Stabilité de sommes pondérées de variables aléatoires vectorielles. (Stability of weighted sums of vector random variables)
This paper studies the stochastic convergence and the almost sure behaviour of weighted sums of random variables of the form \(S_ n=\sum^{k=k_ n}_{k=1}a_{nk}X_ k\) where \(a_{nk}\) are scalars and \(X_ k\) are independent random variables with values in a Banach space B. The following set of assumptions is used: \[ \sum^{k_ n}_{k=1}| a_{nk}|^{\alpha}\leq \Gamma <\infty,\quad \lim_{n\to \infty}\sup_{1\leq k\leq k_ n}| a_{nk}| =0, \] \[ \lim_{s\to \infty}\sup_{n}\sup_{t\geq s}P\{\| X_ n\| \geq t\}=0. \] Then if B is of stable type \(\alpha\) \((1<\alpha <2)\) and if the variables are mean zero, the sequence \(S_ n\) converges to 0 in probability. A similar conclusion is obtained under various variants of the above assumptions. The simpler case \(\alpha <1\) is also considered (then, independence is no longer essential). Moreover, the author gives a condition under which the stochastic convergence (i.e. convergence in probability) of the sequence \(S_ n\) to 0 implies its almost sure convergence to 0.
scientific article; zbMATH DE number 4109713