Monthly Archives: November 2011

Characterization of Kurtz randomness by a differentiation theorem

Title
Characterization of Kurtz randomness by a differentiation theorem

Type
Talk

This talk is in CS Seminar at Kyoto University on 17 Nov 2011.
The same talk will be given in ALC 2011.

Download

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Convergence of random series and the rate of convergence of the strong law of large numbers in game-theoretic probability

News
Oct 25, 2011, Available on line
Oct 18, 2011, Accepted
April 5, 2011, Submitted

Title
Convergence of random series and the rate of convergence of the strong law of large numbers in game-theoretic probability

Type
Fullpaper

Conference and Journal
Stochastic Processes and their Applications, 122:1-30, 2012.
Journal
arXiv

Abstract
We give a unified treatment of the convergence of random series and the rate of convergence of strong law of large numbers in the framework of game-theoretic probability of Shafer and Vovk (2001). We consider games with the quadratic hedge as well as more general weaker hedges. The latter corresponds to existence of an absolute moment of order smaller than two in the measure-theoretic framework. We prove some precise relations between the convergence of centered random series and the convergence of the series of prices of the hedges. When interpreted in measure-theoretic framework, these results characterize convergence of a martingale in terms of convergence of the series of conditional absolute moments. In order to prove these results we derive some fundamental results on deterministic strategies of Reality, who is a player in a protocol of game-theoretic probability. It is of particular interest, since Reality’s strategies do not have any counterparts in measure-theoretic framework, ant yet they can be used to prove results, which can be interpreted in measure-theoretic framework.

You can read a comment by Vovk at Working Papers in
the website of “Probability and Finance”.

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