Advances in Risk Management

JEAN-PAUL PAQUIN, ANNICK LAMBERT AND ALAIN CHARBONNEAU 299

and

̃X=^1

n(1−ρ)

[u ̃ 1 (1−ρn)+u ̃ 2 (1−ρn−^1 )+u ̃ 3 (1−ρn−^2 )+···

+u ̃n− 2 (1−ρ^3 )+u ̃n− 1 (1−ρ^2 )+u ̃n(1−ρ)]

By settingwt= 1 −ρn−t+^1 the mean random variable becomes equal to:

̃X=^1

n(1−ρ)

∑n

t= 1

wtu ̃t (A.5)

Let us now demonstrate that the weighted sum of random variablesu ̃tobeys the CLT.
Given the initial assumptions governing the randomu ̃t’s, we obtain:

E( ̃X)=^1

n(1−ρ)

∑n

t= 1

wtE(u ̃t)= 0

and

V(X ̃)=

1

n^2 (1−ρ)^2

∑n

t= 1

w^2 tV(u ̃t)=

1

n^2 (1−ρ)^2

∑n

t= 1

w^2 t

The CLT will be established once we show that:

nlim→∞

∑n

t= 1

wtu ̃t

√

∑n

t= 1

w^2 t

→N(0, 1)

Given that theu ̃t’s are independent in probability, then the characteristic function is
equal to:

φ∑n

t= 1

wt ̃ut

√∑n

t= 1

w^2 t

(h)=E

(

e

ih√∑∑wt ̃ut

w^2 t

)

=

∏n

t= 1

e

√ihw∑t ̃ut

w^2 t =

∏n

t= 1

φu ̃



√wth

∑

w^2 t





The logarithm of the characteristic function then becomes:

√∑wtu ̃t

∑w 2

t

=

∑n

t= 1

logφu ̃t







wth

√∑

w^2 t







=

∑n

t= 1

log



 1 +i√wt

∑

w^2 t

hμ 1 −

1

2



√wt

∑

w^2 t





2

h^2 μ 2

−

i

3!



√wt

∑

w^2 t





3

h^3 μ 3 +

1

4!



√wt

∑

wt^2





4

h^4 μ 4 +...





