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

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 marker, say Markers A and B, distributed along the DNA sequence. The two primary conditions of interest are (i) many of Marker B (say ≥ m) occur, and (ii) few of Marker B (say ≥ l) occur. We title these the conditional r-scan models, and inquire on the extent to which Marker A clusters or is over-dispersed in regions satisfying condition (i) or (ii). Limiting distributions for the extremal r-scan statistics form the A array satisfying conditions (i) and (ii) are derived by extending the Chen–Stein Poisson approximation method.