convergence in probability is preserved under continuous transformations

convergence in probability is preserved under continuous transformations

Theorem 1

.

Let

g

:

R

k

→

R

l

be a

continuous function

.
If

{

X

n

}

are

R

k

-valued

random variables

converging to

X

in

probability

, then

{

g

⁢

(

X

n

)

}

converge in probability

to

g

⁢

(

X

)

also.

Proof.

Suppose first that

g

is

uniformly continuous

.
Given

ϵ

>

0

, there is

δ

>

0

such that

∥

g

⁢

(

X

n

)

-

g

⁢

(

X

)

∥

<

ϵ

whenever

∥

X

n

-

X

∥

<

δ

.
Therefore,

ℙ

(

∥

g

(

X

n

)

-

g

(

X

)

∥

≥

ϵ

)

≤

ℙ

(

∥

X

n

-

X

∥

≥

δ

)

→

0

as

n

→

∞

.

Now suppose

g

is not necessarily uniformly continuous on

ℝ

k

.
But it will be uniformly continuous on any

compact set

{

x

∈

ℝ

k

:

∥

x

∥

≤

m

}

for

m

≥

0

.
Consequently, if

X

n

and

X

are

bounded

(by

m

), then the proof just
given is applicable. Thus we attempt to reduce the general case
to the case that

X

n

and

X

are bounded.

Let

f

m

⁢

(

x

)

=

{

x

,

∥

x

∥

≤

m

m

⁢

x

/

∥

x

∥

,

∥

x

∥

≥

m

Clearly,

f

m

:

ℝ

k

→

ℝ

k

is

continuous

; in fact, it can be verified that

f

m

is uniformly continuous on

ℝ

k

.
(This is geometrically obvious in the one-dimensional case.)

Set

X

n

m

=

f

m

⁢

(

X

n

)

and

X

m

=

f

m

⁢

(

X

)

,
so that

X

n

m

converge

to

X

m

in probability for each

m

≥

0

.

We now show that

g

⁢

(

X

n

)

converge to

g

⁢

(

X

)

in probability by a four-step

estimate

. Let

ϵ

>

0

and

δ

>

0

be given.
For any

m

≥

0

(which we will later),

ℙ

(

∥

g

(

X

n

)

-

g

(

X

)

∥

≥

δ

)

≤

ℙ

(

∥

g

(

X

n

m

)

-

g

(

X

m

)

∥

≥

δ

)

+

ℙ

(

∥

X

n

∥

≥

m

)

+

ℙ

(

∥

X

∥

≥

m

)

.

Choose

M

such that for

m

≥

M

,

ℙ

(

∥

X

∥

≥

m

)

≤

ℙ

(

∥

X

∥

≥

M

)

<

ϵ

4

.

(This is possible since

lim

m

→

∞

ℙ

(

∥

X

∥

≥

m

)

=

ℙ

(

⋂

m

=

0

∞

{

∥

X

∥

≥

m

}

)

=

ℙ

(

∅

)

=

0

.)

In particular, let

m

=

M

+

1

.
Since

X

n

m

converge in probability to

X

m

and

X

n

m

,

X

m

are bounded,

g

⁢

(

X

n

m

)

converge in probability to

g

⁢

(

X

m

)

.
That means for

n

large enough,

ℙ

(

∥

g

(

X

n

m

)

-

g

(

X

m

)

∥

≥

δ

)

<

ϵ

4

.

Finally, since

∥

X

n

∥

≤

∥

X

n

-

X

∥

+

∥

X

∥

,
and

X

n

converge to

X

in probability,
we have

ℙ

(

∥

X

n

∥

≥

m

=

M

+

1

)

≤

ℙ

(

∥

X

n

-

X

∥

≥

1

)

+

ℙ

(

∥

X

∥

≥

M

)

<

ϵ

4

+

ϵ

4

for large enough

n

.

Collecting the previous

inequalities

together,
we have

ℙ

(

∥

g

(

X

n

)

-

g

(

X

)

∥

≥

δ

)

<

ϵ

for large enough

n

.
∎

Title

convergence in probability is preserved under continuous transformations

Canonical name

ConvergenceInProbabilityIsPreservedUnderContinuousTransformations

Date of creation

2013-03-22 16:15:05

Last modified on

2013-03-22 16:15:05

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

10

Author

stevecheng (10074)

Entry type

Theorem

Classification

msc 60A10

Generated on Sat Feb 10 11:52:57 2018 by

LaTeXML