Central limit theorems of partial sums for large segmental values

Let (Xi,Ui) be i.i.d., Xi real valued and Ui vector valued, bounded random variables or governed by a finite state Markov chain. Assuming that E[X]<0 and P(X> 0) > 0, central limit theorems are derived for [Sigma]iUi on segments conditioned that [Sigma]iXi is increasingly high, going to +[infinity]. While these segments are exponentially rare, they are of importance in many models of stochastic analysis including queueing systems and molecular sequence comparisons. Particular applications give central limit theorems for the empirical frequencies over such segments and for their length.