Related papers: Probability distributions with rational free $R$-t…
In this note we report the solution of the problem of defining the $S$-transform in Free Probability Theory in arbitrary dimensions. This is achieved by generalising the theory and embedding it into an algebraic-geometric framework.…
Let $d$ be a probability distribution. Under certain mild conditions we show that $$ \lim_{x\to\infty}x\sum_{n=1}^\infty \frac{d^{*n}(x)}{n}=1,\qquad\text{where}\quad d^{*n}:=\underbrace{\,d*d*\cdots*d\,}_{n\text{ times}}. $$ For a…
We study unimodality for free multiplicative convolution with free normal distributions $\{\lambda_t\}_{t>0}$ on the unit circle. We give four results on unimodality for $\mu\boxtimes\lambda_t$: (1) if $\mu$ is a symmetric unimodal…
An "element-free" probability distribution is what remains of a probability distribution after we forget the elements to which the probabilities were assigned. These objects naturally arise in Bayesian statistics, in situations where…
We consider the question of existence of a unique invariant probability distribution which satisfies some evolutionary property. The problem arises from the random graph theory but to answer it we treat it as a dynamical system in the…
We study the free product of rooted graphs and its various decompositions using quantum probabilistic methods. We show that the free product of rooted graphs is canonically associated with free independence, which completes the proof of the…
Rational transformations of polynomials are extensively studied in the context of finite fields, especially for the construction of irreducible polynomials. In this paper, we consider the factorization of rational transformations with…
The Kolmogorov axioms for probability functions are placed in the context of signed meadows. A completeness theorem is stated and proven for the resulting equational theory of probability calculus. Elementary definitions of probability…
In this paper we study some analytic properties of bi-free additive convolution, both scalar and operator-valued. We show that using properties of Voiculescu's subordination functions associated to free additive convolution of…
Finite (word) state transducers extend finite state automata by defining a binary relation over finite words, called rational relation. If the rational relation is the graph of a function, this function is said to be rational. The class of…
A rational approximation by a ratio of polynomial functions is a flexible alternative to polynomial approximation. In particular, rational functions exhibit accurate estimations to nonsmooth and non- Lipschitz functions, where polynomial…
This paper continues our earlier investigations into the inversion of random functions in a general (abstract) setting. In Section 2 we investigate a concept of invertibility and the invertibility of the composition of random functions. In…
We extend to arbitrary measures results of Bao, Erd\"os, Schnelli, Moreillon, and Ji on the connectedness of the supports of additive convolutions of measures on \mathbb{R} and of free multiplicative convolutions of measures on…
We study the semantic foundation of expressive probabilistic programming languages, that support higher-order functions, continuous distributions, and soft constraints (such as Anglican, Church, and Venture). We define a metalanguage (an…
We give a precise functional comparison between classical and free convolutions. If $\mu$ and $\nu$ are compactly supported probability measures, we show that the expectation of $f$ over the classical convolution $\mu * \nu$ is at least the…
A finite number of rational functions are compatible if they satisfy the compatibility conditions of a first-order linear functional system involving differential, shift and q-shift operators. We present a theorem that describes the…
Conventional multiclass conditional probability estimation methods, such as Fisher's discriminate analysis and logistic regression, often require restrictive distributional model assumption. In this paper, a model-free estimation method is…
This note contains sufficient conditions for the probability density function of an arbitrary continuous univariate distribution, supported on $(0,\infty),$ such that the corresponding Mills ratio to be reciprocally convex (concave). To…
This work concerns random dynamics of hyperbolic entire and meromorphic functions of finite order and whose derivative satisfies some growth condition at infinity. This class contains most of the classical families of transcendental…
Possible parameter values in a random sampling model are shown by definition to have uniform base-rate prior probabilities. This allows a frequentist posterior probability distribution to be calculated for such possible parameter values…