On complete convergence of triangular arrays of independent random variables

István Berkes , Michel Weber (1)

1 IRMA - Institut de Recherche Mathématique Avancée

Given a triangular array a={an,k,1⩽k⩽kn,n⩾1}a={an,k,1⩽k⩽kn,n⩾1} of positive reals, we study the complete convergence property of View the MathML sourceTn=∑k=1knan,kXn,k for triangular arrays X={Xn,k,1⩽k⩽kn,n⩾1}X={Xn,k,1⩽k⩽kn,n⩾1} of independent random variables. In the Gaussian case we obtain a simple characterization of density type. Using Skorohod representation and Gaussian randomization, we then derive sufficient criteria for the case when Xn,kXn,k are in LpLp, and establish a link between the LpLp-case and L2pL2p-case in terms of densities. We finally obtain a density type condition in the case of uniformly bounded random variables.

Mots clés: Physical Sciences

Domaines: Probabilités [math.PR]

István Berkes, Michel Weber. On complete convergence of triangular arrays of independent random variables. Statistics and Probability Letters, 2009, 77 (10), pp.952. ⟨10.1016/j.spl.2006.12.011⟩. ⟨hal-00538011⟩