By Birger Iversen

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Additional resources for Ecole D'Ete de Probabilites: Processus Stochastiques

Sample text

Soit ( X v } une suite de variables aleatoires independantes ay ant la mime fonction de repartition* Alors enposant S„ = ^y X v , la probability P ( S „ € M pour une infinite de valeurs de n) est egale a o ou i suivant que la serie n= l est convergente ou divergente. Dimonslration. — Le cas de convergence est une consequence immediate du theoreme classique de Borel-Cantelli. Considerons done le cas de divergence. Designonspar E„ l'evenement S „ e M et en general par E' le complement de E. Nous avons, d'une m a n u r e generate, E „ = ( E , + E ; E2 + .

For the sake of brevity we introduce the following symbolic notation: ( 1, if Si is favorable to E2 Ei/Ei = \ 0, if Ex is indifferent to E2 [—1, if Ei is unfavorable to E2. Then by (ii) and (iii) we have Ei/E, = E2/Ex, E[/E2 = E2/E[ = Ei/Et = ft/Ei. = E[/E2 = E'2/E[ = -{EYfE2), Ei/E,, analogous to the rules of signs in the multiplication of integers. i is favorable to E3 ; in fact, it may happen that Ei is unfavorable to E3. For instance, imagine 11 identical balls in a bag marked respectively with the numbers - 1 1 , - 1 0 , - 3 , - 2 , - 1 , 2, 4, 6, 11, 13, 16.

For k = 1, • • • , n — 1 and 1 g m ^ k we have PROOF. Substitute (13) and a similar formula for & + 1 into the two sides respectively. After this substitution we observe that the number of terms is the same on both sides, since (n — m\f \k -m)\k n \ A + l \ _ / n —m + l)\ m ) ~ \k + 1 - \ A \ A \ m)\k)\m)' Also, the number of terms with a given U = (MI , • • • , Mm) unaccented is the same, since — m\ / n — m \ _ / n — m \ / n — m\ n— — mj \k + 1 — m) \k + 1 — mj \k — m)' k Let the sum of all the terms with U unaccented in the two summations be denoted by ak+i =