Advances in Risk Management

JEAN-PAUL PAQUIN, ANNICK LAMBERT AND ALAIN CHARBONNEAU 297

Given that the cumulants of order higher than 2 generally are not equal to zero, it then
follows that:

nlim→∞√∑αtε ̃t

∑α 2

t

=nlim→∞

∑n

t= 1

logφ ̃ε







αth

√∑

α^2 t







=nlim→∞





−

h^2

2

−

i

3!

∑n

t= 1







αt

√∑

α^2 t







3

h^3 K 3

+

1

4!

∑n

t= 1







αt

√∑

α^2 t







4

h^4 K 4 +...





=−

h^2

2

However, when the discount rate is set equal to zero then:

nlim→∞

α^21

∑n

t= 1

α^2 t

=

1

n

and the logarithm of the characteristic function in terms of its cumulants can be written as:

√∑αt ̃εt

∑α 2

t

=

∑n

t= 1

logφε ̃

(

αth

∑

α^2 t

)

=−

h^2

2

−

i

3!

∑n

t= 1

(

1

√

n

) 3

h^3 K 3

+^1

4!

∑n

t= 1

(

√^1

n

) 4

h^4 K 4 +...

Consequently, its limit value can be written as:

nlim→∞√∑αtε ̃t

∑α 2

t

=nlim→∞

∑n

t= 1

logφ ̃ε

(

αth

∑

α^2 t

)

=−

h^2

2

and therefore limn→∞φ ̃ε

(

∑αth

α^2 t

)

=e−

h 22

, which is the characteristic function of the Normal

probability distribution.

APPENDIX 2: THE CLT AND THE FIRST-ORDER

AUTOREGRESSIVE PROCESS

We consider a weighted sum of random cash flowsX ̃tsuch that each variate has an equal
weight. We thus define the random meanX ̃ ̄as the sum ofnequally weighted random
cash flowsX ̃tas:

X ̃ ̄=

∑n

t= 1

X ̃t

n

(A.1)

for whichX ̃t=μX+ ̃εt, fort=1, 2, 3...,n.