition pressures limiting the range of incomes available to most companies and determining the few most successful ones.

The underlying theme is that efficiency, competition, or attention with regards to resources or

information

tends to result in Zipf’s law holding to the ranking of objects or datum of concern.

0.1

References

•

References on Zipf’s law -

http://linkage.rockefeller.edu/wli/zipf/

http://linkage.rockefeller.edu/wli/zipf/

•

Baeza-Yates and Ribeiro-Neto,

Modern Information Retrieval

, ACM Press, 1999.

Title

Zipf’s law

Canonical

name

ZipfsLaw

Date of creation

2013-03-22 13:01:31

Last modified on

2013-03-22 13:01:31

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

7

Author

akrowne (2)

Entry type

Definition

Classification

msc 60E05

Classification

msc 68P20

Classification

msc 94A99

Generated on Sat Feb 10 11:55:38 2018 by

LaTeXML
infinitely divisible random variable

infinitely divisible random variable

Let

n

be a positive integer. A real

random variable

X

defined on a

probability space

(

Ω

,

ℱ

,

P

)

is said to be

1.

n

-

decomposable

if there exist

n

independent

random variables

X

1

,

…

,

X

n

such that

X

is

identically distributed

as the sum

X

1

+

⋯

+

X

n

. A

2

-

decomposable random variable

is also called a

decomposable random variable

;

2.

n

-divisible

if

X

is

n

-decomposable and the

X

i

’s can be chosen so that they are identically distributed;

3.

infinitely divisible

if

X

is

n

-divisible for every positive integer

n

. In other words,

X

can be written as the sum of

n

iid random variables for any

n

.

A

distribution function

is said to be

infinitely divisible

if it is the distribution function of an infinitely divisible random variable.

Remark

. Any

stable random variable

is infinitely divisible.

Some examples of infinitely divisible distribution functions, besides those that are stable, are the

gamma distributions

,

negative binomial distributions

, and compound Poisson distributions.

Title

infinitely divisible random variable

Canonical

name

InfinitelyDivisibleRandomVariable

Date of creation

2013-03-22 16:25:58

Last modified on

2013-03-22 16:25:58

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60E07

Defines

n

-decomposable

Defines

n

-divisible

Defines

infinitely divisible distribution

Defines

infinitely divisible

Defines

decomposable random variable

Generated on Sat Feb 10 11:55:47 2018 by

LaTeXML
stable random variable

stable random variable

A real

random variable

X

defined on a

probability space

(

Ω

,

ℱ

,

P

)

is said to be

stable

if

1.

X

is not

trivial

; that is, the range of the

distribution function

of

X

strictly

includes

{

0

,

1

}

, and

2.

given any positive integer

n

and

X

1

,

…

,

X

n

random variables, iid as

X

:

S

n

:=

X

1

+

⋯

+

X

n

=

t

X

.

In other words, there are real

constants

a

,

b

such that

S

n

and

a

⁢

X

+

b

have the same distribution functions;

S

n

and

X

are of the same type.

Furthermore,

X

is

strictly stable

if

X

is stable and the

b

given above can always be take as

0

. In other words,

X

is strictly stable if

S

n

and

X

belong to the same

scale family

.

A distribution function is said to be

stable

(

strictly stable

) if it is the distribution function of a stable (strictly stable) random variable.

Remarks

.

•

If

X

is stable, then

a

⁢

X

+

b

is stable for any

a

,

b

∈

ℝ

.

•

If

X

and

Y

are

independent

, stable, and of the same type, then

X

+

Y

is stable.

•

X

is stable iff for any independent

X

1

,

X

2

,

identically distributed

as

X

, and any

a

,

b

∈

ℝ

, there exist

c

,

d

∈

ℝ

such that

a

⁢

X

1

+

b

⁢

X

2

and

c

⁢

X

+

d

are identically distributed.

•

A

stable distribution function

is

absolutely continuous

(

http://planetmath.org/AbsolutelyContinuousFunction2

) and

infinitely divisible

.

Some common stable distribution functions are the

normal distributions

and

Cauchy distributions

.

