Appendix A
A.1 Derivation of formulas in section 3.6.2 for Sykes (1969)
The formula for var
( )
Ρ(t) for the additive errors model can be derived in the following way(
(t), (u))
E[ ( (t) E( (t))(
(u) E( (u))
']
Cov Ρ Ρ = Ρ − Ρ Ρ − Ρ
Α
Α
=
∑ ∑
= − −
= −−
u
j
j u j t
i
i t
iS S
1
1 1
1
Α Α
=
∑∑
= =
−
− −
− t
i u
j
j j i u t
iS S
E
1 1
' 1 ' 1
j j u i t t
i u
j
i '
1 , 1
1 1
Α Ψ
Α
= −− − −
= =
∑∑
That indicates that var
( )
Ρ(t) is given by
( )
t t i t j ji t
j
t i
Var 1, 1 '
1 1
)
( = Α Ψ Α
Ρ −− − −
= =
∑∑
For the Branching process formulation Sykes suggests using a conditional argument. The expectation can be given by
(
(t 1))
E{
E(
(t 1)| (t)) }
E Ρ + = Ρ + Ρ
(
(t))
E ΑΡ
=ΑE
( )
Ρ(t)=
) 0 Ρ( Α
= t The conditional variance formula is given by
(
(t 1)) (
EVar (t 1)| (t))
Var(
E(
(t 1)| (t)) )
Var Ρ + = Ρ + Ρ + Ρ + Ρ
Applying the conditional variance formula leads to
(
VarΡ(t+1)|Ρ(t))
=var(
ΑΡ(t))
|Ρ(t)( )
()varΡ t Α
=E
( [ΑΡ(t)][
ΑΡ(t)]
')
−(
E[
ΑΡ(t)] )
2
=
( )
(
( ))
2) ( )
(t t E t
a a
E i i − ΑΡ
Ρ Ρ
=
∑∑
β α βα β α
(
(t) (t)) (
E( )
(t))
2E a
ai i − Α Ρ
Ρ Ρ
=
∑∑
β α βα β α
( ) ( ) ( )
[
cov (t), (t) E (t) E (t)] (
E( )
(t))
2a
ai i − Α Ρ
Ρ Ρ + Ρ Ρ
=
∑∑
β α β α βα β α
( )
[
Ρ Ρ]
+(
Ρ) (
Ρ)
−=
∑∑
ai ai cov (t), (t)∑∑
ai aiβE α(t) E β(t)α β α
β α α β α β
( ) (
ΑE Ρ(t))
2( )
[
cov (t), (t)]
a
ai iβ α β
α β α Ρ Ρ
=
∑∑
It follows that
( )
()[
cov(
( ), ()) ]
var(
( 1))
'varΡ t =
∑∑
ai aiβ Ρα t Ρβ t +Α Ρ t− Αα β α
This can be written as
( )
t i t ii
i
t C
t −−
−
=
−
− Α
Α
=
Ρ
∑
1 ' 10
) 1
(
var ,
where
( )
[
cov (i), (i)]
a a
Ci i iβ α β
α β α Ρ Ρ
=
∑∑
.Sykes’ third approach is that of the random transition matrix. The author introduces the model
(
( ))
( )) 1
(t+ = Α+Κ t Ρ t
Ρ , t=0,1,...,
where Κ(t) is a sequence of independent l×l matrix random variables satisfying E
(
Κ t())
=0,and var
(
Κ )(t)
=Σ, with Σ is a singular l2×l2 matrix. To find E(
Ρ t( +1))
and var(
Ρ t( +1))
Sykes uses the conditional mean to obtain the expectation as follows E
(
Ρ(t+1)|Ρ(t)) (
=E Α+Κ(t))
Ρ(t)|Ρ(t)(
() ())
| ( ))
(t +E Κ t Ρ t Ρ t ΑΡ
=ΑΡ(t)+Ρ(t)E
(
Κ(t)|Ρ(t))
=ΑΡ(t)+Ρ(t)E
(
Κ(t))
= , because of independence
) ΑΡ(t
=
⇒ E
(
Ρ(t+1))
=E{
E(
Ρ(t+1)|Ρ(t)) } (
(t))
E ΑΡ
=ΑE
( )
Ρ(t)=
) 0 Ρ( Α
= t
Sykes makes use of the conditional variance formula to calculate the variance as follows
Var
(
Ρ(t+1))
=Var{ (
Α+Κ(t))
Ρ(t)|Ρ(t)} {
(t) (t)| (t)}
Var Κ Ρ Ρ
=E
(
Κ(t)Ρ(t))(
Κ(t)Ρ(t))
' |Ρ(t)−{
E(
Κ(t)) ( )
E Ρ(t)}
2=E
{
Κ(t)Ρ(t)Ρ'(t)Κ'(t)}
|Ρ(t)=
Ρ
Κ Κ Ρ Ρ
=
∑∑
= =
) (
| ) ( ) ( ) ( ) (
1 1
t t t t t E
p
v
w p
w
v w v
where the curly brackets indicate that expression inside them is the ijth element of the matrix considered. It follows that
( ) ∑∑ ( )
= =
Ρ Ρ Κ Κ
= Ρ +
Ρ l
v
w l
w
v w
v t t t t
t t
Var
1 1
) ( ) ( ) ( ) ( cov )
(
| ) 1 (
This implies that
(
( 1))
cov(
( ) ( )) (
( ) ( )) (
())
1 1
t Var t
t E t t t
Var
l
v l
w
w v w
v Κ Ρ Ρ + ΑΡ
Κ
= +
Ρ
∑∑
= =
(
Κ Κ) ( [
Ρ Ρ) (
+ Ρ) (
Ρ) ]
+=
∑∑
= = l
v l
w
w v
w v w
v t t t t E t E t
1 1
) ( ) ( )
( ), ( cov ) ( ) ( cov
(
(t))
Var ΑΡ
(
Κ Κ) ( [
Ρ Ρ) (
+ Ρ) (
Ρ) ]
+=
∑∑
= = l
v l
w
w v
w v w
v t t t t E t E t
1 1
) ( ) ( )
( ), ( cov ) ( ) ( cov
( )
Ρ( )Α' ΑVar t That indicates that(
Κ − Κ −) (
Ρ − Ρ −)
+=
Ρ
∑∑
= = l
v l
w
w v
w
v t t t t
t Var
1 1
) 1 ( ), 1 ( [cov ) 1 ( ) 1 ( cov )
(
(
Ρ (t−1)) (
E Ρ (t−1))
+ΑVar(
Ρ(t−1))
Α']E v w
Let
( ) ( [ ) ( ) ( ) ]
∑∑
= =Ρ Ρ
+ Ρ Ρ Κ
Κ
=
Ο l
v l
w
w v
w v w
v t t t t E t E t
t
1 1
) ( ) ( )
( ), ( cov ) ( ) ( cov )
(
Then
(
( 1))
') 1 ( )
( =Ο − +Α Ρ − Α
Ρ t t Var t
Var
i t i t
i
i
t −−
−
=
−
− Ο Α
Α
=
∑
1 ' 10 1
Writing out Ο gives
( ) ( [ ) ( ) ( ) ]
t ll
v l
w
w v
w v w
v t
l
l
t t t t t E t E t
t
Var −−
= =
−
=
−
− Κ Κ Ρ Ρ + Ρ Ρ Α
Α
=
Ρ
∑ ∑∑
' 11 1
1
0
1 cov () ( ) cov ( ), ( ) () ()
) (
A.2 Derivation of the formula of
d(t)in section 3.6.2 for Lee (1974)
Equation 3.2.1 can be written in the following way ) ( )
(t B Bd t
B = +
(
( ) ( , ))(
( ))
1
j t Bd B t j x j n
Upper
j
− +
−
=
∑
=
( )
∑
= + − + + −= Upper
j
j t d t j x t j x j t d j n j n B
1
) ( ) , ( ) , ( ) ( ) ( ) (
That means that
1 ) ( ) , ( ) , ( ) ( ) ( )
( )
(
1 1
−
− +
+
− +
=
∑ ∑
=
=
j t d t j x t j x j t d j n j
n t
d
Upper
j Upper
j
( ) ∑
∑
∑
= = =− +
− +
−
= Upper
j Upper
j Upper
j
j t d t j x j
n t j n j
t d j n
1 1
1
) ( ) , ( )
( ) , ( )
( )
(
∑
∑
= =− +
+
−
= Upper
j Upper
j
j t d t j x t
j t d j n
1 1
) ( ) , ( )
( ) ( )
( ε
Based on earlier results Lee ignores the last term and approximates d(t) by the following AR process
