Pedro Terán

• Ph. D.
• Professor (Associate) at University of Oviedo

Shige Peng’s sublinear expectations generalize ordinary linear expectation operators. It is shown that the behaviour of sample averages of Peng i.i.d. variables may be very different from the probabilistic intuition. In particular, Peng’s generalization of the Monte Carlo method is shown to be wrong. It is also observed that sublinear expectations...

This paper discusses the definition of independence used in the proof of a law of large numbers for upper expectations [Z.Chen, P.Wu, B.Li (2013). Int. J. Approx. Reasoning 54, 365-377]. It is shown to be trivial for very simple examples of upper expectations. When the upper probability is the maximum of two probability measures $P$ and $Q$ which...

It is a celebrated result in the theory of random sets that, in a locally compact second countable Hausdorff space, distributions of random closed sets are in one-to-one correspondence with certain capacities. In this paper we generalize the correspondence to locally compact sigma-compact Hausdorff spaces, showing that second countability is not ne...

We prove by counterexample that a large deviation principle established by Chen and Feng [Comm. Statist. Theory Methods 45 (2016), 400-412] in the framework of sublinear expectations is incorrect. That implies that the rate function cannot, in general, be obtained by computing the Fenchel transform of the cumulant generating function, as is the cas...

The recently defined concept of statistical depth function for fuzzy sets provides a theoretical framework for ordering fuzzy sets with respect to the distribution of a fuzzy random variable. One of the most used and studied statistical depth functions for multivariate data is simplicial depth, based on multivariate simplices. We introduce a notion...

Tukey depth is the first and one of the most notorious statistical depth functions. It has very good properties and measures in a proper way the centrality of points with respect to a distribution. In a forthcoming paper by the authors, the Tukey depth function is extended to the fuzzy setting. Here, we comment on this extension and some interestin...

Statistical depth functions order the elements in a space with respect to their centrality in a probability distribution. Their study has substantially grown since the notion of depth for multivariate data was introduced in 2000 and nowadays it is an important tool in non-parametric statistics. González-de la Fuente et al. (2022) propose two genera...

Some nice properties of a possible definition of random sets in certain non-metrizable spaces are studied. These concern specially applications to products of random sets and to random functions.

We study whether convergence in distribution of fuzzy random variables, defined as the weak convergence of their probability distributions, is consistent with the additional structure of spaces of fuzzy sets. Positive results are obtained which reinforce the viability of that definition.

We study a statistical data depth with respect to compact convex random sets, which is consistent with the multivariate Tukey depth and the Tukey depth for fuzzy sets. In addition, it provides a different perspective to the existing halfspace depth with respect to compact convex random sets. In studying this depth function, we provide a series of p...

We study a statistical data depth with respect to compact convex random sets which is consistent with the multivariate Tukey depth and the Tukey depth for fuzzy sets. In doing so, we provide a series of properties for statistical data depth with respect to compact convex random sets. These properties are an adaptation of properties that constitute...

The recently defined concept of a statistical depth function for fuzzy sets provides a theoretical framework for ordering fuzzy sets with respect to the distribution of a fuzzy random variable. One of the most used and studied statistical depth function for multivariate data is simplicial depth, based on multivariate simplices. We introduce a notio...

A Vitali convergence theorem is proved for subspaces of an abstract convex combination space which admits a complete separable metric. The convergence may be in that metric or, more generally, in a quasimetric satisfying weaker properties. Versions for convergence in probability and in distribution are given. As applications, we show that some domi...

Statistical depth functions provide a way to order the elements of a space by their centrality in a probability distribution. That has been very successful for generalizing non-parametric order-based statistical procedures from univariate to multivariate and (more recently) to functional spaces. We introduce two general definitions of statistical d...

Versions of several results from the theory of random variables are proved for fuzzy random variables: the Skorokhod representation theorem, the Vitali convergence theorem, the dominated convergence theorem, the continuous mapping theorem, existence of regular conditional distributions, and a few others.

The operation of taking the α-cut of a compact convex fuzzy set is shown to be jointly measurable with respect to both α and the fuzzy set. As a consequence, a number of mappings on product spaces which are induced by a fuzzy random variable are shown to be jointly measurable. Some applications to the relationships between fuzzy random variables an...