Title

stable random variable

Canonical

name

StableRandomVariable

Date of creation

2013-03-22 16:25:56

Last modified on

2013-03-22 16:25:56

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 60E07

Defines

stable distribution function

Defines

strictly stable random variable

Defines

strictly stable distribution function

Generated on Sat Feb 10 11:55:56 2018 by

LaTeXML
Paul Lévy continuity theorem

Paul Lévy continuity theorem

Let

F

1

,

F

2

,

…

be

distribution functions

with

characteristic functions

φ

1

,

φ

2

,

…

, respectively. If

φ

n

converges

pointwise

to a limit

φ

, and if

φ

⁢

(

t

)

is

continuous at

t

=

0

, then
there exists a distribution function

F

such that

F

n

→

F

weakly (

http://planetmath.org/ConvergenceInDistribution

), and the characteristic function associated to

F

is

φ

.

Remark.

The reciprocal of this

theorem

is a corollary to the

Helly-Bray theorem

; hence

F

n

→

F

weakly if and only if

φ

n

→

φ

pointwise; but this theorem says something stronger than the

sufficiency

of that : it says that the limit of a

sequence

of characteristic functions is a characteristic function whenever it is continuous at 0.

Title

Paul Lévy continuity theorem

Canonical name

PaulLevyContinuityTheorem

Date of creation

2013-03-22 13:14:31

Last modified on

2013-03-22 13:14:31

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Theorem

Classification

msc 60E10

Generated on Sat Feb 10 11:56:00 2018 by

LaTeXML
a simple method for comparing real functions

a simple method for comparing real functions

Theorem

:

Let

f

⁢

(

x

)

and

g

⁢

(

x

)

be real-valued, twice

differentiable
functions

on

[

a

,

b

]

, and let

x

0

∈

[

a

,

b

]

.

If

f

⁢

(

x

0

)

=

g

⁢

(

x

0

)

,

f

′

⁢

(

x

0

)

=

g

′

⁢

(

x

0

)

,

f

′′

⁢

(

x

)

≤

g

′′

⁢

(

x

)

for all

x

in

[

a

,

b

]

, then

f

⁢

(

x

)

≤

g

⁢

(

x

)

for all

x

in

[

a

,

b

]

.

Proof.

Let

h

⁢

(

x

)

=

g

⁢

(

x

)

-

f

⁢

(

x

)

; by our hypotheses,

h

⁢

(

x

)

is a twice differentiable
function on

[

a

,

b

]

, and by the Taylor formula with Lagrange form

remainder

(

http://planetmath.org/RemainderVariousFormulas

) one has for any

x

∈

[

a

,

b

]

:

h

⁢

(

x

)

=

h

⁢

(

x

0

)

+

h

′

⁢

(

x

0

)

⁢

(

x

-

x

0

)

+

1

2

⁢

h

′′

⁢

(

ξ

)

⁢

(

x

-

x

0

)

2

where

ξ

=

ξ

⁢

(

x

)

∈

[

x

,

x

0

]

.

Then by hypotheses,

h

⁢

(

x

0

)

=

g

⁢

(

x

0

)

-

f

⁢

(

x

0

)

=

0

h

′

⁢

(

x

0

)

=

g

′

⁢

(

x

0

)

-

f

′

⁢

(

x

0

)

=

0

h

′′

⁢

(

ξ

)

=

g

′′

⁢

(

ξ

)

-

f

′′

⁢

(

ξ

)

≥

0

so that

h

⁢

(

x

)

=

1

2

⁢

h

′′

⁢

(

ξ

)

⁢

(

x

-

x

0

)

2

≥

0

whence the thesis.
∎

Title

a

simple method for comparing real functions

Canonical

name

ASimpleMethodForComparingRealFunctions

Date of creation

2013-03-22 16:10:47

Last modified on

2013-03-22 16:10:47

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

10

Author

Andrea Ambrosio (7332)

Entry type

Result

Classification

msc 60E15

Generated on Sat Feb 10 11:56:03 2018 by

LaTeXML
absolute moments bounding (necessary and sufficient condition)

absolute moments bounding (necessary and sufficient condition)

