$r$-scan statistics of a marker array in multiple sequences derived from a common progenitor

Abstract

This study is motivated by problems of molecular sequence comparisons for biological traits conserved or lost over evolution time.A marker of interest is distributed in the genome of the ancestor and inherited among $l$ offspring species which descend from this common ancestor. Each marker will be retained or lost during the evolution of the descendent species. The objective of the analysis here is to ascertain probabilities of clustering or overdispersion of the marker array among the sequences of the descendent species. Limiting distributions for the extremal $r$-scan statistics (defined in text) of the trait distributed among the $l$ dependent offspring processes are derived by adapting the Chen–Stein Poisson approximation method. Results that accommodate new occurrences of the trait (gene) arising from duplications and transposition occurrences are also described.The $r$-scan statistical analysis is further applied to a multi sequence combined Poisson model where ${B_1,\dots, B_l}$ are generated from $m$ independent Poisson processes ${A_1,\dots, A_m}$ such that $B_k = \bigcup_{i\epsilonZ_k}A_i$, where ${Z_k}_1\leqk\leql$ are subsets of ${1, 2,\dots,m}$.