Given a triangular array a={an,k,1[less-than-or-equals, slant]k[less-than-or-equals, slant]kn,n[greater-or-equal, slanted]1} of positive reals, we study the complete convergence property of for triangular arrays of independent random variables. In the Gaussian case we obtain a simple...

We prove a.s. limit theorems corresponding to the classical Darling-Erdös theorem for the maxima of normalized partial sums of i.i.d. random variables. Our results yield the analogue of the a.s. central limit theorem for the Darling-Erdös max functional and its variants. Unlike in standard...

The discovery of the almost sure central limit theorem (Brosamler, Math. Proc. Cambridge Philos. Soc. 104 (1988) 561-574; Schatte, Math. Nachr. 137 (1988) 249-256) revealed a new phenomenon in classical central limit theory and has led to an extensive literature in the past decade. In particular,...

We investigate the law of large numbers with exceptional n-sets, i.e. when the theorem is required to hold only for almost all n, in the sense of a suitable measure on the integers. We prove the surprising result that in the presence of such exceptional sets, the weak and strong laws of large...