Conditional limit theorems for exponential families and finite versions of de Finetti's theorem (Q1109411)

Let \(\{P_{\lambda}\}\) be an exponential family of probability measures, maximal, smooth, and with uniformly bounded, standardized 4 th moments. Consider i.i.d. random variables \(X_ i\), distributed according to \(P_{\lambda}\). Let \(P^ k_{\lambda}\) be the joint distribution of \(X_ 1\),..., \(X_ k\) and \(Q_{nsk}\) the joint distribution of \(X_ 1\),..., \(X_ k\) given that \(X_ 1+...+X_ n=s\). Choose \(\lambda\) such that \(E_{\lambda}X_ 1=s/n\) and let \(k\to \infty\), \(n\to \infty\) such that k/n\(\to 0\). Then \[ | Q_{nsk}-P^ k_{\lambda}| =\gamma k/n+o(k/n) \] uniformly in s with \(\gamma =E| 1-Z^ 2| /2\), Z standard normal. Similar results hold for k/n\(\to \infty\) and for mixtures of the \(P^ k_{\lambda}\). Versions of de Finetti's theorem are obtained as corollaries.