Boundary crossing probabilities by nondifferentiable processes and applications to two-phase regression (Q1198994)

Let \(X_ 1,X_ 2,\dots,X_ m\) be independent random vectors with mean vector \(\mu\) and covariance matrix \(\Sigma\). Let \(S_ n=X_ 1+X_ 2+\dots+X_ n\), \(1\leq n\leq m\). The author derives an approximation for the following probability: \(P\{\| S_ n+\xi\|>a\) for some \(1\leq n\leq m\}\), where \(a=O(\sqrt{m})\) and \(\xi=O(\sqrt{m})\). Simulations are performed to assess the accuracy of the approximation. The accuracy improves as the sample size increases.