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

Let (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=∫Bx dμ(x). Nummelinʹs conditional weak law of large numbers establishes that under suitable conditions on (D⊂B,μ) and for every ε>0, limn P(∥Sn/n−a0∥<ε|Sn/n∈D)=1, with a0 the dominating point of D and Sn=∑j=1n Xj. We study the rates of convergence of such laws, i.e., we examine limn P(∥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 Nummelinʹs 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.