Stabilité de sommes pondérées de variables aléatoires vectorielles

AbstractStochastic convergence and almost sure convergence are studied for weighted sums Sn = Σk = 1kn ankXk of random variables taking values in a Banach space B. These convergence properties are obtained under various regularity assumptions on the sequence (Xn). For instance, the following set of hypotheses is put on the random variables (Xn): (i) Σk = 1kn |ank|α ≤ Π < ∞; (ii) limn→∞supk=1kn|ank| = 0; (iii) lims→∞supnsupt ⩾- s tαP[∥Xn∥ ≥ t] = 0; (iv) B is of α-stable type