The aim of this course is to visit the basis needed to obtain different scaling limits present in the probability literature. Scaling limit here is understood as the convergence of measures in processes where the space and/or time is/are re-scaled in order to obtain limiting objects.
The idea is to start with topology properties associated with the convergence of measures as are, for example, the Portmanteau theorem and Prokhorov theorem. As a proof of concept, we will revisit the central limit theorem and the Brownian motion construction from random walks (already studied in previous courses). As new applications, we will study scaling limits of random planar maps, random matrices and if time permits the Neural Tangent Kernel.