2023/09/14 by Douglas T. Pfeffer, Pfeffer, Douglas T., J. Darby Smith +3
Computer Science · Mathematics · #60E05 #Analytic Number Theory Research #Benford’s Law and Fraud Detection #Chaos-based Image/Signal Encryption #FOS: Mathematics #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2309.07366
openalex publication_date 2023/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we analyze the probability distributions associated with rolling (possibly unfair) dice infinitely often. Specifically, given a q-sided die, if xi∈\0,…,q-1\ denotes the outcome of the ith toss, then the distribution function is F(x)=ℙ[X≤ x], where X = ∑i=1^∞ xi 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.