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.

Dynamical Borel–Cantelli lemma for recurrence theory

We study the dynamical Borel–Cantelli lemma for recurrence sets in a measure-preserving dynamical system $(X, \mu , T)$ with a compatible metric d. We prove that under some regularity conditions, the $\mu $ -measure of the following set $$\begin{align*}R(\psi)= \{x\in X : d(T^n x, x) < \psi(n)\ \text{for infinitely many}\ n\in\mathbb{N} \} \end{align*}$$ obeys a zero–full law according to the convergence or divergence of a certain series, where $\psi :\mathbb {N}\to \mathbb {R}^+$ . The applications of our main theorem include the Gauss map, $\beta $ -transformation and homogeneous self-similar sets.

The Borel-Cantelli Lemmas, and Their Relationship to Limit Superior and Limit Inferior of Sets (or, Can a Monkey Really Type Hamlet?)

The purpose of this chapter is to show that if a monkey types infinitely, Shakespeare’s Hamlet and any other works one may wish to add to the list will each be typed, not once, not twice, but infinitely often with a probability of 1. This dramatic fact is a simple consequence of the Borel-Cantelli lemma and will come as no surprise to anyone who has taken a graduate-level course in Probability. The proof of this result, however, is quite accessible to anyone who has but a rudimentary understanding of the concept of independence, together with the notion of limit superior and limit inferior of a sequence of sets.

Marstrand's Approximate Independence of Sets and Strong Differentiation of the Integral

A constructive proof is given for the existence of a function belonging to the product Hardy space $H^1(\mathsf{R} \times \mathsf{R})$ and the Orlicz space $L(\log L)^{\epsilon}(\mathsf{R}^{2})$ for all $0<\epsilon <1$, for all whose integral is not strongly differentiable almost everywhere on a set of positive measure. It consists of a modification of a non-negative function created by J. M. Marstrand. In addition, we generalize the claim concerning "approximately independent sets" that appears in his work in relation to hyperbolic-crosses. Our generalization, which holds for any sets with boundary of sufficiently low complexity in any Euclidean space, has a version of the second Borel-Cantelli Lemma as a corollary.