fundamental theorem of demography, proof of

fundamental theorem of demography, proof of

∙

First we will prove that there exist

m

,

M

>

0

such that

m

≤

∥

x

k

+

1

∥

∥

x

k

∥

≤

M

(1)

for all

k

,
with

m

and

M

of the

sequence

. In to show this we
use the primitivity of the matrices

A

k

and

A

∞

.
Primitivity of

A

∞

implies that there exists

l

∈

ℕ

such that

A

∞

l

≫

0

By continuity, this implies that there exists

k

0

such that, for all

k

≥

k

0

, we have

A

k

+

l

⁢

A

k

+

l

-

1

⁢

⋯

⁢

A

k

≫

0

Let us then write

x

k

+

l

+

1

as a function of

x

k

:

x

k

+

l

+

1

=

A

k

+

l

⁢

⋯

⁢

A

k

⁢

x

k

We thus have

∥

x

k

+

l

+

1

∥

≤

C

l

+

1

⁢

∥

x

k

∥

(2)

But since the matrices

A

k

+

l

,…,

A

k

are strictly

positive

for

k

≥

k

0

, there exists a

ε

>

0

such that each
of these matrices is superior or equal to

ε

. From this we deduce that

∥

x

k

+

l

+

1

∥

≥

ε

⁢

∥

x

k

∥

for all

k

≥

k

0

.
Applying (

2

), we then have that

C

l

⁢

∥

x

k

+

1

∥

≥

ε

⁢

∥

x

k

∥

which yields

∥

x

k

+

1

∥

≥

ε

C

l

⁢

∥

x

k

∥

for all

k

≥

0

,
and so we indeed have (

1

).

∙

Let us denote by

e

k

the (normalised) Perron

eigenvector

of

A

k

. Thus

A

k

⁢

e

k

=

λ

k

⁢

e

k

∥

e

k

∥

=

1

Let us denote by

π

k

the

projection

on the supplementary space of

{

e

k

}

invariant

by

A

k

. Choosing a proper norm, we can find

ε

>

0

such that

|

A

k

⁢

π

k

|

≤

(

λ

k

-

ε

)

for all

k

.

∙

We shall now prove that

⟨

e

k

+

1

*

,

x

k

+

1

⟩

⟨

e

k

*

,

x

k

⟩

→

λ

∞

⁢

when

⁢

k

→

∞

In order to do this, we compute the

inner product

of the sequence

x

k

+

1

=

A

k

⁢

x

k

with the

e

k

’s:

⟨

e

k

+

1

*

,

x

k

+

1

⟩

=

⟨

e

k

+

1

*

-

e

k

*

,

A

k

⁢

x

k

⟩

+

λ

k

⁢

⟨

e

k

*

,

x

k

⟩

=

o

⁢

(

⟨

e

k

*

,

x

k

⟩

)

+

λ

k

⁢

⟨

e

k

*

,

x

k

⟩

Therefore we have

⟨

e

k

+

1

*

,

x

k

+

1

⟩

⟨

e

k

*

,

x

k

⟩

=

o

⁢

(

1

)

+

λ

k

∙

Now assume

u

k

=

π

k

⁢

x

k

⟨

e

k

*

,

x

k

⟩

We will verify that

u

k

→

0

when

k

→

∞

. We have

u

k

+

1

=

(

π

k

+

1

-

π

k

)

⁢

A

k

⁢

x

k

⟨

e

k

+

1

*

,

x

k

+

1

⟩

+

⟨

e

k

*

,

x

k

⟩

⟨

e

k

*

,

x

k

+

1

⟩

⁢

A

k

⁢

π

k

⁢

x

k

⟨

e

k

*

,

x

k

⟩

and so

|

u

k

+

1

|

≤

|

π

k

+

1

-

π

k

|

⁢

C

′

+

⟨

e

k

*

,

x

k

⟩

⟨

e

k

+

1

*

,

x

k

+

1

⟩

⁢

(

λ

k

-

ε

)

⁢

|

u

k

|

We deduce that there exists

k

1

≥

k

0

such that, for all

k

≥

k

1

|

u

k

+

1

|

≤

δ

k

+

(

λ

∞

-

ε

2

)

⁢

|

u

k

|

where we have noted

δ

k

=

(

π

k

+

1

-

π

k

)

⁢

C

′

We have

δ

k

→

0

when

t

→

∞

, we thus finally deduce that

|

u

k

|

→

0

⁢

when

⁢

k

→

∞

Remark that this also implies that

z

k

=

π

k

⁢

x

k

∥

x

k

∥

→

0

⁢

when

⁢

k

→

∞

∙

We have

z

k

→

0

when

k

→

∞

, and

x

k

/

∥

x

k

∥

can be written

x

k

∥

x

k

∥

=

α

k

⁢

e

k

+

z

k

Therefore, we have

α

k

⁢

e

k

→

1

when

k

→

∞

, which implies
that

α

k

tends to 1, since we have chosen

e

k

to be
normalised (

i.e.

,

∥

e

k

∥

=

1

).

We then can conclude that

x

k

∥

x

k

∥

→

e

∞

⁢

when

⁢

k

→

∞

and the proof is done.

Title

fundamental theorem of demography

, proof of

Canonical name

FundamentalTheoremOfDemographyProofOf

Date of creation

2013-03-22 13:24:42

Last modified on

2013-03-22 13:24:42

Owner

aplant (12431)

Last modified by

aplant (12431)

Numerical id

10

Author

aplant (12431)

Entry type

Proof

Classification

msc 92D25

Classification

msc 37A30

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