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

AbstractLet (B,∥·∥) be a real separable Banach space of dimension 1⩽d⩽∞, and assume X,X1,X2,… are i.i.d. B valued random vectors with law μ=L(X) and mean m=∫Bxdμ(x). Nummelin's conditional weak law of large numbers establishes that under suitable conditions on (D⊂B,μ) and for every ε>0,limnP(∥Sn/n−a0∥<ε|Sn/n∈D)=1, with a0 the dominating point of D and Sn=∑j=1nXj. We study the rates of convergence of such laws, i.e., we examine limnP(∥Sn/n−a0∥<t/nr|Sn/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