fundamental theorem of demography, proof of

First we will prove that there exist m,M>0 such that

mxk+1xkM (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 such that


By continuity, this implies that there exists k0 such that, for all kk0, we have


Let us then write xk+l+1 as a function of xk:


We thus have

xk+l+1Cl+1xk (2)

But since the matrices Ak+l,…,Ak are strictly positivePlanetmathPlanetmath for kk0, there exists a ε>0 such that each of these matrices is superior or equal to ε. From this we deduce that


for all kk0. Applying (2), we then have that


which yields


for all k0, and so we indeed have (1).

Let us denote by ek the (normalised) Perron eigenvectorMathworldPlanetmathPlanetmathPlanetmath of Ak. Thus


Let us denote by πk the projection on the supplementary space of {ek} invariant by Ak. Choosing a proper norm, we can find ε>0 such that


for all k. We shall now prove that

ek+1*,xk+1ek*,xkλ when k

In order to do this, we compute the inner productMathworldPlanetmath of the sequence xk+1=Akxk with the ek’s:

ek+1*,xk+1 = ek+1*-ek*,Akxk+λkek*,xk
= o(ek*,xk)+λkek*,xk

Therefore we have


Now assume


We will verify that uk0 when k. We have

uk+1 = (πk+1-πk)Akxkek+1*,xk+1+ek*,xkek*,xk+1Akπkxkek*,xk

and so


We deduce that there exists k1k0 such that, for all kk1


where we have noted


We have δk0 when t, we thus finally deduce that

|uk|0 when k

Remark that this also implies that

zk=πkxkxk0 when k

We have zk0 when k, and xk/xk can be written


Therefore, we have αkek1 when k, which implies that αk tends to 1, since we have chosen ek to be normalised (i.e.,ek=1).

We then can conclude that

xkxke 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