Identifying several biased coins encountered by a hidden random walk

首次通过时间的有偏简单随机游走的胜率

Win rates at first-passage times for biased simple random walks

研究有偏简单随机游走首次通过时间的胜率，并通过闭式表达式设计无偏硬币翻转估计器来估计π。

Streaming Algorithms for Optimal Generation of Random Bits

Generating random bits from a source of biased coins (the biased is unknown) is a classical question that was originally studied by von Neumann. There are a number of known algorithms that have asymptotically optimal information efficiency, namely, the expected number of generated random bits per input bit is asymptotically close to the entropy of the source. However, only the original von Neumann algorithm has a `streaming property' - it operates on a single input bit at a time and it generates random bits when possible, alas, it does not have an optimal information efficiency. The main contr…

Detecting the trail of a random walker in a random scenery

Suppose that the vertices of the Euclidean lattice Z^d are endowed with a random scenery, obtained by tossing a fair coin at each vertex. A random walker, starting from the origin, replaces the coins along its path by i.i.d. biased coins. For which walks and dimensions can the resulting scenery be distinguished from the original scenery? We find the answer for simple random walk, where it does not depend on dimension, and for walks with a nonzero mean, where a transition occurs between dimensions three and four. We also answer this question for other types of graphs and walks, and raise severa…

Trees of self-avoiding walks

We consider the biased random walk on a tree constructed from the set of finite self-avoiding walks on a lattice, and use it to construct probability measures on infinite self-avoiding walks. The limit measure (if it exists) obtained when the bias converges to its critical value is conjectured to coincide with the weak limit of the uniform SAW. Along the way, we obtain a criterion for the continuity of the escape probability of a biased random walk on a tree as a function of the bias, and show that the collection of escape probability functions for spherically symmetric trees of bounded degree…

Concentration of measures via size biased couplings

Let $Y$ be a nonnegative random variable with mean $μ$ and finite positive variance $σ^2$, and let $Y^s$, defined on the same space as $Y$, have the $Y$ size biased distribution, that is, the distribution characterized by E[Yf(Y)]=μE f(Y^s) for all functions $f$ for which these expectations exist. Under a variety of conditions on the coupling of Y and $Y^s$, including combinations of boundedness and monotonicity, concentration of measure inequalities hold. Examples include the number of relatively ordered subsequences of a random permutation, sliding window statistics including the number of m…

Estimating Bernoulli trial probability from a small sample

The standard textbook method for estimating the probability of a biased coin from finite tosses implicitly assumes the sample sizes are large and gives incorrect results for small samples. We describe the exact solution, which is correct for any sample size.

Random Coin Tossing with unknown bias

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…

Bernoulli trials of fixed parity, random and randomly oriented graphs

Suppose you can color $n$ \emph{biased} coins with $n$ colors, all coins having the same bias. It is forbidden to color both sides of a coin with the same color, but all other colors are allowed. Let $X$ be the number of different colors after a toss of the coins. We present a method to obtain an upper bound on a median of $X$. Our method is based on the analysis of the probability distribution of the number of vertices with even in-degree in graphs whose edges are given random orientations. Our analysis applies to the distribution of the number of vertices with odd degree in random sub-graphs…

Voting Rights, Markov Chains, and Optimization by Short Bursts

Finding outlying elements in probability distributions can be a hard problem. Taking a real example from Voting Rights Act enforcement, we consider the problem of maximizing the number of simultaneous majority-minority districts in a political districting plan. An unbiased random walk on districting plans is unlikely to find plans that approach this maximum. A common search approach is to use a biased random walk: preferentially select districting plans with more majority-minority districts. Here, we present a third option, called short bursts, in which an unbiased random walk is performed for…