300 NPV PROBABILITY DISTRIBUTION OF RISKY INVESTMENTSWe may express the logarithm of the characteristic function in terms of its cumulants:
√∑wtu ̃t
∑w 2
t=∑nt= 1logφ ̃εt
wth
√∑
w^2 t
=∑n
t= 1
i√wt
∑
w^2 thK 1 −
1
2
√wt
∑
w^2 t
2
h^2 K 2−
i
3!
wt
√∑
w^2 t
3
h^3 K 3 +
1
4!
αt
√∑
α^2 t
4
h^4 K 4 +...
By assumption, we have setK 1 =0etK 2 =1 whereas
∑n
t= 1
(
√∑wt
w^2 t) 2
=1. Therefore: √∑wtu ̃t
∑w 2
t=∑nt= 1logφε
√wth
∑
w^2 t
=−h^2
2
−
i
3!∑nt= 1
√wt
∑
w^2 t
3
h^3 K 3+
1
4!∑n
t= 1
wt
√∑
w^2 t
4
h^4 K 4 +...The limit value of the logarithm of the characteristic function becomes:
nlim→∞ √∑wtu ̃t
∑w 2
t=nlim→∞∑nt= 1logφ ̃ε
wth
√∑
w^2 t
=nlim→∞
−h^2
2
−
i
3!∑nt= 1
wt
√∑
w^2 t
3
h^3 K 3+
1
4!∑n
t= 1
wt
√∑
w^2 t
4
h^4 K 4 +...
Furthermore, given 0≤ρ<1, it follows that:
nlim→∞w^21
∑n
t= 1w^2 t=nlim→∞
(1−ρn−t+^1 )
∑n
t= 1(1−ρn−t+^1 )^2=lim
n→∞(1−ρn−t+i)
∑n
t= 1(1+ρ2(n−t+1)− 2 ρn−t+^1 )=nlim→∞
1
n
= 0