) ( ) ( ) ( )
(
1
t j t d j n t
d
Upper
j
ε +
−
=
∑
=
A.3 Derivation of formulas in section 3.6.2 for Alho and Spencer (1991)
The covariance between the prediction error of population size in ages i and j at time t, l
j i t≤ ≤ ≤
≤
1 , is given by
− + − +
− + − +
=
∑ ∑
−=
−
=
1
0
~ 1 ~
0
~
~
~
~
) , (
) 0 , ( , ) , (
) 0 , ( cov )
, ( ), , ( cov
t
n t
m
n n t j sv t
j p m m t i sv t
i p t
j p t i p
− + − +
+
− −
=
∑
−∑
=
−
= 1
0
1
0
~
~
~
~
) , (
), , (
cov ) 0 , ( ), 0 , ( cov
t
m
t
n
n n t j sv m m t i sv t
j p t i p
) , , ,
( )
, , (
1
0 1
0
n m n t j m t i t
j i
t
n t
m
sv − + − +
+
=
∑∑
−=
−
=
σ σ
The Taylor series expansion used by Alho and Spencer (1991) can be described as follows.
Consider the function
=
∑
=
) exp(
log ) ,..., (
1 1
n
i
i
n y
y y h
Then a linear Taylor series representation for h at z ,...,1 zn is given by
(
( ,...,)
* exp( )( )exp ) ,..., ( ) ,..., (
1 1
1
1 i i
n
i
i n
n
n h z z h z z z y z
y y
hL = + −
∑
−=
Alho and Spencer apply Taylor series expansion by taking
Β
=log ( , )
~
t j
yi and
(
( , ))
log j t
zi = Β . Therefore
{
log ( , )}
log ( , ) (0, )exp log ) ,..., (
44
15 44
15
1 z j t j t p t
z h
j j
n =
∑
Β =∑
Β ==
=
,
(
( ,...,)
1/ (0, )exp −h z1 zn = p t , and
∑
∑
= =
Β − Β
Β
=
− 44
15 44 ~
15
)) , ( log(
)) , ( log(
) , ( )
)(
exp(
j j j j
j y z j t j t j t
z
Β − Β
Ρ Β +
⇒ =
∑
= 44
15
~
~
)) , ( log(
)) , ( log(
) , ) (
, 0 ( ) 1 , 0 ( ) , 0 (
j
t j t
j t
t j t
p t p
The covariance between the jump-off populations at two different times is given by
Β
Ρ
Β − Β
Β
≈ Ρ
∑ ∑
=
=
44
15 44
15
~
) , ) (
, 0 1 ( , )) , ( log(
)) , ( log(
) , ) (
, 0 1 ( cov ) , , 0 , 0 (
j i
u u j
t i t
i t
t i u
σ t
log(Β( , ))−log(Β( , ))
~
u j u
j
Β
Β Β
Β
Ρ
≈ Ρ
∑ ∑
= =
) , ( log , ) , ( log cov ) , ( ) , ) (
, 0 1 ( ) , 0
1 ( 44 ~ ~
15 44
15
u j t
i u
j t u i
t i j
Since
− + + − + + + −
=
Β
∑
−=
) 1 , ( ) , 1 (
) 0 , 1 (
exp ) , (
2 ~
0
~
~
~
t k ft n n t k sv t
k p t
k
t
n
it follows that, using the independence assumption between the jump-off population and vital rates and between fertility and survival rates,
∑ ∑
= = Β Β − + − + +
Ρ
≈ Ρ 44
15 44
15
~
~
)) 0 , 1 (
), 0 , 1 (
)(cov(
, ( ) , ) (
, 0 ( ) , 0 1 ( ) , , 0 , 0 (
i j
u j p t
i p t
j t u j
