Lineability within probability theory settings

Abstract

The search of lineability consists on finding large vector spaces of mathematical objects with special properties. Such examples have arisen in the last years in a wide range of settings such as in real and complex analysis, sequence spaces, linear dynamics, norm-attaining functionals, zeros of polynomials in Banach spaces, Dirichlet series, and non-convergent Fourier series, among others.In this paper we present the novelty of linking this notion of lineability to the area of Probability Theory by providing positive (and negative) results within the framework of martingales, random variables, and certain stochastic processes.

Lineability, algebrability, and sequences of random variables

“…Remark Notice that Theorem 2.2 is a stronger version of Theorem 1 in [24] in two respects: here algebrability instead of lineability is obtained and an algebraically independent set of cardinality $$\mathfrak {c}$$ is constructed as well. Also, notice that additional assumptions on the probability space are necessary for Theorem 1 in [24]. In fact, for the probability space $$\big (\lbrace 0,1\rbrace ,2^{\lbrace 0,1\rbrace },\frac{1}{2} (\delta _{0} + \delta _{1})\big )$$, e.g., the set $$\mathcal {S}_1$$ is empty.…”

“…𝑛 converges to 0 pointwise on [0,1], and, arguing analogously as for inequality (2.2), we get the existence of infinitely many 𝑛 ∈ ℕ and infinitely many 𝑛 ∈ ℕ such that In a nutshell, Theorem 2.2, Theorem 2.4 and Theorem 2.5 show that, without a dominating integrable function, the family of sequences not fulfilling the Dominated Convergence Theorem (see [11]) is very large. We continue in this direction and derive additional results related to the interchangeability of the integral and the limit (also compare with Theorem 2 in [24]). Doing so we start with the family  4 , defined by Let us recall that, given 𝜅 a finite or infinite cardinality, a subset 𝑀 of a linear space 𝑋 is called positively 𝜅-coneable (see [1]) in 𝑋 if there exists an 𝜅-dimensional set 𝑀 such that 𝛼𝑀 ⊂ 𝑀 for every 𝛼 > 0.…”

“…is positively 𝔠-coneable (also see Theorem 3 in [24]). In other words: The family of sequences (𝑋 𝑛 ) 𝑛∈ℕ in  0,1 for which the inequality in Fatou's Lemma is sharp is very large.…”

Undominated Sequences of Integrable Functions

“…For results dealing with lineability of families of sequences of measurable functions [0, 1] → R or [0, +∞) → R -where several kinds of convergence are considered-see [1, Section 7] and [11]. See also [6] and [12] for lineability facts related to expect values of sequences of random variables defined on a probability space.…”

Lineability and modes of convergence

“…In the probability theory setting, Conejero et al (see [11]) studied in 2017 some lineability and algebrability problems, as for example, convergent not L 1 -unbounded martingales, pointwise convergent random variables whose means do no not converge to the expected value, stochastic processes that are L 2 bounded and convergent but not pointwise convergent in a null set.…”