fundamental theorem of demography, proof of
∙ First we will prove that there exist m,M>0 such that
m≤∥xk+1∥∥xk∥≤M | (1) |
for all k, with m and M of the sequence. In to show this we use the primitivity of the matrices Ak and A∞. Primitivity of A∞ implies that there exists l∈ℕ such that
Al∞≫0 |
By continuity, this implies that there exists k0 such that, for all k≥k0, we have
Ak+lAk+l-1⋯Ak≫0 |
Let us then write xk+l+1 as a function of xk:
xk+l+1=Ak+l⋯Akxk |
We thus have
∥xk+l+1∥≤Cl+1∥xk∥ | (2) |
But since the matrices Ak+l,…,Ak are strictly positive
for k≥k0, there exists a ε>0 such that each
of these matrices is superior or equal to
ε. From this we deduce that
∥xk+l+1∥≥ε∥xk∥ |
for all k≥k0. Applying (2), we then have that
Cl∥xk+1∥≥ε∥xk∥ |
which yields
∥xk+1∥≥εCl∥xk∥ |
for all k≥0, and so we indeed have (1).
∙ Let us denote by ek the (normalised) Perron eigenvector of
Ak. Thus
Akek=λkek |
Let us denote by the projection on the supplementary space of invariant by . Choosing a proper norm, we can find such that
for all . We shall now prove that
In order to do this, we compute the inner product of the sequence
with the ’s:
Therefore we have
Now assume
We will verify that when . We have
and so
We deduce that there exists such that, for all
where we have noted
We have when , we thus finally deduce that
Remark that this also implies that
We have when , and can be written
Therefore, we have when , which implies that tends to 1, since we have chosen to be normalised (i.e.,).
We then can conclude that
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 |