18.177 course project: Invariance Principles

We think of this as saying that under some mild conditions on our probability space, the distribution of the linear form Q is roughly the same, regardless of the actual underlying distribution. There are, of course, a family of different central limit theorems, with different “mildness conditions” leading to different notions of the closeness of the resulting distributions. As “central limit theorems,” however, these classical results all aimed to show that a linear form on some (arbitrary) distribution was close to the single Gaussian corresponding to that linear form. When these theorems are stated in the form given here, it is natural to wonder if they hold for a richer class of functions, where although the final distribution may not be Gaussian, it may be close to the distribution produced by a function on Gaussian space. Recently, such generalizations of the Berry-Essen Theorem to multilinear polynomials, i.e., Q of the form