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 ${1, 2,\dots,m}$.