Relaxing Independence in the Marchenko-Pastur Law for Random Matrices and the Application to Approximate Embeddings
This dissertation adds to the collection of works studying the Marchenko-Pastur law in two ways. First it considers two new models of random column vectors that have weaker independence hypotheses than the well-known i.i.d. hypothesis and shows that random matrices, formed by concatenating random column vectors of a model, still follow the Marchenko-Pastur law. The two models of random column vectors are block columns and vectorized tensor columns. The block column vectors will be made up of n blocks each of length d, with d=o(n). If the entries are mean zero, variance one, have uniformly bounded fourth moments, entries within a block are uncorrelated, and entries in different blocks are independent, then the Marchenko-Pastur theorem holds as n tends to infinity. Furthermore, if additionally an exchangeability criteria is satisfied, then the theorem holds without requiring d=o(n). The vectorized tensor columns will be made up of a vectorized t-tensor of an i.i.d. vector of length n which has entries that are mean zero, variance one, uniformly bounded fourth moments, and t^3=o(n), and again the Marchenko-Pastur theorem holds as n tends to infinity.
The second contribution of this dissertation is in studying a particular type of an approximate embedding of vectors. For any collection of vectors, a general lower bound for the least dimension required for an approximate embedding is given. For vectors which are column vectors of a matrix that follows the Marchenko-Pastur law, an asymptotic formula for the exact value of the least dimension required is given. Numerical results show this asymptotic formula holds quite well, even for relatively small dimensions. Because this works so well for small dimensions, this gives an easy numerical test that provides evidence for answering the question, "Does a specific covariance structure have a limiting spectral distribution or not?" We consider a particular covariance structure which relates to the number of terms needed in the Karhunen-Loeve expansion to approximate a random field within a specified tolerance.