On the Cramér transform, large deviations in boundary value problems, and the conditional invariance principle
Let \(S(n)\) denote the sum of \(n\) independent, identically distributed random variables \((\xi(i), i=1,2, \dots)\) in \(\mathbb{R}^d\) and let \(\Delta(x)\) be a \(d\)-dimensional cube around \(x\). The author gives a probabilistic interpretation of the Cramér transform in terms of the limit \(\lim_{n\to\infty}P[\xi(k(n))\in B\mid S(n)\in\Delta(x(n))]\). He then applies this result to the study of asymptotics of \(P[S(n)\in\Delta(x)\), \(S(1)\notin W_1,\dots,S(n-1)\notin W_{n-1}]\) and other problems with boundaries. In addition, he generalises the conditional invariance principle.
scientific article; zbMATH DE number 923516