The Relations between Several Kinds of Convergence of Random Variables

2017 3rd International Conference on Computer Science and Mechanical Automation (CSMA 2017) ISBN: 978-1-60595-506-3

The Relations between Several Kinds of Convergence of

Random Variables

Qi ZHAO

Department of Basic Courses, Shandong Yingcai University, Jinan, Shandong, 250104, P. R. China

Email:[email protected]

Keywords: Convergence almost surely, Convergence in probability, Convergence in distribution, Convergence in mean, Cauchy convergence.

Abstract. In this paper, we study the convergence of random variables. We outline the definition and fundamental properties of various kinds convergence and study their relations. We focus on proving several theorem, these theorems show the relations between the different types of convergence, and we give some counter example to show that some relations are not true. Convergence in ther−th_{mean is the simplest in form, its relationship with others is the weakest.} Convergence almost surely is the strongest, but it doesn’t imply convergence in ther−th_mean. Both convergence almost surely and convergence in ther−th _{mean imply convergence in} probability, convergence in probability implies convergence in distribution. Cauchy convergence can judge the convergence of sequences of random variables by using certain characteristics of sequences.

Introduction

The convergence of random variables is the important content in the theory of probability. It is the theoretical basis of limit theorems of probability theory. In fact, different definitions of the convergence of random variables will lead to different limit theorem. Different forms of convergence have the difference, also have a contact. Here is to discuss the relations between several kinds of convergence of random variables.

The Definitions and Properties of Some Convergence of Random Variables

Convergence Everywhere

Definition 2.1 The sequence of random variables{Xn}defined in the same probability space. For each sample point

ω

∈ Ω

(Ωdenotes the sample space), there exists a sequence{X_n}_{..If there exists}

lim

n

( )

( )

n

X

X

ω

ω

ω

→∞

=

∀ ∈Ω

_{Then the sequence of random variables{ }}

n

X converges surely are

very where.

We note here that convergence everywhere is too harsh, in many situation, the case is not satisfied

Convergence Almost Surely

Definition 2.2 A sequence of random variables {X_n}converges almost surely to a random variableX, if

{

lim _n( ) ( )

}

1

n

P X ω X ω

→∞

= =

which is denoted as

X

_n

( )

ω

→

a s. .

X

( )

ω

_.

1 0 1

(|

n

( )

( ) |

)

(|

n

( )

( ) |

)

k n k k n k

X

X

X

X

ε

ω

ω

ε

ω

ω

ε

∞ ∞ ∞ ∞

= = > = =



 



−

≥

⊂

−

≥



 





∩∪

 

∪∩∪



. .

( ) a s ( )

n

X ω →X ω

, so 1

(| n( ) ( ) | ) 0 k n k

P X

ω

X

ω

ε

∞ ∞ = =   − ≥ =   

∩∪

 Properties:

(1)If Xn→a s. . X and {Xnk,k∈N} is any one subsequence of {Xn:n N}

∈ _{, then}

. . ,

k a s n

X →X

k

→ ∞

_.

(2)If

. .

a s n

X

→

X

and

. . '

,

a s

n

X

→

X

then

X

=

X

' almost surely.

(3)If

. .

a s n

X

→

X

and Xn=Yn _{almost surely,}

X

=

Y

_{almost surely, then}

. .

.

a s n

Y

→

Y

(4)If

. .

a s n

X →X

and

. .

.

a s n

Y

→

Y

then . .

,

a s n n

X ±Y →X±Y

X Y

_{n n}

→

a s. .

XY

,

a s. .

n

cX →cX

(for any constantc) ,

. .

, 0,

r a s r n

X → X ∀ >r n a s. . n

X X

Y Y

 → _{( Where { 0} 0, { 0} 0}

n

P Y = = P Y = = _).

Convergence in Probability

Definition 2.3If for allε >0_,lim

{

_n( ) ( )

}

0 n P X X

ω ω ε

→∞

− ≥ =

Then the sequence{Xn}converges in probability towards the random variables X,which is denoted

as Xn( )ω P→X( )ω . Properties:

(1)If

P n

X



→

X

and {Xnk,k N} ∈

is any one subsequence of {Xn:n∈N}_,then

,

k P n

X →X

k

→ ∞

;

(2)If →

P n

X X _and Xn P→X',_then

' ( ) 0

P X ≠ X =

(3)If

P n

X →X

and

P X

(

n

≠

Y

n

)

=

0,

_then . P n

Y →Y

(4)If ( ) P ( )

n

X ω →X ω and ( ) P ( )

n

Y ω →Y ω , then

( )

( )

P

0

n

X

ω

−

X

ω



→

_, P

,

n

cX



→

cX

_{(for any real number}

c

_),

,

P

n n

X

±

Y



→

X

±

Y

P

,

n n

X Y



→

XY

n P ,

n

X X

Y Y

→ ₍

,

₀

n

Y Y

ω

∀

≠

_).