Different approaches to robustly measure the location of data associated with a random experiment have been proposed in the literature, with the aim of avoiding the high sensitivity to outliers or data changes typical for the mean. In particular, M-estimators and trimmed means have been studied in general spaces, and can be used to handle Hilbert-v...

We prove a measurability result which implies that the measurable events concerning the values of a fuzzy random variable, in two related mathematical approaches wherein the codomains of the variables are different spaces, are the same (provided both approaches apply). Further results on the perfectness of probability distributions of fuzzy random...

A fundamental long-standing problem in the theory of random sets is concerned with the possible characterization of the distributions of random closed sets in Polish spaces via capacities. Such a characterization is known in the locally compact case (the Choquet theorem) in two equivalent forms: using the compact sets and the open sets as test sets...

The law of large numbers for coherent lower previsions (specifically, Choquet integrals against belief measures) can be applied to possibility measures, yielding that sample averages are asymptotically confined in a compact interval. This interval differs from the one appearing in the law of large numbers from possibility theory. In order to unders...

We prove a strong law of large numbers for random closed sets in a separable Banach space. It improves upon and unifies the laws of large numbers with convergence in the Wijsman, Mosco and slice topologies, without requiring extra assumptions on either the properties of the space or the kind of sets that can be taken on by the random set as values.

Chareka [Statistics and Probability Letters 79, 1456–1462] presented two limit theorems in which the additivity requirement of a probability measure is weakened to a one-sided version of the inclusion-exclusion formula. Counterexamples are presented which show that both his Central Limit Theorem and his Weak Law of Large Numbers are false.

The Aumann and Herer expectations represent two different approaches to defining the expectation of a random set (analytical versus geometrical). This paper investigates their relationships in the setting of Banach spaces, which are shown to be related to the geometry of the dual unit ball, to the bornological differentiability properties of the no...

An approximation scheme for estimating a fixed, unknown fuzzy set from random samples taken from the nested random set defined by its alpha-level sets is presented. Its strong consistency is studied, giving rates of convergence in four metrics. A simulation study suggests that the behaviour for moderately small samples is coherent with the theoreti...

We prove a strong law of large numbers for random closed sets in a separable Banach space. It improves upon and unifies the laws of large numbers with convergence in the Wijsman, Mosco and slice topologies, without requiring extra assumptions on either the properties of the space or the kind of sets that can be taken on by the random set as values.

The law of large numbers is studied under a weakening of the axiomatic properties of a probability measure. Averages do not generally converge to a point, but they are asymptotically confined in a limit set for any random variable satisfying a natural 'finite first moment' condition. It is also shown that their behaviour can depart strikingly from...

A law of large numbers for the possibilistic mean value of a variable in a possibility space is presented. An example shows that the convergence in distribution (under a definition involving the possibilistic mean value) of the sample average to a variable with a certain distribution cannot be replaced, in general, by convergence either almost sure...

Jensen's inequality is extended to metric spaces endowed with a convex combination operation. Applications include a dominated convergence theorem for both random elements and random sets, a monotone convergence theorem for random sets, and other results on set-valued expectations in metric spaces and on random probability measures.

Random elements of non-Euclidean spaces have reached the forefront of statistical research with the extension of continuous process monitoring, leading to a lively interest in functional data. A fuzzy set is a generalized set for which membership degrees are identified by a [0, 1]-valued function. The aim of this review is to present random fuzzy s...

Random elements of non-Euclidean spaces have reached the forefront of statistical research with the extension of continuous process monitoring, leading to a lively interest in functional data. A fuzzy set is a generalized set for which membership degrees are identified by a [0, 1]-valued function. The aim of this review is to present random fuzzy s...

Consider the Strong Law of Large Numbers for t-normed averages of fuzzy random variables in the uniform metric d ∞ . That probabil-sitic property is known to hold when the t-norm is the minimum and to fail when the t-norm is the product. We prove that it is character-ized by an algebraic property of the t-norm (that of being eventually idempotent)...

If a sequence of random closed sets X n in a separable complete metric space converges in distribution in the Wijsman topology to X, then the corresponding sequence of cores (sets of probability measures dominated by the capacity functional of X n) converges to the core of the capacity of X. Core convergence is achieved not only in the Wijsman topo...

It is a celebrated result in the theory of random sets that, in a locally compact second countable Hausdorff space, distributions of random closed sets are in one-to-one correspondence with certain capacities. In this paper we generalize the correspondence to locally compact a-compact Hausdorff spaces, showing that second countability is not necess...

