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 mu ltiple marker arrays with correlated distributions. In this paper, the mode l assumes two (or more) kinds of markers, say Markers A and B, distributed along the DNA sequence. The two primary conditions of interest are (i) many of Marker B (say greater than or equal to m) occur, and (ii) few of Marker B (say less than or equal to l) occur. We title these the conditional r-sc an models, and inquire on the extent to which Marker A clusters or is over- dispersed in regions satisfying condition (i) or(ii). Limiting distribution s for the extremal r-scan statistics from the A array satisfying conditions (i) and (ii) are derived by extending the Chen-Stein Poisson approximation method.