Related papers: Random Dirichlet series arising from records

Beta regression has been extensively used by statisticians and practitioners to model bounded continuous data and there is no strong and similar competitor having its main features. A class of normalized inverse-Gaussian (N-IG) process was…

Simplex-valued data appear throughout statistics and machine learning, for example in the context of transfer learning and compression of deep networks. Existing models for this class of data rely on the Dirichlet distribution or other…

We develop a local theory of lacunary Dirichlet series of the form $\sum\limits_{k=1}^{\infty}c_k\exp(-zg(k)), \Re(z)>0$ as $z$ approaches the boundary $i\RR$, under the assumption $g'\to\infty$ and further assumptions on $c_k$. These…

According to a theorem of S. Schumacher, for a diffusion X in an environment determined by a stable process that belongs to an appropriate class and has index a, it holds that X_t/(log t)^a converges in distribution, as t goes to infinity,…

A new family of probability distributions $\beta_{M, N},$ $M=0\cdots N,$ $N\in\mathbb{N}$ on the unit interval $(0, 1]$ is defined by the Mellin transform. The Mellin transform of $\beta_{M, N}$ is characterized in terms of products of…

We consider a class of random loop models (including the random interchange process) that are parametrised by a time parameter $\beta\geq 0$. Intuitively, larger $\beta$ means more randomness. In particular, at $\beta=0$ we start with loops…

Dirichlet process mixtures are particularly sensitive to the value of the precision parameter controlling the behavior of the latent partition. Randomization of the precision through a prior distribution is a common solution, which leads to…

Conjugate pairs of distributions over infinite dimensional spaces are prominent in statistical learning theory, particularly due to the widespread adoption of Bayesian nonparametric methodologies for a host of models and applications. Much…

For an observed response that is composed by a set - or vector - of positive values that sum up to 1, the Dirichlet distribution (Bol'shev, 2018) is a helpful mathematical construction for the quantification of the data-generating mechanics…

Let $s$ be the sum-of-digits function in base $2$, which returns the number of $\mathtt 1$s in the base-2 expansion of a nonnegative integer. For a nonnegative integer $t$, define the asymptotic density \[ c_t=\lim_{N\rightarrow \infty}…

In the present paper new insights into the study of the Non-central Dirichlet distribution are provided. This latter is the analogue of the Dirichlet distribution obtained by replacing the Chi-Squared random variables involved in its…

We study lower and upper bounds for the density of a diffusion process in ${\mathbb{R}}^n$ in a small (but not asymptotic) time, say $\delta$. We assume that the diffusion coefficients $\sigma_1,\ldots,\sigma_d$ may degenerate at the…

The set of indices that correspond to the positive entries of a sequence of numbers is called its positivity set. In this paper, we study the density of the positivity set of a given linear recurrence sequence, that is the question of how…

We consider random walks in a random environment that is given by i.i.d. Dirichlet distributions at each vertex of Z^d or, equivalently, oriented edge reinforced random walks on Z^d. The parameters of the distribution are a 2d-uplet of…

A beautiful theorem of Zeckendorf states that every integer can be written uniquely as the sum of non-consecutive Fibonacci numbers $\{ F_i \}_{i = 1}^{\infty}$. A set $S \subset \mathbb{Z}$ is said to satisfy Benford's law if the density…

Let $S= (s_1<s_2<\dots)$ be a strictly increasing sequence of positive integers and denote $\mathbf{e}(\beta)=\mathrm{e}^{2\pi i \beta}$. We say $S$ is good if for every real $\alpha$ the limit $\lim_N \frac1N\sum_{n\le N}…

It is proved that the scattering amplitude $A(\beta, \alpha_0, k_0)$, known for all $\beta\in S^2$, where $S^2$ is the unit sphere in $\mathbb{R}^3$, and fixed $\alpha_0\in S^2$ and $k_0>0$, determines uniquely the surface $S$ of the…

Directional data require specialized probability models because of the non-Euclidean and periodic nature of their domain. When a directional variable is observed jointly with linear variables, modeling their dependence adds an additional…

Discovery problems often require deciding whether additional sampling is needed to detect all categories whose prevalence exceeds a prespecified threshold. We study this question under a Bernoulli product (incidence) model, where categories…

Nonparametric density estimation for compositional data supported on the simplex is examined under a missing at random mechanism. Rather than imputing missing values and estimating the density from a completed data set, we adopt a strategy…