Related papers: Bernoulli Sieve
We give a nonstandard analytic proof of de Finetti's theorem for an exchangeable sequence of Bernoulli random variables. The theorem postulates that such a sequence is uniquely representable as a mixture of iid sequences of Bernoulli random…
A class of random discrete distributions $P$ is introduced by means of a recursive splitting of unity. Assuming supercritical branching, we show that for partitions induced by sampling from such $P$ a power growth of the number of blocks is…
We introduce new method for generating correlated or uncorrelated Bernoulli random variables by using the binary expansion of a continuous random variable with support on the unit interval. We show that when this variable has a symmetric…
A sequential importance sampling algorithm is developed for the distribution that results when a matrix of independent, but not identically distributed, Bernoulli random variables is conditioned on a given sequence of row and column sums.…
The bijection between composition structures and random closed subsets of the unit interval implies that the composition structures associated with $S \cap [0,1]$ for a self-similar random set $S\subset {\mathbb R}_+$ are those which are…
We say that a string of length $d$ occurs, in a Bernoulli sequence, if a success is followed by exactly $(d-1)$ failures before the next success. The counts of such $d$-strings are of interest, and in specific independent Bernoulli…
Scale invariant scattering suggests that all Bernoulli numbers B_{2n} can be naturally partitioned, i.e., written as particular finite sums of same-signed, monotonic, rational numbers. Some properties of these rational numbers are discussed…
It is known that backward iterations of independent copies of a contractive random Lipschitz function converge almost surely under mild assumptions. By a sieving (or thinning) procedure based on adding to the functions time and space…
The asymptotics, as $n\to\infty$, for the expected number of distinct part sizes in a random composition of an integer n is obtained.
In this paper we show the distributions of sliding block patterns for Bernoulli processes with finite alphabet, which is not based on the induction on sample size. We show a new inclusion-exclusion formula in multivariate generating…
A cornerstone of human statistical learning is the ability to extract temporal regularities / patterns from random sequences. Here we present a method of computing pattern time statistics with generating functions for first-order Markov…
Let $ A_n $ be an $n \times n$ random matrix with i.i.d Bernoulli($p$) entries. For a fixed positive integer $\beta$, suppose $p$ satisfies $$ \frac{ \log(n) }{ n } \le p \le c_\beta $$ where $c_\beta \in ( 0, 1/2 )$ is a…
A Bernoulli factory is an algorithmic procedure for exact sampling of certain random variables having only Bernoulli access to their parameters. Bernoulli access to a parameter $p \in [0,1]$ means the algorithm does not know $p$, but has…
If $S$ is a cofinite set of positive integers, an "$S$-restricted composition of $n$" is a sequence of elements of $S$, denoted $\vec{\lambda}=(\lambda_1,\lambda_2,...)$, whose sum is $n$. For uniform random $S$-restricted compositions, the…
The topic of this paper is the distributed and incremental generation of long executions of concurrent systems, uniformly or more generally with weights associated to elementary actions. Synchronizing sequences of letters on alphabets…
The fragmentation processes of exchangeable partitions have already been studied by several authors. In this paper, we examine rather fragmentation of exchangeable compositions, that means partitions of $\mathbb{N}$ where the order of the…
A novel multinomial theorem for commutative idempotents is shown to lead to new results about the moments, central moments, factorial moments, and their generating functions for any random variable $X = \sum_{i} Y_i $ expressible as a sum…
Sylvester showed that the partition of an integer into a set of positive integers can be represented as a sum of the polynomial term and quasiperiodic components called the Sylvester waves. The wave itself is a weighted sum of the…
Consider the random quadratic form $T_n=\sum_{1 \leq u < v \leq n} a_{uv} X_u X_v$, where $((a_{uv}))_{1 \leq u, v \leq n}$ is a $\{0, 1\}$-valued symmetric matrix with zeros on the diagonal, and $X_1,$ $X_2, \ldots, X_n$ are i.i.d.…
A nested occupancy scheme in random environment is a generalization of the classical Karlin infinite balls-in-boxes occupancy scheme in random environment (with random probabilities). Unlike the Karlin scheme in which the collection of…