DescriptionThe Law of Large Numbers (LLN) is a limit statement about the Cesàro mean of a family of independent and identically distributed (i.i.d.) random variables. More specifically, given a sequence of real numbers \(\mathbf{x}=(x_n, n \in \mathbb{N})\), The Cesàro mean is defined by \(c_n(\mathbf{x}) := \frac{1}{n} \sum_{k = 1}^n x_k\). The LLN for i.i.d. random variables \(X_i: \Omega \to \mathbb{R}\) with finite mean \(\mu \in \mathbb{R}\) states that \(\mathbb{P}\big(\lim_{n \to \infty}c_n(\mathbf{X}) = \mu \big) = 1, \text{ where } \mathbf{X}(\omega) = (X_n(\omega), n \in \mathbb{N}).\) While the LLN provides a robust foundation for understanding the convergence of averages of i.i.d. random variables, it is limited in its ability to address more complex situations that arise in various applications. For example, in finance, time series analysis, and signal processing, different observations might have different importances, which requires the use of weighted averages or more general summation methods. Voronoi mean. We consider the following weighted summation. Let three real sequences \(\mathbf{p} = (p_n,n \in \mathbb{N})\), \(\mathbf{q} = (q_n,n \in \mathbb{N})\), and \(\mathbf{u} = (u_n,n \in \mathbb{N})\) be given, with \(u_n \neq 0\) for \(n \geq 0\). For a real sequence \(\mathbf{s} = (s_n, n \in \mathbb{N})\) for each \(n\in \mathbb{N}\), define \( v_n = v_n\big((\mathbf{p},\mathbf{q},\mathbf{u}),\mathbf{s}\big):=\frac{1}{u_n}\sum_k p_{(n-k)} q_k s_k.\) The real sequence \(\mathbf{s}\) has a Voronoi mean \(s\) if : \(\lim_{n \to \infty}v_n = s \). Voronoi averages allow for very flexible averaging schemes and one can choose the parameters \((\mathbf{p},\mathbf{q},\mathbf{u})\) to match various averaging schemes such as: Weighted sums of random variables allow us to explore the impact of non-linear weighting schemes on the convergence properties of the sequences. This can lead to a more nuanced understanding of the interplay between the weights and the underlying random variables. Generalizing simple averages to weighted sums or more complex summation methods, such as Voronoi sums, enables us to study the robustness and sensitivity of the convergence results to changes in the underlying assumptions or weighting schemes. The goal of this project is to start from the study of Voronoi sums and explore its relation with broad topics in probability theory such as: PrerequisitesProbability II. Resources |
email: Conrado da Costa