Induced Distributions from Generalized Unfair Dice

Abstract

In this paper we analyze the probability distributions associated with rolling (possibly unfair) dice infinitely often. Specifically, given a $q$-sided die, if $x_i\in\{0,\ldots,q-1\}$ denotes the outcome of the $i^{\text{th}}$ toss, then the distribution function is $F(x)=\mathbb{P}[X\leq x]$, where $X = \sum_{i=1}^\infty x_i q^{-i}$. We show that $F$ is singular and establish a piecewise linear, iterative construction for it. We investigate two ways of comparing $F$ to the fair distribution -- one using supremum norms and another using arclength. In the case of coin flips, we also address the case where each independent flip could come from a different distribution. In part, this work aims to address outstanding claims in the literature on Bernoulli schemes. The results herein are motivated by emerging needs, desires, and opportunities in computation to leverage physical stochasticity in microelectronic devices for random number generation.