Random sampling of continuous objects

Abstract

Randomization is a very powerful tool which can reduce the complexity of a lot of algorithms in practice. However, to implement a randomized algorithm, sampling procedure from the fixed probability distribution is essential. In this paper, we formalized the sampling procedure from continuous probability distribution in the sense of Type-2 Theory of Effectivity which is theoretical framework to handle the computation over continuous data. We first show that every Borel probability measure on second countable $T_0$ spaces is just a push-forward measure of Canonical probability measure on Cantor Space. This is the extension of the result by Simpson and Schr$\ddot{o}$der, 2006. Second result is concerned with Brownian motion as the probability measure on the space of continuous function $f$ :[0,1]-> $\mathbb{R}$ with $f$(0)=0. We figure out the condition when such a measure can be sampled. This measure can be sampled if and only if its family of modulus of continuity can be sampled.