Some Results of Probabilities
Nikolaos Limnios · 2007
Some Results of ProbabilitiesThis appendix contains the elements of the theory of probabilities that are necessary in the statement of the Monte Carlo method.Proposition C.1 (Chebychev inequality) Consider a square intergrable r.v X.Then for all ε > 0, we have: Definition C.2 (i.i.d sequences)A sequence of r.v (X n , n ≥ 1) defined on the probability space (Ω, F, )ސ is said to be i.i.d, when any finite sequence extracted from this sequence is of the i.i.d type Set Proposition C.3 (Week law of large numbers: Khintchine theorem)Let there be a sequence of r.v (X n , n ≥ 1) i.i.d with |ޅX 1 | < ∞.When n → ∞, we have