Some Weak and Strong Laws of Large Numbers for D(0,1) - Valued Random Variables

摘要

Pointwise Weak Law of Large Numbers and Weak Law of Large Numbers in the norm topology of D(0,1) are shown to be equivalent under uniform convex tightness and uniform integrability conditions for weighted sums of a sequence of random elements in D(0,1). Uniform convex tightness and uniform integrability conditions are jointly characterized. Marcinkiewicz-Zygmund-Kolmogorov's and Brunk-Chung's Strong Laws of Large Numbers are derived in the setting of D(0,1) - space under uniform convex tightness and uniform integrability conditions. Equivalence of pointwise convergence, convergence in the Skorokhod topology and convergence in the norm topology for sequences in D(0,1) is studied. (Author)