Gaussian Approximation for a Large Number of Linear or Non-linear Forms with Applications to Statistical Inference

There are many problems in statistics and industry where the distribution of the probabilistic model is not well-studied. One way to deal with the following issue is to approximate the unknown distribution with the good one. In our case ”the good one” is the Gaussian distribution. What is the point of Gaussian approximation? Let us given a sample of random vectors. We want to evaluate the probability of hitting some measurable set A by a sum of these vectors S_n. Gaussian approximation result (or GAR) allows us to replace these unconvenient probability (in distribution sense) with the probability of hitting the same set A by the Gaussian random vector with the same mean and covariance structure. Our work is dedicated to deriving the Gaussian approximation result for the case where A is defined as an intersection of ellipsoids, or, which is the same, as an event "the value of S_n over the set of quadratic forms is no greater than x". Also we suggest the application of our result to the multiplier bootstrap procedure.