This paper aims at formalizing the intuitive idea that some points are more central in a probability distribution than others. Our proposal relies on fuzzy events to define a fuzzy set of central points for a distribution (or a family of distributions, including imprecise probability models). This framework has a natural interpretation in terms of...

We show that two probabilistic interpretations of fuzzy sets via random sets and large deviation principles have a common feature: they regard the fuzzy set as a depth function of a random object. Conversely, some depth functions in the literature can be regarded as the fuzzy sets of central points of appropriately chosen random sets.

Recently, Balaji and Xu studied the consistency of stationary points, in the sense of the Clarke generalized gradient, for the sample average approximations to a one-stage stochastic optimization problem in a separable Banach space with separable dual. We present an alternative approach, showing that the restrictive assumptions that the dual space...

We are concerned with some properties of the family of all subsets of a Banach space that can be written as an intersection of balls. A space with the Mazur Intersection Property (MIP) always satisfies those properties, so they can be regarded as weakenings of the MIP in different directions. The 'ball hull' function (mapping a set to the intersect...

We prove a strong law of large numbers for sequences of pairwise i.i.d. fuzzy random variables. Addition is given by the extension principle associated with a general continuous triangular norm. Our result includes the SLLN for levelwise sums, obtained when the chosen triangular norm is the minimum.

Spaces where the Aumann and Herer notions of expectation of a random set coincide are exactly those having the Mazur Intersection Property (the closed convex hull of a bounded set is the intersection of all balls covering it). For a random compact set, more can be said: its Herer expectation is always the intersection of all closed balls covering i...

In this paper, we strengthen the convergence in a uniform law of large numbers for random upper semicontinuous multifunctions of Shapiro and Xu. The proof is based on an abstract law of large numbers in a metric space endowed with a convex combination operation. Convergence in the Hausdorff metric is obtained, whereas the original result presented...

We present a variant of the Analytic Hierarchy Process intended to facilitate consensus search in group decision making. This soft methodology combines fuzzy sets and probabilistic information to provide judgements oriented by the actors’ attitude towards negotiation. A Monte Carlo approach is taken to derive a preference structure distribution whi...

We aim at clarifying the relationship between laws of large numbers for fuzzy sets or possibility distributions and laws of large numbers for fuzzy or possibilistic variables. We contend that these two frameworks are different and present the relationships between them that explain why this fact was unrecognized so far.

If a sequence of random closed sets X n in a separable complete metric space converges in distribution in the Wijsman topology to X, then the corresponding sequence of cores (sets of probability measures dominated by the capacity functional of X n) converges to the core of the capacity of X. Core convergence is achieved not only in the Wijsman topo...

We propose fuzzy random variables as a tool for modelling measurements when both aleatory and fuzzy uncertainty have to be taken into account. Uncertainty propagation follows the ordinary scheme for the random part and uses a t-normed extension principle for the fuzzy part. We concentrate on the probabilistic theoretical underpinnings of the model,...

We consider a separable complete metric space equipped with a convex combination operation. For such spaces, we identify the corresponding convexification operator and show that the invariant elements for this operator appear naturally as limits in the strong law of large numbers. It is shown how to uplift the suggested construction to work with su...

A Banach space of dimension at least 2 does not admit an equi-Lipschitzian family of additive mappings parametrizing all non-empty compact convex sets. Examples of linear Lipschitzian as well as positively homogeneous equi-Lipschitzian parametrizations exist in the literature.

Fuzzy random variables are a special case of measurable mappings taking on values in a non-separable metric space. Due to that non-separability, the two main definitions in the literature turn out not to be equivalent. We give an easy necessary and sufficient condition for their equivalence. An intermediate result of independent value is a characte...

We argue that the optimal stopping model which has been used by Yoshida to discuss option pricing can many times lead to an overoptimistic evaluation of payoffs (put option prices). This effect is due to the method used to compare fuzzy payoffs, using a Sugeno integral. It is shown that each fuzzy payoff can be associated to an indifferent non-fuzz...

We present a general law of large numbers in a (separable complete) metric space endowed with an abstract convex combination operation. Spaces of fuzzy sets are shown to be particular cases of that framework. We discuss the compatibility of the usual definition of expectation with the abstract one. We close the paper with two applications to the t...