u t σ t
)) 1 , ( ), 1 , ( cov(
) , 1 (
), , 1 (
cov
~ 2 ~
0 2 ~
0
~ − + +
∑
− + + + − −∑
−=
−
=
u j ft t i ft n
n u j sv m m t
i sv
t
n t
m
Therefore, σ(0,0,t,u) is given by
=∑ ∑
= Β Β − + − + +
Ρ
≈ Ρ 44
15 44
15
) 1 ,
1 (
)(
, ( ) , ) (
, 0 ( ) , 0 1 ( ) , , 0 , 0 (
i j
u j t i t j t u j
u t
t σ
σ
)) 1 , 1 , , ( ) , , 1 2
0 2 0
, 1
( − + + + − −
∑−
= ∑−
= − + + i j t u
n ft m n u j t
m u n
m t
sv i σ
σ
For the surviving births, max
[
0,t−16]
≤ j≤i<t,
(, ), ( , ) cov
~
~
t j p t i
p , the covariance is
given by
− + − + − + − +
=
∑ ∑
−=
−
=
1
0
~ 1 ~
0
~
~
~
~
), ,
( )
, 0 ( ), ,
( )
, 0 ( cov )
, ( ), , ( cov
j
m i
n
m j t m sv j
t p n i t n sv i
t p t
j p t i p
+ +
− +
− +
−
−
=
∑∑
−=
−
=
) ,
, , ( )
, , 0 , 0 (
1
0 1
0
m j t n i t m n j
t i t
i
n j
m
σsv
σ
− − +
+
−
∑
− +∑
−=
−
=
1
0
~ 1 ~
0
~
~
, ( ), , 0 ( cov ,
( ), , 0 ( cov
j
n j
m
n i t n sv j t p m
j t m sv i t p
Using the Taylor series expansion explained earlier
−
Β −
− Β
−
= Ρ
−
∑
− +∑
=
−
=
) , ( log ) , ) (
, 0 1 ( cov )
, ( ), , 0 ( cov
44 ~
15 1
0
~
~
i t k i
t i k
m t j t m sv i t p
k j
m
+
−
Β −
∑
−= 1
0
~
) ,
( , ) , ( log
j
m
m j t m sv i
t k
− +
Β −
− Β
−
≈ Ρ
∑ ∑
−=
=
1
0
~ 44 ~
15
) ,
( , ) , ( log cov ) , ) (
, 0
1 ( j
m k
m j t m sv i
t k i
t i k
t
But
− + + + − + + + + − −
=
−
Β
∑
−−=
) 1 ,
( ) , 1 (
) 0 , 1 (
exp ) , (
2 ~
0
~
~
~
i t k ft n n i t k sv i
t k p i
t k
i t
n
Therefore
− + + + − + + + +
≈
+
−
Β −
∑ ∑
−−=
−
=
) , 1 (
) 0 , 1 (
cov )
, ( , ) , ( log cov
2
0
~ 1 ~
0
~
~
n n i t k sv i
t k p m
j t m sv i
t k
i t
n j
m
+
−
−
−
∑
−= 1
0
~
~
) ,
( ), 1 ,
(
j
m
m j t m sv i
t k ft
[
k t i n m n t j m]
i t i k
t
i t
n j
m sv k
+
− +
+ +
−
−
− Β
≈ Ρ
∑ ∑∑
−−=
−
=
=
, , , 1 )
, ) (
, 0
1 ( 2
0 1
0 44
15
σ
That means that
+ +
− +
− +
−
−
≈
(), ( ) (0,0, , )
∑∑
− − ( , , , )cov
1 1
~
~
n j t m i t n m j
t i t j
p i p
i
m j
n
σsv
σ
[
k t i n m n t j m]
i t i k
t
i t
n j
m sv k
+
− +
+ +
−
−
− Β
Ρ
∑ ∑∑
−−=
−
=
=
, , , 1 )
, ) (
, 0
1 ( 2
0 1
0 44
15
σ
(A.3.1)
Alho and Spencer set the covariance between the survival rates of the mothers giving birth at time t and the survival rates of their own mothers (17 or more years earlier) to zero, leading to the cancellation of the last term in the formula above.
