The Borel–Cantelli Lemmas, Probability Laws and Kolmogorov Complexity

Statistical Properties of Martin-Löf Random Sequences

“…The aim of this paper is to explore and prove these properties for Martin-Löf random sequences and to do some basic comparison with other definitions of randomness. Similar results can be found in previously published papers such as [1] and [7], but we hope that the reader will find our exposition to be more unified and accessible.…”

Computable Measure Theory and Algorithmic Randomness

“…Moreover, having an upper bound on this deficiency gives important information about x that usually cannot be recovered from x. For instance, Davie [27] showed how this additional information can be used to compute the speed of convergence in the strong law of large numbers or the law of the iterated logarithm, or to bound the number of events in the Borel-Cantelli lemma. More generally one can prove an effective version of Egorov's theorem, involving again Martin-Löf random points and their deficiencies [67].…”

Gambling strategies for random sequences

“…Ons kan selde vir 'n gegewe algoritmiese toevalsry, vir c vind, wat nie verbasend is nie, gegee dat sulke rye maksimaal onberekenbaar is. Dit is inderdaad nogal vreemd dat ons vir enige algoritmiese toevalsrye c kan vind, maar ons kan in sekere belangrike gevalle dit wel doen 10 .…”