TABLE OF CONTENTS This appendix contains the elements of the theory of probabilities that are necessary in the statement of the Monte Carlo method.
Consider a square intergrable r.v X. Then for all ? > 0, we have:
| (C.1) |  |
A sequence of r.v ( X n, n ? 1) defined on the probability space ( ?, 
, ?) is said to be i.i.d, when any finite sequence extracted from this sequence is of the i.i.d type Set S n = X 1 + ... + X n. n ? 1.
Let there be a sequence of r.v ( X n, n ? 1) i.i.d with 
. When n ? ?, we have
| (C.2) |  |
Let there be a sequence of r.v ( X n, n ? 1) i.i.d. Then, 
is a necessary and sufficient condition such that the sequence ( X n) verifies the law of large numbers, i.e. when n ? ?, we have
| (C.3) |  |
For a series of independent events ( A n, n ? 1) and of the same probability p, the strong law of large numbers says that the frequencies 
converge almost surely (a.s) towards p when n ? ?. This result justifies the estimation of the probabilities through the frequencies.