Convergence in Distribution

Definition 2.4 Suppose F xn( ) and F x( ) are the cumulative distribution functions of random

variables Xn andX, respectively. If lim_n→ ∞F xn( ) F x( ) =

Then the sequence of random variables{Xn} is said to converge in distribution which is denoted by ( ) L ( )

n

X ω →X ω _{.We note here that}F x( )_{is continuous for every number}x∈R_.

Convergence in Mean

Definition 2.5 For random variablesXn,Xand any numberr>0. let | | , | | ,

r r

n

E X < ∞ E X < ∞ _If

lim | n |r 0

n→∞E X X

− = _{then the sequence{ }}

n

which is denoted as X_n( )ω r→X( )ω _.

The most important case for convergence in_r−_thmean is whenr=2, At this point.Xnis said to

converges in mean square toX.

Cauchy Convergence

Definition 2.6 Forn→∞and _m>₀,the sequence_{{ }} n

X is said to be Cauchy convergence.

If

0

n n m

X −X + →

The Relations between Several Kinds of Convergences

The sequence of random variables {X_n} converges to X in some sense, the convergence

depends only on the asymptotic properties of {X_n}, which all abide by the corresponding of Cauchy convergence criterion. For example, for all

ε

>

0

, there exists n0 such that for all

n

>

n

0

and m∈N_, E X

{

_{n m+} −X_nr

}

<ε_{, then the sequence of random variables} {X_n} _{converges in the}

r th− mean.

From the different definitions of the convergence, we can find that convergence inr−th_{mean is} the simplest in form, it only relates a single sequence. For convergence almost surely, we have a family sequences. For each sample pointω, it corresponds to a sequence. For convergence in probability, we also have a family sequence. For each ", it corresponds to a sequence. For convergence in distribution, we discuss convergence of a column function. For almost each real number x, it corresponds to a sequence{F xn( )}.There are both differences and relations between the different types of convergence. Next, we will study the relations between several kinds of convergence about the sequence of random variables.

Theorem 3.1[1]If ( ) a s. . ( )

n

X ω →X ω _{, then ( )} P _{( )} n

X

ω

→X

ω

Proof. For ε>0_{, we have that}

1 0 1

(| _n( ) ( ) | ) (| _n( ) ( ) | )

k n k k n k

X X X X

ε

ω

ω

ε

ω

ω

ε

∞ ∞ ∞ ∞

= = > = =

   

− ≥ ⊂ − ≥

   



∩∪

 

∪∩∪



Form ( ) a s. . ( )

n

X ω →X ω _{, We get that} 1

(| _n( ) ( ) | ) 0

k n k

P X

ω

X

ω

ε

∞ ∞

= =

 

− ≥ =

 



∩∪



Noticing that

1

(| _n( ) ( ) | ) 0 lim (| _n( ) ( ) | ) 0

k

k n k n k

P

X

ω

X

ω

ε

P

X

ω

X

ω

ε

∞ ∞ ∞

→∞

= = =









− ≥ = ⇔ − ≥ =











∩ ∪





∪



and{| _k( ) ( ) | } (| _n( ) ( ) |

n k

X ω X ω ε X ω X ω ε

∞

=

 

− ≥ ⊂_ − ≥ _



∪



so ,

lim {|

_k

( )

( ) |

}

0

k

P

X

X

ω

ω

ε

→∞

−

≥

=

, i.e. ( ) P ( )

n

X ω →X ω

In general, and ( ) P ( )

n

X ω →X ω _and ( ) r ( )

n

X ω →X ω _{cannot imply ( )} a s. . _{( )}

n

X ω →X ω _.

We give a example illustrating this case.

Example 3.2[1] LetΩ =(0,1]_{, FisBorel}

σ

−

_{algebra on}Ω, P is Lebesguemeasure,X( )

ω

≡0_, 1

1, ,

( ) 1, 2,..., ; 1, 2,... 1

0, ,

ki

i i k k

i k k

i i k k

ω ξ ω

ω

  − 

∈_{ }



  

=_ = =

−

   _∉

 

 _{ }

Define

1

( )

11

( ),

2

( )

21

( ),

3

( )

22

( ),

4

( )

31

( ),

5

( )

32

( ),...

X

ω

=

ξ

ω

X

ω

=

ξ

ω

X

ω

=

ξ

ω

X

ω

=

ξ

ω

X

ω

=

ξ

ω

In generalXn( )ω =ξki( ),ω Where

( 1) 2 k k

n= +i − _{. For each}ω∈(0,1]_{, there exist two infinitekandi}

such that Xn( ) 1,ω = at the same time , there also exist two infinite kandisuch that Xn( )ω =0,i.e.

