Related papers: Bernoulli Sieve
The feedback class of a locally Brunovsky linear system is fully determined by the decomposition of state space as direct sum of system invariants [4]. In this paper we attack the problem of enumerating all feedback classes of locally…
We consider the problem of distilling uniform random bits from an unknown source with a given $p$-entropy using linear hashing. As our main result, we estimate the expected $p$-divergence from the uniform distribution over the ensemble of…
Randomness processing in the Bernoulli factory framework provides a concrete setting in which quantum resources can outperform classical ones. We experimentally demonstrate an entanglement-assisted quantum Bernoulli factory based on…
We propose an aproach for asymptotic analysis of plane partition statistics related to counts of parts whose sizes exceed a certain suitably chosen level. In our study, we use the concept of conjugate trace of a plane partition of the…
Consider a discrete time Markov process $X^\varepsilon$ on $\mathbb R^d$ that makes a deterministic jump prescribed by a map $\varphi \colon \mathbb R^d \to \mathbb R^d$, and then takes a small Gaussian step of variance $\varepsilon^2$. For…
A result of Chebyshev (1864) and Hoeffding1956}, on bounding an expectation of a given function with respect to a Bernoulli convolution (also called Poisson binomial law, or law of the number of successes in independent trials) with any…
We investigate the spacing distribution of sequence \[S_n=\left\{0,\frac{1}{n},\frac{2}{n},\dots,\frac{n-1}{n},1\right\}\] after Bernoulli sampling. We describe the closed form expression of the probability mass function of the spacings,…
Quantum superposition, a cornerstone of quantum mechanics, enables systems to exist in multiple states simultaneously, giving rise to probabilistic outcomes. In quantum information science, conditional entropy has become a key metric for…
We study the asymptotic density of the set of subscripts of the Bernoulli numbers having a given denominator. We also study the distribution of distinct Bernoulli denominators and some related problems.
Let $\mu$ be a probability measure (or corresponding random variable) such that all moments $\mu_n$ exist. Knowledge of the moments is not sufficient to determine infinite divisibility of the measure; we show also that infinitely divisible,…
The pentagonal number theorem is extended to the sequence of the number of integer partitions with all parts equal. The new pentagonal number theorem implies that the distribution of the primes is just a specific detail of the application…
We consider Markov chains on the space of (countable) partitions of the interval $[0,1]$, obtained first by size biased sampling twice (allowing repetitions) and then merging the parts with probability $\beta_m$ (if the sampled parts are…
One can consider $\mu$-Martin-L\"of randomness for a probability measure $\mu$ on $2^{\omega}$, such as the Bernoulli measure $\mu_p$ given $p \in (0, 1)$. We study Bernoulli randomness of sequences in $n^{\omega}$ with parameters $p_0,…
Stern's diatomic sequence with its intrinsic repetition and refinement structure between consecutive powers of $2$ gives rise to a rather natural probability measure on the unit interval. We construct this measure and show that it is purely…
We present a new sieve that allows us to find the prime numbers by using only regular patterns and, more importantly, avoiding any duplication of elements between them.
The factorially normalized Bernoulli polynomials $b_n(x) = B_n(x)/n!$ are known to be characterized by $b_0(x) = 1$ and $b_n(x)$ for $n >0$ is the antiderivative of $b_{n-1}(x)$ subject to $\int_0^1 b_n(x) dx = 0$. We offer a related…
There has been a resurgence of interest in incomplete U-statistics that only sum over a subset of kernel evaluations, due to their computational efficiency and asymptotic normality which can be leveraged to quantify the uncertainty of…
We introduce and study an infinite random triangulation of the unit disk that arises as the limit of several recursive models. This triangulation is generated by throwing chords uniformly at random in the unit disk and keeping only those…
We speculate on the distribution of primes in exponentially growing, linear recurrence sequences $(u_n)_{n\geq 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we…
We derive a central limit theorem for the mean-square of random waves in the high-frequency limit over shrinking sets. Our proof applies to any compact Riemannian manifold of arbitrary dimension, thanks to the universality of the local Weyl…