Poisson approximations for conditionalr-scan lengths of multiple renewal processes and application to marker arrays in biomolecular sequences

Abstract

This study is motivated by problems of molecular sequence comparison for multiple marker arrays with correlated distributions. In this paper, the model assumes two (or more) kinds of markers, say MarkersAandB, distributed along the DNA sequence. The two primary conditions of interest are (i) many of MarkerB(say ≥m) occur, and (ii) few of MarkerB(say ≤l) occur. We title these theconditional r-scan models, and inquire on the extent to which MarkerAclusters or is over-dispersed in regions satisfying condition (i) or (ii). Limiting distributions for the extremalr-scan statistics from theAarray satisfying conditions (i) and (ii) are derived by extending the Chen-Stein Poisson approximation method.