The covariance between the surviving births in age i, max{0,t− }16 ≤i<t, at time t and the survivors of the jump-off population at age j at time u (u≤ j) is given by
∑
∑
−=
−
=
+
− +
− +
− +
−
=
1
0
~ 1 ~
0
~
~
~
~
) , (
) 0 , ( ), ,
( )
, 0 ( cov[
) , ( ), , ( cov
u
m i
n
m m u j sv u
j p n i t n sv i
t p u
j p t i p
+
− −
=cov (0, ), ( ,0)
~
~
u j p i t p
+
∑
− − +∑
− +=
−
= 1
0
1
0
~
~
) , (
), ,
( cov
i
n
u
m
m m u j sv n i t n sv
− + −
+
−
∑
− +∑
−=
−
=
1
0
~ 1 ~
0
~
~
) 0 , ( ), ,
( cov
) , (
), , 0 ( cov
i
n u
m
u j p n i t n sv m
m u j sv i t p
Using Taylor series expansion, (0, )
~
i t
p − can be written as
∑
= Β − − Β −
− Β
− + Ρ
−
≈
− 44
15
~
~
)) , ( log(
)) , ( log(
) , ) (
, 0 1 ( ) , 0 ( ) , 0 (
k
i t k i
t k i
t i k
i t t p i t p
with
− + + + − + + + + − −
=
−
Β
∑
−−=
) 1 ,
( ) , 1 (
) 0 , 1 (
exp ) , (
2 ~
0
~
~
~
i t k ft r r i t k sv i
t k p i
t k
i t
r
Therefore, the covariance between the surviving births in age i at time t and the survivors of the jump-off population at age j at time u(u≤ j) can be given by
+ +
− +
−
≈
∑∑
−=
−
=
) , ,
, ( )
, ( ), , ( cov
1
0 1
0
~
~
m n i t m u j n u
j p t i p
i
n n
m
σsv
Β − − + + − +
Ρ
∑
=
) 0 , , 1 (
)[
, ) (
, 0
1 ( 44
15
u j i t k i t t k
k
σ
+
− + + +
∑∑
−− −=
−
= 2
0 1
0
)]
, , ,
1 (
i t
r u
m
sv k t i r j u m r m
σ (A.3.2)
Alho and Spencer set the covariance between the errors in the jump-off population and the errors in births of the mothers giving birth at time t to zero. The authors also set the covariance between the errors in the fertility forecasts for year t−1 and the past births to zero. These simplifications lead to the cancellation of the last term in A.3.2.
