arXiv subjects

Paolo Leonetti

Publications and source records attributed to Paolo Leonetti.

At least 55 records · Page 3Linked to original sources

arXiv2024

Expected multi-utility representations of preferences over lotteries

Let $\succsim$ be a binary relation on the set of simple lotteries over a countable outcome set $Z$. We provide necessary and sufficient conditions on $\succsim$ to guarantee the existence of a set $U$ of von Neumann--Morgenstern utility functions $u: Z\to \mathbf{R}$ such that $$ p\succsim q \,\,\,\Longleftrightarrow\,\,\, \mathbf{E}_p[u] \ge \mathbf{E}_q[u] \,\text{ for all }u \in U $$ for all simple lotteries $p,q$. In such case, the set $U$ is essentially unique. Then, we show that the analogue characterization does not hold if $Z$ is uncountable. This provides an answer to an open question posed by Dubra, Maccheroni, and Ok in [J. Econom. Theory~\textbf{115} (2004), no.~1, 118--133]. Lastly, we show that different continuity requirements on $\succsim$ allow for certain restrictions on the possible choices of the set $U$ of utility functions (e.g., all utility functions are bounded), providing a wide family of expected multi-utility representations.

math.CA↗

arXiv2023

Regular matrices of unbounded linear operators

Let $X,Y$ be Banach spaces, and fix a linear operator $T \in \mathcal{L}(X,Y)$, and ideals $\mathcal{I}, \mathcal{J}$ on $ω$. We obtain Silverman--Toeplitz type theorems on matrices $A=(A_{n,k}: n,k \in ω)$ of linear operators in $\mathcal{L}(X,Y)$, so that $$ \mathcal{J}\text{-}\lim Ax=T(\hspace{.2mm}\mathcal{I}\text{-}\lim x) $$ for every $X$-valued sequence $x=(x_0,x_1,\ldots)$ which is $\mathcal{I}$-convergent [and bounded]. This allows us to establish the relationship between the classical Silverman--Toeplitz characterization of regular matrices and its multidimensional analogue for double sequences, its variant for matrices of linear operators, and the recent version (for the scalar case) in the context of ideal convergence. As byproducts, we obtain characterizations of several matrix classes and a generalization of the classical Hahn--Schur theorem. In the proofs we will use an ideal version of the Banach--Steinhaus theorem which has been recently obtained by De Bondt and Vernaeve in [J.~Math.~Anal.~Appl.~\textbf{495} (2021)].

math.FA↗

arXiv2023

Quasi-arithmetic means ad libitum

Let $α_1, \ldots, α_m$ be two or more positive reals with sum $1$, let $C\subseteq \mathbb{R}^k$ be an open convex set, and $f: C\to \mathbb{R}^k$ be a continuous injection with convex image. For each nonempty set $S\subseteq C$, let $\mathscr{M}(S)$ be the family of quasi-arithmetic means of all $m$-tuples of vectors in $C$ with respect to $f$ and the weights $α_1,\ldots,α_m$, that is, the family $$ \mathscr{M}(S)= \left\{ f^{-1}\left(α_1f(x_1)+\cdots+α_mf(x_m)\right): x_1,\ldots,x_m \in S \right\}. $$ We provide a simple necessary and sufficient condition on $S$ for which the infinite iteration $\bigcup_{n}\mathscr{M}^n(S)$ is relatively dense in the convex hull of $S$.

math.CA↗

arXiv2023

Capacities and Choquet Averages of Ultrafilters

We show that a normalized capacity $ν: \mathcal{P}(\mathbf{N})\to \mathbf{R}$ is invariant with respect to an ideal $\mathcal{I}$ on $\mathbf{N}$ if and only if it can be represented as a Choquet average of $\{0,1\}$-valued finitely additive probability measures corresponding to the ultrafilters containing the dual filter of $\mathcal{I}$. This is obtained as a consequence of an abstract analogue in the context of Archimedean Riesz spaces.

math.FA↗

arXiv2023

Rough families, cluster points, and cores

We define the notion of ideal convergence for sequences $(x_n)$ with values in topological spaces $X$ with respect to a family $\{F_η: η\in X\}$ of subsets of $X$ with $η\in F_η$. Each set $F_η$ quantifies the degree of accuracy of the convergence toward $η$. After proving that this is really a new notion, we provide some properties of the set of limit points and characterize the latter through the ideal cluster points and the ideal core of $(x_n)$.