Let

X

be a

random variable

; then

E

⁢

[

|

X

|

k

]

≤

M

k

⁢

⁢

∀

k

≥

1

,

k

∈

𝐍

if and only if,

∀

i

≥

0

,

i

∈

𝐍

E

⁢

[

|

X

|

k

]

≤

E

⁢

[

|

X

|

i

]

⁢

M

k

-

i

⁢

⁢

∀

k

≥

i

,

k

∈

𝐍

Proof.

a)

(

E

[

|

X

|

k

]

≤

E

[

|

X

|

i

]

M

k

-

i

⟹

E

[

|

X

|

k

]

≤

M

k

)

It’s enough to take

i

=

0

and the thesis follows easily.

b)

(

E

[

|

X

|

k

]

≤

M

k

⟹

E

[

|

X

|

k

]

≤

E

[

|

X

|

i

]

M

k

-

i

)

Let

1

≤

i

≤

k

(the case

i

=

0

is trivial). Then, using

Cauchy-Schwarz
inequality

N

times, one has:

E

⁢

[

|

X

|

k

]

=

E

⁢

[

|

X

|

i

2

⁢

|

X

|

k

-

i

2

]

≤

E

⁢

[

|

X

|

i

]

1

2

⁢

E

⁢

[

|

X

|

2

⁢

k

-

i

]

1

2

=

E

⁢

[

|

X

|

i

]

1

2

⁢

E

⁢

[

|

X

|

i

2

⁢

|

X

|

2

⁢

k

-

3

2

⁢

i

]

1

2

≤

E

⁢

[

|

X

|

i

]

(

1

2

+

1

4

)

⁢

E

⁢

[

|

X

|

4

⁢

k

-

3

⁢

i

]

1

4

≤

E

⁢

[

|

X

|

i

]

(

1

2

+

1

4

+

1

8

)

⁢

E

⁢

[

|

X

|

(

8

⁢

k

-

7

⁢

i

)

]

1

8

…

≤

E

⁢

[

|

X

|

i

]

(

∑

m

=

1

N

1

2

m

)

⁢

E

⁢

[

|

X

|

2

N

⁢

k

-

(

2

N

-

1

)

⁢

i

]

1

2

N

=

E

⁢

[

|

X

|

i

]

(

1

-

1

2

N

)

⁢

E

⁢

[

|

X

|

2

N

⁢

(

k

-

i

)

+

i

]

1

2

N

≤

E

⁢

[

|

X

|

i

]

(

1

-

1

2

N

)

⁢

M

(

k

-

i

)

+

i

2

N

,

and since this must hold for any

N

, we obtain

E

⁢

[

|

X

|

k

]

≤

E

⁢

[

|

X

|

i

]

⁢

M

k

-

i

∎

Title

absolute moments

bounding (necessary and sufficient condition)

Canonical name

AbsoluteMomentsBoundingnecessaryAndSufficientCondition

Date of creation

2013-03-22 16:13:58

Last modified on

2013-03-22 16:13:58

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

5

Author

Andrea Ambrosio (7332)

Entry type

Theorem

Classification

msc 60E15

Generated on Sat Feb 10 11:56:07 2018 by

LaTeXML
Bennett inequality

Bennett inequality

Theorem

:(

Bennett inequality

, 1962):

Let

{

X

i

}

i

=

1

n

be a

collection

of

independent

random
variables

satisfying the conditions:

a)

E

⁢

[

X

i

2

]

<

∞

∀

i

, so that one can write

∑

i

=

1

n

E

⁢

[

X

i

2

]

=

v

2

b)

Pr

⁡

{

|

X

i

|

≤

M

}

=

1

∀

i

.

