Suppose that a coin with bias theta is tossed at renewal times of a renewal process, and a fair coin is tossed at all other times. Let mu_θbe the distribution of the observed sequence of coin tosses, and let u_n denote the chance of a renewal at time n. Harris and Keane showed that if sum_{n=1}^infty u_n^2=\infty, then mu_theta and μ_0 are singular, while if sum_{n=1}^{infty} u_n^2<infty and theta is small enough, then mu_theta is absolutely continuous with respect to mu_0. They conjectured that absolute continuity should not depend on theta, but only on the square-summability of {u_n}. We show that in fact the power law governing the decay of {u_n} is crucial, and for some renewal sequences {u_n}, there is a {phase transition at a critical parameter theta_c in (0,1): for |theta|<theta_c the measures mu_theta$ and mu_0 are mutually absolutely continuous, but for |theta|>theta_c, they are singular. We also prove that when u_n=O(n^{-1}), the measures mu_theta for theta in [-1,1] are all mutually absolutely continuous.
A classical result of K. L. Chung and W. Feller deals with the partial sums $S_k$ arising in a fair coin-tossing game. If $N_n$ is the number of "positive" terms among $S_1, S_2,\dots,S_n$ then the quantity $P(N_{2n}=2r)$ takes an elegant form. We lift the restriction on an even number of tosses and give a simple expression for $P(N_{2n+1}=r)$, $r=0,1,2,\dots,2n+1$. We get to this result by adapting the Feynman-Kac methodology.
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 $1/2 + θ$, $θ\in [-1/2,1/2]$, $θ$ 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 $θ$ 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 co…
We prove a functional limit theorem for Markov chains that, in each step, move up or down by a possibly state dependent constant with probability $1/2$, respectively. The theorem entails that the law of every one-dimensional regular continuous strong Markov process in natural scale can be approximated with such Markov chains arbitrarily well. The functional limit theorem applies, in particular, to Markov processes that cannot be characterized as solutions to stochastic differential equations. Our results allow to practically approximate such processes with irregular behavior; we illustrate thi…
What is the average number of tosses needed before a particular sequence of heads and tails turns up? We solve the problem didactically, starting with doubles, finding that a tail, followed by a head, turns up on the average after only four tosses, while six tosses are needed for two successive heads. The method is extended to encompass the triples head-tail-tail and head-head-tail, but head-tail-head and head-head-head are surprisingly more recalcitrant. However, the general case is finally solved by a new algorithm that allows a simple computation that can be done by hand, even for relativel…
Identifying several biased coins encountered by a hidden random walk
假设在Z中的每一个位置z上都有一个带有偏差theta（z）的硬币，并且这些硬币中只有20%的硬币具有非零偏差。允许一个简单的随机步行者通过在每次移动时抛出与当前位置相关的硬币来生成观察结果。然后，我们可以确定偏差{theta（z）：z in Z}，只使用这些硬币投掷的结果，而不使用关于随机行走者的路径的信息，直到Z的移动和反射。这推广了Harris和Keane的一个结果。
We determine the convergence speed of a numerical scheme for approximating one-dimensional continuous strong Markov processes. The scheme is based on the construction of coin tossing Markov chains whose laws can be embedded into the process with a sequence of stopping times. Under a mild condition on the process' speed measure we prove that the approximating Markov chains converge at fixed times at the rate of $1/4$ with respect to every $p$-th Wasserstein distance. For the convergence of paths, we prove any rate strictly smaller than $1/4$. Our results apply, in particular, to processes with …
In this paper we study the small scale equidistribution property of random waves whose coefficients are determined by an unfair coin. That is the coefficients take value $+1$ with probability $p$ and $-1$ with probability $1-p$. Random waves whose coefficients are associated with a fair coin are known to equidistribute down to the wavelength scale. We obtain explicit requirements on the deviation from the fair ($p=0.5$) coin to retain equidistribution.