{Xn( )}ω does not converge almost surely to X. However, on the other hand, for all ε >0,

1

{|

ki

( ) |

}

P

k

ξ

ω

≥

ε

≤

_{, since} ( 1) ( 1) _,

2 2

− −

=k k + ≤k k +

n i k son→ ∞_impliesk→ ∞.

Hence,lim {| ( ) | } lim {| ( ) | } 0 ( 1) 2

n ki

n n

k k

P X

ω

ε

P

ξ

ω

ε

n i

→∞ →∞

−

 

≥ = ≥ = _ = + _

 .

That is Xn( )

ω

P→X( )

ω

.It is not hard to show that

(|

| )

0

r n

E

X

−

X

→

_.

Under special circumstances, however, have the following result.

Theorem 3.3 [2]If{X_n}is a monotone decreasing sequence of random variables, then

. .

0

0

P a s

n n

X



→

⇒

X

→

Proof.Because{Xn} is a monotone decreasing sequence of random variables, Xnhave limit as

n

→ ∞

_{, and} {lim n } { k }

n X X

ε

ε

→∞

< ⊃ < _{, Because} P ₀ n

X → _{, and}

1

{lim

_n

}

lim {

_k

} 1

n k

P

X

ε

P X

ε

→∞ →∞

≥

<

≥

<

=

By X_n >0 _{and arbitrariness of}

ε

_{, we obtain that}

{lim

_n

( )

0} 1

n

P

X

ω

→∞

=

=

_{, i.e.}

. .

0

a s n

X

→

.

Theorem 3.4 If P n

X



→

X

, then there exist a subsequence

{

}

k n

X

of the sequence

{

X

_n

}

such

that . .

k

a s n

X

→

X

,

k

→ ∞

Proof. Notice that X_nP→X _{, so}

,

k

k

N

n

N

∀ ∈

∃

∈

_{, when}

k

n

≥

n

_{, we have}

1 1

2 2

n k k

P__X −X ≥ __<

 

 

Moreover, we can take k

n ↑ , by Borel - Cantellilemma [4], it is not hard to show that

. .

k

a s n

X

 →

X

.

Theorem 3.5[3] r P

n n

X →X ⇒ X →X

Proof. SupposeF x( ) isthe cumulative distribution functions of random variables

X

_n

−

X

_{, for all}

0

ε

>

| | | |

|

|

| |

1

{|

|

}

( )

( )

| |

( )

ε ε

ε

ε

ε

ε

+∞

≥ ≥ −∞

−

−

≥

=

∫

≤

∫

≤

∫

=

r r

r n

n _{x x} r r r

E X

X

x

P X

X

dF x

dF x

x dF x

Hence r P

n n

X



→

X

⇒

X



→

X

.

In general, P n

X



→

X

and a s. .

n

X

→

X

cannot imply r

n

X



→

X

.

We give a example illustrating this case.

1 1

, 0 ( )

1

0 , 1

ω ω

ω



< ≤



= _

 _{< ≤}



r

n

n

n X

n

Clearly , for all

ω

∈ Ω

,

X

_n

( )

ω

→

X

( ),

ω

_so a s. .

n

X

→

X

.On the

other hand , for all

ε

>

₀

,

P X

{|

n

( )

X

( ) |

}

1

n

ω

−

ω

≥

ε

≤

_.So P

n

X



→

X

_.However,

1

1

|

|

r

(

r

)

r

1

n

E X

X

n

n

−

=

⋅

=

_{, i.e. the sequence}

{

}

n

X

doesn’t converges in the

r th

−

mean

towards the random variablesX .Under special circumstances, the following result is held.

Theorem 3.7 [4]If the sequence of random variables {Xn}is uniform bounds, then

P n

X →X ⇒ r n

X →X

Proof. Notice that P n

X



→

X

,

|

|

n

X

≤

M

, so

|

X

|

≤

M a s

. .

( because exist)

. .

k

a s n

X

→

X

Hence,

|

X

_n

−

X

|

r



P

→

0,|

X

_n

−

X

|

r

≤

(2

M

)

r by Control convergence theorem, we

obtain thatlim | n |r 0 n→∞E X X

− = _.

Theorem 3.8 [1]Xn  P→ X ⇒

L n

X



→

X

.

Proof. For '

x

<

x

, we have

' ' ' '

{

X

≤

x

} {

=

X

_n

≤

x X

,

≤

x

} {

+

X

_n

>

x X

,

≤

x

}

⊂

{

X

_n

≤

x

} {

+

X

_n

>

x X

,

≤

x

}

,

so ' '

( )

_n

( )

{

_n

,

}

F x

≤

F x

+

P X

>

x X

≤

x

. Because of

X

_n



P

→

X

,

' '

{

_n

,

}

{|

_n

|

}

0

P X

>

x X

≤

x

≤

P

X

−

X

≥

x

−

x

→

Hence '

( )

lim

_n

( )

n

F x

F x

→∞

≤

. By the same way, for

x

''

>

x

,

we have_____ ''

lim

_n

( )

( )

n→∞

F x

F x

≤

.

Therefore, for ' ''

,

x

<

x

<

x

we have ' _____ ''

( )

lim

_n

( )

lim

_n

( )

( )

n n

F x

F x

F x

F x

→∞ →∞

≤

≤

≤

If

x

is continuous point of

F x

( )

, then ( ) li m n( ) n

F x F x

→ ∞

= _,as ' ''

,

x x

→x_.

In general, L n

X



→

X

doesn’t imply P n

X



→

X

. we give the following example illustrating this case.

Example 3.9 [1] Let

Ω =

{ ,

ω ω

_{1 2}

}

be a simple space satisfying 1 2

1

( ) ( ) ,

2

Pω =Pω = _{the random}

variables

X

( )

ω

_{is defined as follows:}

1 2

( ) 1, ( ) 1,

X

ω

= − X

ω

= _{the probability distribution of} X( )ω _is described as follows:

1

1

1

1

2

2

−

















Let

X

_n

( )

ω

= −

X

( )

ω

_{For all}

n

_{, it is clearly that ( )} n

X ω _{and X( )}ω _{have the same probability}

{|

n

( )

( ) |

}

{ } 1

P

X

ω

−

X

ω

>

ε

=

P

Ω =

_{. Therefore,{}X_n( )}ω _{doesn’t converge in probability towards}

( )

X

ω

_.

Because the different distributions of random variables can correspond to the same distribution function, in general, convergence in distribution does not imply the other convergence. For example, forX( )

ω

_{of the previous example, define the sequence of random variables}

{

( )}

n

X

ω

_{as follows:}

2n

( )

( ),

2n 1

( )

( )

X

ω

X

ω

X

ω

X

ω

+

=

= −

It is clear thatX_n L→X _.but{X_n( )}ω _{hasn’t other convergence.}

however, under the special circumstances, we have the following result.

Theorem 3.10 [5] If

C

is constant, then

X

_n



L

→

C

⇒

X

_n



P

→

C

Proof. For all

ε

>

₀

, we have that

{|

_n

|

}

{

_n

}

{

_n

}

P X

−

C

≥

ε

=

P X

≥

C

+

ε

+

P X

≤

C

−

ε

1

F C

n

(

ε

)

F C

(

ε

0)

1 1 0

0

(

n

)

= −

+

+

−

+

→ − +

=

→ ∞

To sum up, the relations of convergence almost surely, convergence in probability, convergence in distribution, and convergence in mean can be succinctly stated as follows:

. .

a s

P L

n

n n

r n

X

X

X

X

X

X

X

X



→

⇒



→

⇒



→





→



At last, only to convergence in probability for example, illustrate random variables sequences are subject to the corresponding Cauchy convergence criterion.

Theorem 3.11 Let

{

}

n

X

be a sequence of random variables. The sufficiently and necessary condition of P

n

X



→

X

is that the sequence{ }X_n is Cauchy convergence in probability.

Proof.For all

ε

>

₀

,the following inequality holds: ({| | })

2 2

P X−Y ≥ε ≤PX−Z ≥ε+PZ−Y ≥ε

   

   

If

X

_n



P

→

X

, for all

ε

>

0

,in view of the above inequality, then, for all

m

≥

1,

m

∈

N

_,

({| | }) 0

2 2

n m n n m n

P X X

ε

P X X

ε

P X X

ε

+ +

   

− ≥ ≤ _ − ≥ _+ _ − ≥ _→

   

    .

Hence, the sequence

{

X

_n

}

is Cauchy convergence in probability.

If the sequence

{

X

_n

}

is Cauchy convergence in probability, there exist_X and a subsequence

{

}

{

}

k

n n

X

⊂

X

_{such that}

(

)

k P n

X



→

X k

→ ∞

_{. So, for all}

ε

>

₀

,by taking

n

_k

>

n

,

and

n

→ ∞

,we have that ({| | }) 0

2 2

k k

n n n n

P X −X ≥ε ≤P__X −X ≥ε__+P_X −X ≥ε__→

   

    ,That is ,

P n

X



→

X

_.

Acknowledgments

References

[1] J. G. Wang Foundation of the theory of modern probability, shanghai: Fudan University Press, 1998.

[2] Yu. Yin. Several kinds of convergence of random variable sequences, Association for science and technology BBS, 2012, 6:109-111.

[3] Sh.T. Li. Four kinds of convergence of random variables, Journal of western Hubei University, 1987, 3:70-74.

[4] S. J. Yan, X. F. Liu measure and probability, Beijing: Beijing normal university press, 2003.