On Some Laws of Large Numbers for Uncertain Random Variables

Baoding Liu created uncertainty theory to describe the information represented by human language. In turn, Yuhan Liu founded chance theory for modelling phenomena where both uncertainty and randomness are present. The first theory involves an uncertain measure and variable, whereas the second one introduces the notions of a chance measure and an uncertain random variable. Laws of large numbers (LLNs) are important theorems within both theories. In this paper, we prove a law of large numbers (LLN) for uncertain random variables being continuous functions of pairwise independent, identically distributed random variables and regular, independent, identically distributed uncertain variables, which is a generalisation of a previously proved version of LLN, where the independence of random variables was assumed. Moreover, we prove the Marcinkiewicz–Zygmund type LLN in the case of uncertain random variables. The proved version of the Marcinkiewicz–Zygmund type theorem reflects the difference between probability and chance theory. Furthermore, we obtain the Chow type LLN for delayed sums of uncertain random variables and formulate counterparts of the last two theorems for uncertain variables. Finally, we provide illustrative examples of applications of the proved theorems. All the proved theorems can be applied for uncertain random variables being functions of symmetrically or asymmetrically distributed random variables, and symmetrical or asymmetrical uncertain variables. Furthermore, in some special cases, under the assumption of symmetry of the random and uncertain variables, the limits in the first and the third theorem have forms of symmetrical uncertain variables.

Uniform Stochastic Convergence

This chapter concerns random sequences of functions on metric spaces. The main issue is the distinction between convergence at all points of the space (pointwise) and uniform convergence, where limit points are also taken into account. The role of the stochastic equicontinuity property is highlighted. Generic uniform convergence conditions are given and linked to the question of uniform laws of large numbers.

Stochastic Convergence

The modes of convergence introduced in Chapter 12 are studied in detail. Conditions for almost‐sure convergence are derived via the Borel–Cantelli lemma. Convergence in probability is contrasted, and then a number of results for convergence of transformed series are given. Convergence in LP‐norm is introduced as a sufficient condition for convergence in probability. Examples are given, and the chapter concludes with a preliminary look at the laws of large numbers.

Weak Laws of Large Numbers for Negatively Superadditive Dependent Random Vectors in Hilbert Spaces

Let $\{X_{n}, {n}\in \mathbb{N}\}$ be a sequence of negatively superadditive dependent random vectors taking values in a real separable Hilbert space. In this paper, we present the weak laws of large numbers for weighted sums (with or without random indices) of $\{X_{n}, {n}\in \mathbb{N}\}$.