Uniform law of large numbers (Q1075683)

A new concept, called uniform law of large numbers, is introduced in this paper. The basic concept is: \[ n^{-1}\sum^{n+j-1}_{k=j}(X_ k- EX_ k)\to 0\quad in\quad \Pr.\quad (a.s.),\quad as\quad n\to \infty,\quad uniformly\quad in\quad j. \] By means of this new concept, the authors generalize some known results on the convergence problem of matrix sums of independent random variables.