The existence and abundance of ghost ancestors in biparental populations

In a randomly-mating biparental population of size $N$ there are, with high probability, individuals who are genealogical ancestors of every extant individual within approximately $\log_2(N)$ generations into the past. We use this result of J. Chang to prove a curious corollary under standard models of recombination: there exist, with high probability, individuals within a constant multiple of $ \log_2(N)$ generations into the past who are simultaneously (i) genealogical ancestors of {\em each} of the individuals at the present, and (ii) genetic ancestors to {\em none} of the individuals at the present. Such ancestral individuals - ancestors of everyone today that left no genetic trace -- represent `ghost' ancestors in a strong sense. In this short note, we use simple analytical argument and simulations to estimate how many such individuals exist in finite Wright-Fisher populations.