Then, for any

ε

≥

0

,

Pr

⁡

{

∑

i

=

1

n

(

X

i

-

E

⁢

[

X

i

]

)

>

ε

}

≤

exp

⁡

[

-

v

2

M

2

⁢

θ

⁢

(

ε

⁢

M

v

2

)

]

≤

exp

⁡

[

-

ε

2

⁢

M

⁢

ln

⁡

(

1

+

ε

⁢

M

v

2

)

]

where

θ

⁢

(

x

)

=

(

1

+

x

)

⁢

ln

⁡

(

1

+

x

)

-

x

Remark:
Observing that

(

1

+

x

)

⁢

ln

⁡

(

1

+

x

)

-

x

≥

9

⁢

(

1

+

x

3

-

1

+

2

3

⁢

x

)

≥

3

⁢

x

2

2

⁢

(

x

+

3

)

∀

x

≥

0

, and plugging these expressions into the
bound, one obtains immediately the

Bernstein inequality

under the hypotheses of
boundness of random variables, as one might expect. However, Bernstein
inequalities, although weaker, hold under far more general hypotheses than
Bennett one.

Title

Bennett inequality

Canonical name

BennettInequality

Date of creation

2013-03-22 16:12:25

Last modified on

2013-03-22 16:12:25

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

10

Author

Andrea Ambrosio (7332)

Entry type

Theorem

Classification

msc 60E15

Generated on Sat Feb 10 11:56:09 2018 by

LaTeXML
Bernstein inequalities

Bernstein inequalities

1) Let

{

X

i

}

i

=

1

n

be a

collection

of

independent

random
variables

satisfying the conditions:

a)

E

⁢

[

X

i

2

]

<

∞

∀

i

, so that one can write

∑

i

=

1

n

E

⁢

[

X

i

2

]

=

v

2

b)

∃

c

∈

ℝ

:

∑

i

=

1

n

E

⁢

[

|

X

i

|

k

]

≤

1

2

⁢

k

!

⁢

v

2

⁢

c

k

-

2

for all

integers

k

≥

3

Then, for any

ε

≥

0

,

Pr

⁡

{

∑

i

=

1

n

(

X

i

-

E

⁢

[

X

i

]

)

>

ε

}

≤

exp

⁡

[

-

v

2

c

2

⁢

(

1

+

c

⁢

ε

v

2

-

1

+

2

⁢

c

⁢

ε

v

2

)

]

≤

exp

⁡

(

-

ε

2

2

⁢

(

v

2

+

c

⁢

ε

)

)

Pr

⁡

{

|

∑

i

=

1

n

(

X

i

-

E

⁢

[

X

i

]

)

|

>

ε

}

≤

2

⁢

exp

⁡

[

-

v

2

c

2

⁢

(

1

+

c

⁢

ε

v

2

-

1

+

2

⁢

c

⁢

ε

v

2

)

]

≤

2

⁢

exp

⁡

(

-

ε

2

2

⁢

(

v

2

+

c

⁢

ε

)

)

2) Let

{

X

i

}

i

=

1

n

be a collection of independent, almost
surely absolutely

bounded

(

http://planetmath.org/AlmostSurelyBoundedRandomVariable

) random variables, that is

Pr

⁡

{

|

X

i

|

≤

M

}

=

1

⁢

⁢

∀

i

.

Then, for any

ε

≥

0

,

Pr

⁡

{

∑

i

=

1

n

(

X

i

-

E

⁢

[

X

i

]

)

>

ε

}

≤

exp

⁡

[

-

9

⁢

v

2

M

2

⁢

(

1

+

M

⁢

ε

3

⁢

v

2

-

1

+

2

⁢

M

⁢

ε

3

⁢

v

2

)

]

≤

exp

⁡

(

-

ε

2

2

⁢

(

v

2

+

M

3

⁢

ε

)

)

Pr

⁡

{

|

∑

i

=

1

n

(

X

i

-

E

⁢

[

X

i

]

)

|

>

ε

}

≤

2

⁢

exp

⁡

[

-

9

⁢

v

2

M

2

⁢

(

1

+

M

⁢

ε

3

⁢

v

2

-

1

+

2

⁢

M

⁢

ε

3

⁢

v

2

)

]

≤

2

⁢

exp

⁡

(

-

ε

2

2

⁢

(

v

2

+

M

3

⁢

ε

)

)

Title

Bernstein inequalities

Canonical name

BernsteinInequalities

Date of creation

2013-03-22 16:09:08

Last modified on

2013-03-22 16:09:08

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

21

Author

Andrea Ambrosio (7332)

Entry type

Theorem

Classification

msc 60E15

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

LaTeXML
Chernoff-Cramer bound

Chernoff-Cramer bound

The Chernoff-Cramèr

inequality

is a very general and powerful way of bounding

random variables

. Compared with the famous Chebyshev bound, which implies

inverse

polynomial

decay inequalities, the Chernoff-Cramèr method yields exponential decay inequalities, at the cost of needing a few more
hypotheses on random variables’ .

