Rates of convergence for the Nummelin conditional weak law of large numbers

Abstract

Let ( B,∥·∥) be a real separable Banach space of dimension 1⩽ d⩽∞, and assume X, X 1, X 2,… are i.i.d. B valued random vectors with law μ= L(X) and mean m= ∫ B x dμ(x) . Nummelin's conditional weak law of large numbers establishes that under suitable conditions on ( D⊂ B, μ) and for every ε>0, lim n P(∥S n/n−a 0∥<ε|S n/n∈D)=1 , with a 0 the dominating point of D and S n= ∑ j=1 n X j . We study the rates of convergence of such laws, i.e., we examine lim n P(∥S n/n−a 0∥<t/n r|S n/n∈D) as d, r, t and D vary. It turns out that the limit is sensitive to variations in these parameters. Additionally, we supply another proof of Nummelin's law of large numbers. Our results are most complete when 1⩽ d<∞, but we also include results when d=∞, mainly in Hilbert space. A connection to the Gibbs conditioning principle is also examined.