Rates of convergence for the Nummelin conditional weak law of large numbers. (Q1766064)

Let \(B\) be a separable Banach space and let \(X_i\) be i.i.d.\ random vectors. Nummelin's conditional weak law of large numbers studies in which situations \[ \lim _{n\to \infty } P( \| S_n/n-a \| <\varepsilon \mid S_n/n\in D)=1 \] for all \(\varepsilon >0\), where \(D\subset B\) is an open convex set, \(a\in B\), and \(S_n/n = \sum _{i=1}^n X_i\). The authors discuss the rates of convergence, i.e.\ \(\varepsilon \) is substituted by \(t/n^r\) for \(t,r>0\). The results can be divided into two groups. One group solves the case when the mean \(m=\mathbf E X_i\) belongs to \(\partial D\), the other deals with the case \(m\notin \overline {D}\). In the latter case the set \(D\) is an open ball. The discussion is more complete for finite-dimensional spaces in both groups of results. At the end the authors mention the possibility to obtain the Gibbs conditioning principle via Nummelin conditional law in certain situations.