Theorem: (Chernoff-Cramèr inequality)

Let

{

X

i

}

i

=

1

n

be a collection of

independent

random
variables such that

E

⁢

[

exp

⁡

(

t

⁢

X

i

)

]

<

+

∞

∀

i

in a right

neighborhood

of

t

=

0

, i.e. for any

t

∈

(

0

,

c

)

(Cramèr condition).

Then a zero-valued for

x

=

0

, positive,

strictly increasing

,

strictly convex function

Ψ

⁢

(

x

)

:

[

0

,

∞

)

↦

R

+

exists
such that:

Pr

⁡

{

∑

i

=

1

n

(

X

i

-

E

⁢

[

X

i

]

)

>

ε

}

≤

exp

⁡

(

-

Ψ

⁢

(

ε

)

)

⁢

∀

ε

≥

0

Namely, one has:

Ψ

⁢

(

x

)

=

sup

0

<

t

<

c

⁡

(

t

⁢

x

-

ψ

⁢

(

t

)

)

ψ

⁢

(

t

)

=

∑

i

=

1

n

ln

⁡

E

⁢

[

e

t

⁢

(

X

i

-

E

⁢

X

i

)

]

=

∑

i

=

1

n

(

ln

⁡

E

⁢

[

e

t

⁢

X

i

]

-

t

⁢

E

⁢

[

X

i

]

)

that is,

Ψ

⁢

(

x

)

is the Legendre of the

cumulant generating function

of the

∑

i

=

1

n

(

X

i

-

E

⁢

[

X

i

]

)

random variable.

Remarks:

1) Besides its importance for theoretical questions, the Chernoff-Cramér bound is also the starting point to derive many deviation
or concentration inequalities, among which Bernstein (

http://planetmath.org/BernsteinInequality

), Kolmogorov, Prohorov (

http://planetmath.org/ProhorovInequality

), Bennett (

http://planetmath.org/BennettInequality

),
Hoeffding and Chernoff ones are worth mentioning. All of these
inequalities are obtained imposing various further conditions on

X

i

random variables, which turn out to affect the general form of the cumulant generating function

ψ

⁢

(

t

)

.

2) Sometimes, instead of bounding the sum of n independent random variables, one needs to estimate they ” i.e. the quantity

1

n

⁢

∑

i

=

1

n

X

i

; in to reuse Chernoff-Cramér bound, it’s enough to note that

Pr

⁡

{

1

n

⁢

∑

i

=

1

n

(

X

i

-

E

⁢

[

X

i

]

)

>

ε

′

}

=

Pr

⁡

{

∑

i

=

1

n

(

X

i

-

E

⁢

[

X

i

]

)

>

n

⁢

ε

′

}

so that one has only to replace, in the above stated inequality,

ε

with

n

⁢

ε

′

.

3) It turns out that the

Chernoff-Cramer bound

is asymptotically sharp, as CramÃÂ¨r theorem shows.

Title

Chernoff-Cramer bound

Canonical name

ChernoffCramerBound

Date of creation

2013-03-22 16:09:02

Last modified on

2013-03-22 16:09:02

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

46

Author

Andrea Ambrosio (7332)

Entry type

Theorem

Classification

msc 60E15

Related topic

HoeffdingInequalityForBoundedIndependentRandomVariables

Generated on Sat Feb 10 11:56:14 2018 by

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Hoeffding inequality for bounded independent random variables

Hoeffding inequality for bounded independent random variables

Let

X

1

,

X

2

,

…

,

X

n

be

independent

random variables

, such that

Pr

⁡

(

a

k

≤

X

k

≤

b

k

)

=

1

for all

k

, where

a

k

and

b

k

are constant,

a

k

<

b

k

. Let

S

n

be the sum

X

1

+

…

+

X

n

. Then

Pr

⁡

(

S

n

-

E

⁢

[

S

n

]

>

ϵ

)

≤

exp

⁡

(

-

2

⁢

ϵ

2

∑

k

=

1

n

(

b

k

-

a

k

)

2

)

,

Pr

⁡

(

|

S

n

-

E

⁢

[

S

n

]

|

>

ϵ

)

≤

2

⁢

exp

⁡

(

-

2

⁢

ϵ

2

∑

k

=

1

n

(

b

k

-

a

k

)

2

)

.

