Random Coin Tossing with unknown bias

TL;DR

This paper investigates whether the bias of a coin, determined by a renewal process with unknown parameters, can be almost surely estimated from observed sequences, providing confidence intervals and convergence conditions.

Contribution

It introduces a method to estimate the unknown bias in a coin-tossing experiment with renewal processes, solving an open problem and generalizing to $L^{2}$ random variables.

Findings

Confidence interval for bias $ heta$ constructed.

Almost sure convergence depends on recurrence properties.

Results apply to generalized $L^{2}$ sampling scenarios.

Abstract

Consider a coin tossing experiment which consists of tossing one of two coins at a time, according to a renewal process. The first coin is fair and the second has probability $1/2 + \theta$, $\theta \in [-1/2,1/2]$, $\theta$ unknown but fixed, of head. The biased coin is tossed at the renewal times of the process, and the fair one at all the other times. The main question about this experiment is whether or not it is possible to determine $\theta$ almost surely as the number of tosses increases, given only the probabilities of the renewal process and the observed sequence of heads and tails. We will construct a confidence interval for $\theta$ and determine conditions on the process for its almost sure convergence. It will be shown that recurrence is in fact a necessary condition for the almost sure convergence of the interval, although the convergence still holds if the process is null recurrent but the expected number of renewals up to and including time $N$ is $O(N^{1/2+\alpha}), 0 \leq \alpha < 1/2$. It solves an open problem presented by Harris and Keane (1997). We also generalize this experiment for random variables on $L^{2}$ which are sampled according to a renewal process from either one of two distributions.