math.GN↗

arXiv2023

Subcommutativity of integrals and quasi-arithmetic means

Let $(X, \mathscr{L}, λ)$ and $(Y, \mathscr{M}, μ)$ be finite measure spaces for which there exist $A \in \mathscr{L}$ and $B \in \mathscr{M}$ with either $0 < λ(A) < 1 < λ(X)$ and $0 < μ(B) < μ(Y)$, or the other way around. In addition, let $I \subseteq \mathbb{R}$ be a non-empty open interval, and suppose that $f,g\colon I \to \mathbb{R}_{+}$ are homeo\-morphisms with $g$ increasing. We prove that the functional inequality $$ f^{-1}\!\left(\int_X f\!\left(g^{-1}\!\left(\int_Y g \circ h\;dμ\right)\right)d λ\right)\! \le g^{-1}\!\left(\int_Y g\!\left(f^{-1}\!\left(\int_X f \circ h\;dλ\right)\right)d μ\right) $$ is satisfied by every $\mathscr{L} \otimes \mathscr{M}$-measurable simple function $h: X \times Y \to I$ if and only if $f=a g^b$ for some $a,b \in \mathbb{R}_{+}$ with $b\ge 1$. An analogous characterization is given for probability spaces.

math.FA↗

arXiv2023

Dense lineability and spaceability in certain subsets of $\ell_{\infty}$

We investigate dense lineability and spaceability of subsets of $\ell_\infty$ with a prescribed number of accumulation points. We prove that the set of all bounded sequences with exactly countably many accumulation points is densely lineable in $\ell_\infty$, thus complementing a recent result of Papathanasiou who proved the same for the sequences with continuum many accumulation points. We also prove that these sets are spaceable. We then consider the same problems for the set of bounded non-convergent sequences with a finite number of accumulation points. We prove that such set is densely lineable in $\ell_\infty$ and that it is nevertheless not spaceable. The said problems are also studied in the setting of ideal convergence and in the space $\mathbb{R}^ω$.

math.FA↗

arXiv2023

On the iterates of shifted Euler's function

Let $φ$ be the Euler's function and fix an integer $k\ge 0$. We show that, for every initial value $x_1\ge 1$, the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+1}=φ(x_n)+k$ for all $n\ge 1$ is eventually periodic. Similarly, for every initial value $x_1,x_2\ge 1$, the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+2}=φ(x_{n+1})+φ(x_n)+k$ for all $n\ge 1$ is eventually periodic, provided that $k$ is even.

math.NT↗

arXiv2023

Representations of cones and applications to decision theory

Let $C$ be a cone in a locally convex Hausdorff topological vector space $X$ containing $0$. We show that there exists a (essentially unique) nonempty family $\mathscr{K}$ of nonempty subsets of the topological dual $X^\prime$ such that $$ C=\{x \in X: \forall K \in \mathscr{K}, \exists f \in K, \,\, f(x) \ge 0\}. $$ Then, we identify the additional properties on the family $\mathscr{K}$ which characterize, among others, closed convex cones, open convex cones, closed cones, and convex cones. For instance, if $X$ is a Banach space, then $C$ is a closed cone if and only if the family $\mathscr{K}$ can be chosen with nonempty convex compact sets. These representations provide abstract versions of several recent results in decision theory and give us the proper framework to obtain new ones. This allows us to characterize preorders which satisfy the independence axiom over certain probability measures, answering an open question in [Econometrica~\textbf{87} (2019), no. 3, 933--980].

math.CA↗

arXiv2022

Almost all sets of nonnegative integers and their small perturbations are not sumsets

Fix $α\in (0,1/3)$. We show that, from a topological point of view, almost all sets $A\subseteq \mathbb{N}$ have the property that, if $A^\prime=A$ for all but $o(n^α)$ elements, then $A^\prime$ is not a nontrivial sumset $B+C$. In particular, almost all $A$ are totally irreducible. In addition, we prove that the measure analogue holds with $α=1$.

math.NT↗

arXiv2022

On the density of sumsets, II

Arithmetic quasi-densities are a large family of real-valued set functions partially defined on the power set of $\mathbb{N}$, including the asymptotic density, the Banach density, the analytic density, etc. Let $B \subseteq \mathbb{N}$ be a non-empty set covering $o(n!)$ residue classes modulo $n!$ as $n\to \infty$ (e.g., the primes or the perfect powers). We show that, for each $α\in [0,1]$, there is a set $A\subseteq \mathbb{N}$ such that, for every arithmetic quasi-density $μ$, both $A$ and the sumset $A+B$ are in the domain of $μ$ and, in addition, $μ(A + B) = α$. The proof relies on the properties of a little known density first considered by Buck in 1946.

math.NT↗

arXiv2022

Most numbers are not normal

We show, from a topological viewpoint, that most numbers are not normal in a strong sense. More precisely, the set of numbers $x \in (0,1]$ with the following property is comeager: for all integers $b\ge 2$ and $k\ge 1$, the sequence of vectors made by the frequencies of all possibile strings of length $k$ in the $b$-adic representation of $x$ has a maximal subset of accumulation points, and each of them is the limit of a subsequence with an index set of nonzero asymptotic density. This extends and provides a streamlined proof of the main result given by Olsen in [Math. Proc. Cambridge Philos. Soc. 137 (2004), 43--53]. We provide analogues in the context of analytic P-ideals and regular matrices.