References

1

W. Hoeffding, “

Probability inequalities

for sums of bounded random variables”,

J. Amer. Statist. Assoc.

, vol. 58, pp.13-30, 1963.

Title

Hoeffding inequality for bounded independent random variables

Canonical name

HoeffdingInequalityForBoundedIndependentRandomVariables

Date of creation

2013-03-22 17:46:02

Last modified on

2013-03-22 17:46:02

Owner

kshum (5987)

Last modified by

kshum (5987)

Numerical id

8

Author

kshum (5987)

Entry type

Theorem

Classification

msc 60E15

Related topic

ChernoffCramerBound

Defines

Hoeffding’s inequality

Generated on Sat Feb 10 11:56:16 2018 by

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Kolmogorov’s inequality

Kolmogorov’s inequality

Let

X

1

,

…

,

X

n

be

independent

random variables

in a

probability space

, such that

E

⁡

[

X

k

]

=

0

and

Var

⁡

[

X

k

]

<

∞

for

k

=

1

,

…

,

n

. Then, for each

λ

>

0

,

P

(

max

1

≤

k

≤

n

|

S

k

|

≥

λ

)

≤

1

λ

2

Var

[

S

n

]

=

1

λ

2

∑

k

=

1

n

Var

[

X

k

]

,

where

S

k

=

X

1

+

⋯

+

X

k

.

Title

Kolmogorov’s inequality

Canonical name

KolmogorovsInequality

Date of creation

2013-03-22 13:13:15

Last modified on

2013-03-22 13:13:15

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

9

Author

Koro (127)

Entry type

Theorem

Classification

msc 60E15

Related topic

ChebyshevsInequality2

Related topic

MarkovsInequality

Related topic

ChebyshevsInequality

Generated on Sat Feb 10 11:56:18 2018 by

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Prohorov inequality

Prohorov inequality

Theorem

(

Prohorov inequality

, 1959):

Let

{

X

i

}

i

=

1

n

be a

collection

of

independent

random
variables

satisfying the conditions:

a)

E

⁢

[

X

i

2

]

<

∞

∀

i

, so that one can write

∑

i

=

1

n

E

⁢

[

X

i

2

]

=

v

2

b)

Pr

⁡

{

|

X

i

|

≤

M

}

=

1

∀

i

.

Then, for any

ε

≥

0

,

Pr

⁡

{

∑

i

=

1

n

(

X

i

-

E

⁢

[

X

i

]

)

>

ε

}

≤

exp

⁡

[

-

ε

2

⁢

M

⁢

arsinh

⁡

(

ε

⁢

M

2

⁢

v

2

)

]

Pr

⁡

{

|

∑

i

=

1

n

(

X

i

-

E

⁢

[

X

i

]

)

|

>

ε

}

≤

2

⁢

exp

⁡

[

-

ε

2

⁢

M

⁢

arsinh

⁡

(

ε

⁢

M

2

⁢

v

2

)

]

(See here (

http://planetmath.org/AreaFunctions

) for the meaning of

arsinh

⁡

(

x

)

)

Title

Prohorov inequality

Canonical name

ProhorovInequality

Date of creation

2013-03-22 16:12:56

Last modified on

2013-03-22 16:12:56

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

17

Author

Andrea Ambrosio (7332)

Entry type

Theorem

Classification

msc 60E15

Synonym

Prokhorov inequality

Generated on Sat Feb 10 11:56:20 2018 by

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proof of Kolmogorov’s inequality

proof of Kolmogorov’s inequality

For

k

=

1

,

2

,

…

,

n

, let

A

k

be the event that

|

S

k

|

≥

λ

but

|

S

i

|

<

λ

for all

i

=

1

,

2

,

…

,

k

-

1

. Note that the events

A

1

,

A

2

,

…

,

A

n

are

disjoint

, and

⋃

k

=

1

n

A

k

=

{

max

1

≤

k

≤

n

|

S

k

|

≥

λ

}

.

