Abstract: The first part of this preliminary examination talk I will dedicate to conditional central limit theorems, which, roughly speaking, are statements about sums of conditional random variables converging to the normal distribution. I will provide necessary background for the problem, discuss the approach of studying explicit rates of convergence in dependent settings via Stein's method, and mention current results in application to some examples.
In the second part, I will introduce a model of bond percolation on locally finite Borel graphs on a standard probability space. This model resembles classical Bernoulli percolation with an addition of some dependencies between the edges. The main motivation for this model is the fact that percolation theory on countable graphs often allows for a construction of subgraphs with desired properties and it is of strong interest in measured group theory and measured graph combinatorics to extend it to Borel graphs. I will discuss a spectacular example of this: a measured group theoretic approach to the Day–von Neumann question, known as the Gaboriau–Lyons theorem.