math.NT↗

arXiv2022

The maximum domain of attraction of multivariate extreme value distributions is small

Consider the set of Borel probability measures on $\mathbf{R}^k$ and endow it with the topology of weak convergence. We show that the subset of all probability measures which belong to the domain of attraction of some multivariate extreme value distributions is dense and of the first Baire category. In addition, the analogue result holds in the context of free probability theory.

math.PR↗

arXiv2022

On some locally convex FK spaces

We provide necessary and/or sufficient conditions on vector spaces $V$ of real sequences to be a Fréchet space such that each coordinate map is continuous, that is, to be a locally convex FK space. In particular, we show that if $c_{00}(\mathcal{I})\subseteq V\subseteq \ell_\infty(\mathcal{I})$ for some ideal $\mathcal{I}$ on $ω$, then $V$ is a locally convex FK space if and only if there exists an infinite set $S\subseteq ω$ for which every infinite subset does not belong to $\mathcal{I}$.

math.FA↗

arXiv2022

A Characterization of the Vector Lattice of Measurable Functions

Given a probability measure space $(X,Σ,μ)$, it is well known that the Riesz space $L^0(μ)$ of equivalence classes of measurable functions $f: X \to \mathbf{R}$ is universally complete and the constant function $\mathbf{1}$ is a weak order unit. Moreover, the linear functional $L^\infty(μ)\to \mathbf{R}$ defined by $f \mapsto \int f\,\mathrm{d}μ$ is strictly positive and order continuous. Here we show, in particular, that the converse holds true, i.e., any universally complete Riesz space $E$ with a weak order unit $e>0$ which admits a strictly positive order continuous linear functional on the principal ideal generated by $e$ is lattice isomorphic onto $L^0(μ)$, for some probability measure space $(X,Σ,μ)$.

math.FA↗

arXiv2022

On the density of sumsets

Recently introduced by the authors in [Proc. Edinb. Math. Soc. 60 (2020), 139-167], quasi-densities form a large family of real-valued functions partially defined on the power set of the integers that serve as a unifying framework for the study of many known densities (including the asymptotic density, the Banach density, the logarithmic density, the analytic density, and the Pólya density). We further contribute to this line of research by proving that (i) for each $n \in \mathbf N^+$ and $α\in [0,1]$, there is $A \subseteq \mathbf{N}$ with $kA \in \text{dom}(μ)$ and $μ(kA) = αk/n$ for every quasi-density $μ$ and every $k=1,\ldots, n$, where $kA:=A+\cdots+A$ is the $k$-fold sumset of $A$ and $\text{dom}(μ)$ denotes the domain of definition of $μ$; (ii) for each $α\in [0,1]$ and every non-empty finite $B\subseteq \mathbf{N}$, there is $A \subseteq \mathbf{N}$ with $A+B \in \mathrm{dom}(μ)$ and $μ(A+B)=α$ for every quasi-density $μ$; (iii) for each $α\in [0,1]$, there exists $A\subseteq \mathbf{N}$ with $2A = \mathbf{N}$ such that $A \in \text{dom}(μ)$ and $μ(A) = α$ for every quasi-density $μ$. Proofs rely on the properties of a little known density first considered by R.C. Buck and the "structure" of the set of all quasi-densities; in particular, they are rather different than previously known proofs of special cases of the same results.

math.NT↗

arXiv2021

Density-Like and Generalized Density Ideals

We show that there exist uncountably many (tall and nontall) pairwise nonisomorphic density-like ideals on $ω$ which are not generalized density ideals. In addition, they are nonpathological. This answers a question posed by Borodulin-Nadzieja, Farkas, and Plebanek in [J. Symb. Log. \textbf{80} (2015), 1268--1289]. Lastly, we provide sufficient conditions for a density-like ideal to be necessarily a generalized density ideal.

math.FA↗

arXiv2021

Another characterization of meager ideals

We show that an ideal $\mathcal{I}$ on the positive integers is meager if and only if there exists a bounded nonconvergent real sequence $x$ such that the set of subsequences [resp. permutations] of $x$ which preserve the set of $\mathcal{I}$-limit points is comeager and, in addition, every accumulation point of $x$ is also an $\mathcal{I}$-limit point (that is, a limit of a subsequence $(x_{n_k})$ such that $\{n_1,n_2,\ldots,\} \notin \mathcal{I}$). The analogous characterization holds also for $\mathcal{I}$-cluster points.

math.GN↗