A strong law of large numbers under conditions irrespective of the joint distribution of the sequence is extended to random sets. The extension is such that the role of events of the form {||Vn|| ≤ bn} (where Vn is a random element of a separable Banach space) is played by events of the form {Xn ⊂ Bn} (where Xn is a random closed bounded set).

In this paper we embed the space of upper semicontinuous convex fuzzy sets on a Banach space into a space of continuous functions on a compact space. The following structures are preserved by the embedding: convex cone, metric, sup-semilattice. The indicator function of the unit ball is mapped to the constant function 1. Two applications are presen...

We show a general method to translate Tauberian theorems for summability methods in $\mathbb R$ into Tauberian theorems for the corresponding forms of statistical convergence in metric spaces. The main tools (distance functions and the Hausdorff metric) come from set-valued analysis.

We obtain necessary and sufficient conditions in the Large Deviation Principle for random upper semicontinuous functions on a separable Banach space. The main tool is the recent work of Arcones on the LDP for empirical processes.

We study sub- and supermartingales with values in the class Fc of all convex fuzzy subsets of a Banach space having compact support. The terms “sub” and “super” are understood with respect to a partial order (not necessarily inclusion). In this general framework, orders with respect to which a Doob-type decomposition can hold are naturally associat...

In this paper we study the problem of estimating a càdlàg function f whose values are compact convex sets. For this purpose a random selection of points in the interval [0,1] is considered and for each selected point x, a random sample in f(x). On the basis of this a sequence of approximants fn,m is constructed (where n and m are the respective sam...

In this paper, we show how to construct Korovkin systems in spaces of continuous mappings whose values are (possibly non-convex) sets or more generally (possibly non-quasiconcave) upper semicontinuous functions. The Korovkin system is constructed from a given Korovkin system of real functions. Furthermore, we show that any Korovkin system in the qu...

We present three counterexamples in the context of the Strong Law of Large Numbers and the Central Limit Theorem for fuzzy random variables (with possibly unbounded support) Sums are defined according to the extension principle based on an arbitrary continuous t-norm. Ordinary addition arises when the t-norm taken is the minimum. Besides, we prove...

In this paper we prove a strong law of large numbers for random upper semicontinuous functions on a separable Banach space possibly having unbounded support. Convergence is in a topology closely related to the Puri–Ralescu d∞ metric. Assumptions on the sequence of random variables are of exchangeability and uncorrelatedness type.

A set in a metric space gives rise to its distance function that associates with every point its distance to the nearest point in the set. This function is called the distance transform of the original set. In the same vein, given a real-valued function f we consider the expected distances from any point to a level set of f taken at a random height...

On the basis of Part (I) of this series some applications to the approximation of set-valued functions are obtained: Korovkin type theorems, a method to extend classical approximation operators to the set-valued setting and a Jackson type estimate. Key wordsset-valued extension–Steiner selection–a Jackson type estimate CLC numberO177.91

A way to extend operators in spaces of continuous functions to spaces of continuous set-valued functions is proposed. This extension is developed through the Steiner selections of the set-valued functions. Their properties and characteristics of the convergence of sequences of operators of this class are studied. In PartII of this series some appli...

In this communication we study the problem of estimating a convex fuzzy set U. We consider a random choice of n points in the interval [0, 1] and for each selected point α, a random sample of size m in Uα. On the basis of this, a double sequence of approximants is constructed. Under general conditions, rates of convergence are obtained for the dp m...

In this paper we show how a technique used by R.A. Vitale to obtain a Korovkin-type approximation theorem for random sets can be exploited to develop a similar result for fuzzy random variables. A convergence theorem for positive linear operators is obtained, and consequences of this theorem in the Bernstein approximation of fuzzy random variables...

In this paper we study some aspects of the approximation of mappings taking values in a special class of upper semicontinuous functions. Some Korovkin type theorems for positive linear operators are obtained, and consequences of these theorems for a special class of operators defined through partial sum stochastic processes are analyzed.

In this paper, we study different properties of Bernstein approximants of a fuzzy random variable, as those in relation with the approximation of fuzzy random variables by means of its n-Bernstein approximant. Moreover, we define and study the concept of ϕ-variation for a fuzzy random variable.

In this communication some results about the φ-variation of fuzzy random variables are presented. In order to preserve analogous properties to those of the φ-variation of random variables, it is hardly ever necessary to demand the fuzzy random variable to be convex-valued.