A.4 Derivation of formulas in section 3.6.2 for Alho (1992a) and Alho (1992b)
Given
∑
−( )
= +
= 1
0
) ( ) ( )
(
t
k
mg
sv k k
t
V ε ε , Alho starts with evaluating the term
∑
−= 1
1
) (
t
k sv k
ε by noting that
) ( ) ( ) (
1
1
1
0
k e k t k
t
k
t
k
sv
∑
− sv∑
=
−
=
− ε =
var ( ) ( ) 2 ( )
1 2
0 2 1
0
t g k
t
k sv
t
k sv t
k
sv σ σ
ε = − =
⇒
∑ ∑
−=
−
=
where g(t)=(2t+1)(t+1)t/6. It follows that
2 2
2 ( )
)) (
var(εJO +V t =σJO +σsv g t +tσmg
Using a Taylor series expansion, the covariance between births in the year t and in the year u, 16
1≤t≤u≤ , is given by
∑ ∑
= =+ +
Β Β
Ρ
Ρ
≈
44
15 44
15
2
~ 2
~
){
, ( ) , ) (
, 0 1 ( ) , 0 1 ( ) , 0 ( ), , 0 ( cov
j k
ft
JO t
u k t u j
u t p t
p σ σ
(
( ), ( ))
}covV t V u
the covariance between V(t) and V(u) is given by
( ) ( ) ( )
+ +
=
∑ ∑
−=
−
=
1
0 1
0
, ) ( ) ( ,
) ( ) ( cov
) ( ), ( cov
u
h
mg sv
t
k
mg
sv k k h h
u V t
V ε ε ε ε
+
=
∑ ∑ ∑
−∑
=
−
=
−
=
−
=
1
0
1
0 1
0
1
0
) ( ), ( cov
) ( ),
( cov
t
k
u
h sv sv
t
k
u
h mg
mg k ε h ε k ε h
ε
− −
+
=
∑
−∑
=
−
= 1
0
1
0
2 cov ( ) ( ), ( ) ( ),
t
a
u
b
sv sv
mg t a e a u b e b
tσ
( )
∑∑
−=
−
= − −
+
= 1
0 1
0
2 ( )( )cov ( ), ( )
t
a u
b
sv sv
mg t a u b e a e b
tσ
∑
−= − − +=tσmg2 σsv2 ta10(t a)(u a) 6 / ) 1 )(
1
2(
2 u t t t
t mg + sv + + +
= σ σ
It follows that
6 / ) 1 )(
1 (
) , 0 ( ), , 0 (
cov 2 2 2 2
~
~
t t t u t
t u
p t
p ≈ JO + ft + mg + sv + + +
σ σ σ σ ,
and
) ( )
, 0 (
var 2 2 2 2
~
t g t
t t
p ≈σJO + σft + σmg +σsv
First, for 1≤t≤16≤u≤32,
Next, Alho considers the second generation of births and their survival. That is when the births generated during first 16 years contribute new births, 17≤ t≤32. The contribution of the jump- off population, mortality and migration to the uncertainty of (0, )
~
t
p is eJO +V(t).
The contribution of fertility, however, consists of two parts. First, there is the direct contribution of fertility. Second, the women of the child bearing ages who were born after the jump-off year contribute to the uncertainty of fertility. This contribution is given by
) 1 (
) , ) (
, 0 1 ( ) (
2
15
−
− Β
= Ρ
∑
−=
j t t t j
t H
t
j
εft
Alho considers two types of covariances. First, for 1≤t≤16≤u≤32,
[
( ) ( ), ( ) ( ) ( )]
cov )
( ), , 0 ( cov
~
~
u H u u
V e t t
V e u
p t
p = JO + + f JO+ + ft +
ε ε
+ +
+ + + +
+
=σJO2 tσmg2 σsv2(u t 1)(t 1)t/6 tσft2
(
( ), ( ))
covεft t H u
The last covariance term is given by
( ) ∑
−[ ( ) ]
=
−
− Ρ Β
= 2
15
) 1 ( ), ( cov ) , ) (
, 0 1 ( ) ( ), ( cov
u
j
ft ft
ft j u t u j
u t H
t ε ε
ε
u u j u
{
t u j}
j
ft − −
Ρ Β
=
∑
−=
1 , min )
, ) (
, 0
1 ( 2
15
σ 2
And for 17≤t≤u≤32 this covariance is given by
[
() ( ) ( ), ( ) ( ) ( )]
cov )
( ), , 0 ( cov
~
~
u H u u
V e t H t t
V e u
p t
p = JO + + ft + JO + + ft +
ε ε
[ ]
++ +
+ + + +
+
=σJO2 tσmg2 σsv2(u t 1)(t 1)t/6 tσft2 covεft(t),H(u)
[
( ), ( )]
