Advances in Risk Management

JEAN-PAUL PAQUIN, ANNICK LAMBERT AND ALAIN CHARBONNEAU 295

To simplify the notation, let us defineαt=(1+kc)−t. Equation (1) becomes:

P ̃=

∑n

t= 1

αtε ̃t

We must therefore verify, in accordance with the CLT, whether this weighted average
of random error terms converges towards a normal probability distribution. Given our
initial assumptions we deduct:

E(P ̃)=

∑n

t= 1

αtE(ε ̃t)= 0

whereas

V(P ̃)=

∑n

t= 1

α^2 tV(ε ̃t)=

∑n

t= 1

α^2 t.

Therefore, to verify the CLT we must demonstrate that:

nlim→∞

∑n

t= 1

αt ̃εt

√

∑n

t= 1

α^2 t

→N(0, 1)

LetφX ̃(h)=E(eihX ̃)= 1 +

∑∞
t= 1
(ih)t
t!μtbe the characteristic function of any random vari-
ableX ̃. Given that theε ̃t’s are independent in probability, we may write the characteristic
function of their weighted sum in term of their argument as:

φ∑n

t= 1

αtε ̃t

√∑n

t= 1

α^2 t

(h)=E

(

e

ih√∑∑αt ̃εt

α^2 t

)

=

∏n

t= 1

e

√ihα∑tε ̃t

α^2 t=

∏n

t= 1

φ ̃ε



√αth

∑

α^2 t





Let us take the logarithm of the characteristic function in term of its arguments and thus
define thefunction:

√∑αt ̃εt

∑α 2

t

=

∑n

t= 1

logφε ̃t







αth

√∑

α^2 t







=

∑n

t= 1

log



 1 +i√αt

∑

α^2 t

hμ 1 −

1

2



√αt

∑

α^2 t





2

h^2 μ 2

−

i

3!



√αt

∑

α^2 t





3

h^3 μ 3 +

1

4!



√αt

∑

α^2 t





4

h^4 μ 4 +...





