Let (Ω,F,P) be a probability space and (Xn) ⊂ L1(P) and (Yn) be sequences of positive random variables which are adapted to a filtration (Fn). Suppose that the infinite sum: Y1+Y2+...< ∞ and E[Xn+1|Fn] ≤ (1 + Yn)*Xn, ∀n ∈N0, P-a.e.
And I have to Show know that (Xn) converges P-a.e.
We allready had the Theorem: Let X = (Xn) be a supermartingale with sup E[|Xn|] < ∞. Then Xn → X∞ a.s. as n →∞, for some X∞ ∈ L1
Is this usefull here? My thougt was to use that Yn converge to Zero so for all c>0 there exist a N(c) such that for every n>N follows E[Xn+1|Fn] ≤ (1 + c)*Xn and now define X´j= X(N(c)+j) so for c running to Zero than X´ is a supermartingale right ? i dont know if that kind of arguing is sensefull, corrections would be appreciated.
and know there would be left to Show that X´ holds sup E(|X´n|) < infinite but i dont know how to Show it.. maybe someone has an idea? or an other solution idea ?
PS. sry for my bad english :P
Best Regards
