Lecture 7 : Triangular arrays and Lévy processes

Previous lectures detailed the central limit theorem for dealing with triangular arrays of random variables: under certain conditions, the row sums of these triangular arrays converge in distribution to a standard normal. By extension, we thus have seen conditions that lead the row sums to converge to N(μ, σ) for general μ, σ. Today, we introduce a more general condition for triangular arrays of random variables and examine a canonical example of convergence for these arrays, convergence to a Poi(λ) random variable. Then, we discuss some of the background and context of stochastic processes, beginning with a look at Brownian motion and Poisson processes, and moving into the more general setting of Lévy processes.