Let

I

A

be the

indicator function

of event

A

. Since

A

1

,

A

2

,

…

,

A

n

are disjoint, we have

0

≤

∑

k

=

1

n

I

A

k

≤

1

.

Hence, we obtain

∑

k

=

1

n

Var

⁢

[

X

k

]

=

E

⁢

[

S

n

2

]

≥

∑

k

=

1

n

E

⁢

[

S

n

2

⁢

I

A

k

]

.

After replacing

S

n

2

by

S

k

2

+

2

⁢

S

k

⁢

(

S

n

-

S

k

)

+

(

S

n

-

S

k

)

2

, we get

∑

k

=

1

n

Var

⁢

[

X

k

]

≥

∑

k

=

1

n

E

⁢

[

(

S

k

2

+

2

⁢

S

k

⁢

(

S

n

-

S

k

)

+

(

S

n

-

S

k

)

2

)

⁢

I

A

k

]

≥

∑

k

=

1

n

E

⁢

[

(

S

k

2

+

2

⁢

S

k

⁢

(

S

n

-

S

k

)

)

⁢

I

A

k

]

=

∑

k

=

1

n

E

⁢

[

S

k

2

⁢

I

A

k

]

+

2

⁢

∑

k

=

1

n

E

⁢

[

S

n

-

S

k

]

⁢

E

⁢

[

S

k

⁢

I

A

k

]

=

∑

k

=

1

n

E

⁢

[

S

k

2

⁢

I

A

k

]

≥

λ

2

⁢

∑

k

=

1

n

E

⁢

[

I

A

k

]

=

λ

2

⁢

∑

k

=

1

n

Pr

⁡

(

A

k

)

=

λ

2

⁢

Pr

⁡

(

⋃

k

=

1

n

A

k

)

=

λ

2

⁢

Pr

⁡

(

max

1

≤

k

≤

n

⁡

|

S

k

|

≥

λ

)

,

where in the third line, we have used the

assumption

that

S

n

-

S

k

is

independent

of

S

k

⁢

I

A

k

.

Title

proof of Kolmogorov’s

inequality

Canonical name

ProofOfKolmogorovsInequality

Date of creation

2013-03-22 17:48:35

Last modified on

2013-03-22 17:48:35

Owner

kshum (5987)

Last modified by

kshum (5987)

Numerical id

4

Author

kshum (5987)

Entry type

Proof

Classification

msc 60E15

Generated on Sat Feb 10 11:56:35 2018 by

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discrete density function

discrete density function

Let

X

be a

discrete random variable

. The

function

f

X

:

ℝ

→

[

0

,

1

]

defined as

f

X

(

x

)

=

P

[

X

=

x

]

is called the

discrete probability function

of

X

. Sometimes the syntax

p

X

⁢

(

x

)

is used, to mark the

difference

between this function and the

continuous density function

.

If

X

has discrete density function

f

X

⁢

(

x

)

, it is said that the

random variable

X

has the

distribution

or

is distributed

f

X

⁢

(

x

)

, and this fact is denoted as

X

∼

f

X

⁢

(

x

)

.

Discrete density functions are required to

satisfy

the following

properties

:

•

f

X

⁢

(

x

)

≥

0

for all

x

•

∑

x

f

X

⁢

(

x

)

=

1

Title

discrete density function

Canonical

name

DiscreteDensityFunction

Date of creation

2013-03-22 11:53:14

Last modified on

2013-03-22 11:53:14

Owner

drini (3)

Last modified by

drini (3)

Numerical id

16

Author

drini (3)

Entry type

Algorithm

Classification

msc 60E99

Classification

msc 00-02

Synonym

discrete probability function

Related topic

Distribution

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