Invariance Principles for Random Processes Generated by Extrema and Partial Sums of Random Variables

101 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1982.Assume that F is a distribution function such that for some non-degenerate distribution function G, and sequences of constants a(,n) (a(,n) positive) and b(,n), we have that, as n increases, F('n)(a(,n)x + b(,n)) converges to G(x) for every real value x. Under these conditions we prove a strong invariance principle for extremal processes analogous to the well-known invariance principle of Strassen for partial sums. In our case, the strong invariance principle yields several weak convergence results for extremal processes. This differs from the case of partial sums in that Strassen's result does not yield the central limit theorem. By a similar procedure, we also obtain a strong invariance principle for point processes in the plane.Using a different method, we prove an invariance principle in probability for extremal processes arising from stationary sequences satisfying certain weak dependence conditions. Of independent interest is a generalization to dependent sequences of a technique of Major used to obtain an universal approximating sequence from a collection of sequences, each of which approximates the desired sequence only to within a fixed positive tolerance.We improve a strong invariance principle for partial sums of a stationary (phi)-mixing sequence of random variables with finite (2 + (delta)) moment (0 < (delta) (LESSTHEQ) 2) due to Berkes and Philipp. The rate of decay, (phi)(n), can be as slow as log n to the power -(1 + (epsilon)) (1 + 2/(delta)), where (epsilon) is some positive value. The error term obtained is sufficient to yield the central limit theorem and upper and lower class refinements to the law of the iterated logarithm.The generalization of a technique of Major mentioned earlier is used to improve an invariance principle in probability due to Philipp for partial sums of (phi)-mixing sequences of Banach space valued random variables in the domain of attraction to a Gaussian law.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD