A relaxed randomized Kaczmarz algorithm is investigated in a least squares regression setting by a learning theory approach. When the sampling values are accurate and the regression function (conditional means) is linear, such an algorithm has been well studied in the community of non-uniform sampling. In this paper, we are mainly interested in the different case of either noisy random measurem...

Cantelli was born in Palermo, Sicily. He attended the local university where he graduated in 1899 in mathematics with a thesis on celestial mechanics. Cantelli’s advisor, Filippo Angelitti, was Director of the Palermo Astronomical Observatory, and he began working as Angelitti’s assistant while still a student. Cantelli remained at the Observatory until 1903. His first publications were about v...

In [6], Walter Philipp wrote that “. . . the law of the iterated logarithm holds for any process for which the Borel-Cantelli Lemma, the central limit theorem with a reasonably good remainder and a certain maximal inequality are valid.” Many authors [1, 2, 4, 5, 9] have followed this plan in proving the law of the iterated logarithm for sequences (or fields) of dependent random variables. We ca...

In the general context of computable metric spaces and computable measures we prove a kind of constructive Borel-Cantelli lemma: given a sequence (recursive in some way) of sets A i with recursively summable measures , there are computable points which are not contained in infinitely many A i. As a consequence of this we obtain the existence of computable points which follow the typical statist...

We present a construction showing that a class of sets C that is Glivenko-Cantelli for an i.i.d. process need not be Glivenko-Cantelli for every stationary ergodic process with the same one dimensional marginal distribution. This result provides a counterpoint to recent work extending uniform strong laws to ergodic processes, and a recent characterization of universal Glivenko Cantelli classes.

In the general context of computable metric spaces and computable measures we prove a kind of constructive Borel-Cantelli lemma: given a sequence (constructive in some way) of sets Ai with effectively summable measures, there are computable points which are not contained in infinitely many Ai. As a consequence of this we obtain the existence of computable points which follow the typical statist...