arXiv subjects

Giacomo Aletti

Publications and source records attributed to Giacomo Aletti.

43 records · Page 3Linked to original sources

arXiv2011

A decomposition theorem for fuzzy set-valued random variables and a characterization of fuzzy random translation

Let $X$ be a fuzzy set--valued random variable (\frv{}), and $\huku{X}$ the family of all fuzzy sets $B$ for which the Hukuhara difference $X\HukuDiff B$ exists $\mathbb{P}$--almost surely. In this paper, we prove that $X$ can be decomposed as $X(ω)=C\Mink Y(ω)$ for $\mathbb{P}$--almost every $ω\inΩ$, $C$ is the unique deterministic fuzzy set that minimizes $\mathbb{E}[d_2(X,B)^2]$ as $B$ is varying in $\huku{X}$, and $Y$ is a centered \frv{} (i.e. its generalized Steiner point is the origin). This decomposition allows us to characterize all \frv{} translation (i.e. $X(ω) = M \Mink \indicator{ξ(ω)}$ for some deterministic fuzzy convex set $M$ and some random element in $\Banach$). In particular, $X$ is an \frv{} translation if and only if the Aumann expectation $\mathbb{E}X$ is equal to $C$ up to a translation. Examples, such as the Gaussian case, are provided.

math.PR↗

arXiv2011

A functional equation whose unknown is P([0,1]) valued

We study a functional equation whose unknown maps a Euclidean space into the space of probability distributions on [0,1]. We prove existence and uniqueness of its solution under suitable regularity and boundary conditions, we show that it depends continuously on the boundary datum, and we characterize solutions that are diffuse on [0,1]. A canonical solution is obtained by means of a Randomly Reinforced Urn with different reinforcement distributions having equal means. The general solution to the functional equation defines a new parametric collection of distributions on [0,1] generalizing the Beta family.

math.PR↗

arXiv2009

Approximation of target problems in Blackwell spaces

On a weakly Blackwell space we show how to define a Markov chain approximating problem, for the target problem. The approximating problem is proved to converge to the optimal reduced problem under different pseudometrics. A computational example of compression of information is discussed.

math.PR↗

arXiv2008

Statistical aspects of birth--and--growth stochastic processes

The paper considers a particular family of set--valued stochastic processes modeling birth--and--growth processes. The proposed setting allows us to investigate the nucleation and the growth processes. A decomposition theorem is established to characterize the nucleation and the growth. As a consequence, different consistent set--valued estimators are studied for growth process. Moreover, the nucleation process is studied via the hitting function, and a consistent estimator of the nucleation hitting function is derived.

stat.AP↗

arXiv2008

A set-valued framework for birth-and-growth process

We propose a set-valued framework for the well-posedness of birth-and-growth process. Our birth-and-growth model is rigorously defined as a suitable combination, involving Minkowski sum and Aumann integral, of two very general set-valued processes representing nucleation and growth respectively. The simplicity of the used geometrical approach leads us to avoid problems arising by an analytical definition of the front growth such as boundary regularities. In this framework, growth is generally anisotropic and, according to a mesoscale point of view, it is not local, i.e. for a fixed time instant, growth is the same at each space point.

math.PR↗

arXiv2006

Compressing redundant information in Markov chains

Given a strongly stationary Markov chain and a finite set of stopping rules, we prove the existence of a polynomial algorithm which projects the Markov chain onto a minimal Markov chain without redundant information. Markov complexity is hence defined and tested on some classical problems.

math.PR↗

arXiv2006

First--order continuous models of opinion formation

We study certain nonlinear continuous models of opinion formation derived from a kinetic description involving exchange of opinion between individual agents. These models imply that the only possible final opinions are the extremal ones, and are similar to models of pure drift in magnetization. Both analytical and numerical methods allow to recover the final distribution of opinion between the two extremal ones.

cond-mat.stat-mech↗