Large deviations for global maxima of independent superadditive processes with negative drift and an application to optimal sequence alignments

We examine the distribution of the global maximum of an independent superadditive process with negative drift. We show that, under certain conditions, the distribution's upper tail decays exponentially at a rate that can be characterized as the unique positive zero of some limiting logarithmic moment generating function. This result extends the corresponding one for random walks with a negative drift. We apply our results to sequence alignments with gaps. Calculating p-values of optimal gapped alignment scores is still one of the most challenging mathematical problems in bioinformatics. Our results provide a better understanding of the tail of the optimal score's distribution, especially at the level of large deviations, and they are in accord with common practice of statistical evaluation of optimal alignment results. However, a complete mathematical description of the optimal score's distribution remains far from reach.