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

Let (B, parallel to . parallel to) be a real separable Banach space of dimension 1 less than or equal to d less than or equal to infinity, and assume X,X-1, X-2,... are i.i.d. B valued random vectors with law mu=L(X) and mean m=integral(B) xdmu(x). Nummelin's conditional weak law of large numbers establishes that under suitable conditions on (D subset of B, mu) and for every epsilon > 0, lim(n) P(parallel toS(n)/n-a(0)parallel to < ε\S-n/n ∈ D)=1, with a(0) the dominating point of D and S-n = Σ(n)(j=1) X-j. We study the rates of convergence of such laws, i.e., we examine lim(n) P(parallel toS(n)/n - a(0)parallel to < 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 < infinity, but we also include results when d=infinity, mainly in Hilbert space. A connection to the Gibbs conditioning principle is also examined. (C) 2001 Elsevier Science B.V. All rights reserved.