A proof of the martingale convergence theorem

Polya’s Urn and the Martingale Convergence Theorem

This paper is about Polya’s Urn and the Martingale Convergence Theorem. I will start with the formal definition, followed by a simple example of martingale and the basic properties of martingale. Then I will explain the Polya’s Urn model and how it contributes to proving the Martingale Convergence Theorem. Finally, I will give a full proof of the Martingale Convergence Theorem.

Algorithmic randomness for Doob's martingale convergence theorem in continuous time

We study Doob’s martingale convergence theorem for computable continuous time martingales on Brownian motion, in the context of algorithmic randomness. A characterization of the class of sample points for which the theorem holds is given. Such points are given the name of Doob random points. It is shown that a point is Doob random if its tail is computably random in a certain sense. Moreover, D...

A Martingale Decomposition Theorem

Another proof of Banaschewski's surjection theorem

We present a new proof of Banaschewski's theorem stating that the completion lift of a uniform surjection is a surjection. The new procedure allows to extend the fact (and, similarly, the related theorem on closed uniform sublocales of complete uniform frames) to quasi-uniformities ("not necessarily symmetric uniformities"). Further, we show how a (regular) Cauchy point on a closed uniform subl...

A Concise, Elementary Proof of Arzelà's Bounded Convergence Theorem

Arzelà's Bounded Convergence Theorem (1885) states that if a sequence of Riemann integrable functions on a compact interval is uniformly bounded and has an integrable pointwise limit function, then the sequence of their integrals tends to the integral of the limit. It is a trivial consequence of measure theory; in particular, of the Dominated Convergence Theorem. However, denying oneself the ma...