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 certain rate that can be characterized as the unique positive zero of some limiting logarithmic moment generating function. This limiting logarithmic moment generating function in some sense describes the asymptotic single step distribution of the process. The result extends in a natural way the corresponding result from the world of random walks. We apply our results to sequence alignments with gaps. Calculating p-values of optimal gapped alignment scores still is 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 on the level of large deviations. Although a mathematical satisfying description of the optimal score’s distribution is still far from reach, our results fit nicely into the common practice of statistical evaluation of optimal alignment results.