cov[
(), ( )]
covH t εft u + H t H u
The covariance terms are given as follows
[
(), ( )]
1 (0, ) ( , ) ( 1 )cov
2
15
2 u j
u u j
u H t
u
j
ft
ft = Ρ
∑
− Β − −=
σ
ε ,
[
( ), ( )]
1 (0, ) ( , ) ( 1 )cov
2
15
2 t j
t t j
u t H
t
j
ft
ft = Ρ
∑
− Β − −=
σ
ε , and
[ ]
= Ρ Ρ∑ ∑
− Β Β − −=
−
=
1 min{
) , ( ) , ) (
, 0 1 ( ) , 0 1 ( ) ( ), ( cov
2
15 2
15
2 t
u k t u j
u t H t H
t
j u
k
σft
j,u−1−k}
Alho considers the third generation of births, i.e. the years t=33,...,48. Again the authors splits the contribution of fertility into direct and indirect parts. The direct part is εft(t). The indirect part is given by
{ }
) 1 ( ) , ) (
, 0 1 ( )
( '
2 , 44 min
15
' j t H t j
t t D
t
j
−
− Ρ Β
=
∑
−=
If t−1− j≤16 then H'(t−1− j)=εft(t−1− j), and if t−1− j>16, then )
1 ( ) 1 ( ) 1
'(
j t H j t j
t
H − − =εft − − + − − . For t−1− j≤16 the covariance is given by
[
( 1 ), ( )]
( 1 )covH' t− − j εft t =σft2 t− − j For t−1− j >16, the covariance is given by
[
( 1 ), ()]
cov[
( 1 ) ( 1 ), ()]
covH' t− − j εft t = εft t− − j +H t− − j εft t
[
( 1 ), ( )]
cov ) 1
(t j H t j ft t
ft ε
σ − − + − −
=
+
−
−
=σft(t 1 j)
) 1 ( ) 1 , ) (
1 , 0
1 ( 2
3
15
j t j t j k
t ft
j t
k
−
−
−
−
− Β
−
Ρ
∑
−−=
σ
A.5 Theorems of population projection
This section discusses theorems of population projection. First, theorems developed by Cohen (1977) are described. Second, the theorems of Heyde and Cohen (1985) are viewed. This is followed by a discussion of results presented in Tuljapurkar (1990).
Cohen (1977) developed the ergodic theorems of demography. Given certain assumptions about the projection matrix, Α(t), ergodic theorems describe the long run behaviour of population size,
)
Ρ(t , and of age structure, q(t), and show that the behaviour of these quantities is independent of the initial conditions.
Cohen starts with a set of assumptions. First, the author assumes a finite number of age classes, l. Second, he considers a population subjected only to birth and death, with no immigration or emigration. Third, only one sex is considered, human females, and the vital rates refer to birth and death rates. It is further assumed that the age-specific vital rates apply to all individuals in an age class uniformly and equally. Finally, Cohen considers only large populations.
The author presents next a set of definitions. Cohen defines Ρ(t) as a non-negative vector representing the age census at time t, with Ρ( tj, ) representing the number of females at time t who will be j years old at their next birthday. The age structure, q(t), of an age census Ρ(t) is given by Ρ(t)/||Ρ(t)||, with ||q(t)||=1. Furthermore, Α(t) is defined as a sequence of operators mapping the non-negative l−vectors at one time to the non-negative vectors at the next time. In other words, Cohen considers the model
