We investigate conditions under which a model with stochastic demography or population structure converges to the coalescent with a linear change in timescale. We argue that this is a necessary condition for the existence of a meaningful effective population size. We find that such a linear timescale change is obtained when demographic fluctuations and coalescence events occur on different timescales. Simple models of population structure and randomly fluctuating population size are used to exemplify the ideas and provide an intuitive feel for the meaning of the conditions.

The ratio of divergence at nonsynonymous and synonymous sites, dN/dS, is a widely used measure in evolutionary genetic studies to investigate the extent to which selection modulates gene sequence evolution. Originally tailored to codon sequences of distantly related lineages, dN/dS represents the ratio of fixed nonsynonymous to synonymous differences. The impact of ancestral and lineage-specific polymorphisms on dN/dS, which we here show to be substantial for closely related lineages, is generally neglected in estimation techniques of dN/dS. To address this issue, we formulate a codon model that is firmly anchored in population genetic theory, derive analytical expressions for the dN/dS measure by Poisson random field approximation in a Markovian framework and validate the derivations by simulations. In good agreement, simulations and analytical derivations demonstrate that dN/dS is biased by polymorphisms at short time scales and that it can take substantial time for the expected value to settle at its time limit where only fixed differences are considered. We further show that in any attempt to estimate the dN/dS ratio from empirical data the effect of the intrinsic fluctuations of a ratio of stochastic variables, can even under neutrality yield extreme values of dN/dS at short time scales or in regions of low mutation rate. Taken together, our results have significant implications for the interpretation of dN/dS estimates, the McDonald–Kreitman test and other related statistics, in particular for closely related lineages.

We study the genealogical structure of samples from a population for which any given generation is made up of direct descendents from several previous generations. These occur in nature when there are seed banks or egg banks allowing an individual to leave offspring several generations in the future. We show how this temporal structure in the reproduction mechanism causes a decrease in the coalescence rate. We also investigate the effects of age-dependent neutral mutations.Our main result gives weak convergence of the scaled ancestral process, with the usual diffusion scaling, to a coalescent process which is equivalent to a time-changed version of Kingman's coalescent. Running head: Seed Bank Coalescents

This is an author-produced version of a chapter published in In and Out of Equilibrium 2. It does not include the final publisher proof-corrections or pagination.Citation for the published chapter: Kaj, I.; Taqqu, M. S. "Convergence to fractional Brownian motion and to the Telecom process: the integral representation approach" In: In and Out of Equilibrium 2. Ed. V. Sidoravicius and M. E. Vares. Basel:Abstract. It has become common practice to use heavy-tailed distributions in order to describe the variations in time and space of network traffic workloads. The asymptotic behavior of these workloads is complex; different limit processes emerge depending on the specifics of the work arrival structure and the nature of the asymptotic scaling. We focus on two variants of the infinite source Poisson model and provide a coherent and unified presentation of the scaling theory by using integral representations. This allows us to understand physically why the various limit processes arise.3. In Section 4, the convergence in finite-dimensional distributions of the continuous flow model is extended to weak convergence in function space. The infinite source Poisson modelInfinite source Poisson models are arrival processes with M/G/∞ input obtained by integrating the standard M/G/∞ queueing system size. The resulting class of Poisson shot noise processes are widely used traffic models which describe the amount of workload accumulating over time. Such models have been suggested as realistic workload processes for Internet traffic, where is is natural to assume that while web sessions are inititated according to a Poisson process, duration lengths and transmission rates could vary considerably. More exactly, the aggregated traffic consists of sessions with starting points distributed according to a Poisson process on the real time line. Each session lasts a random length of time and involves workload arriving at a random transmission rate. There are two slightly different sets of assumptions that are natural to make regarding the precise traffic pattern during a session. The first is that the workload arrives continuously at a randomly chosen transmission rate, which is fixed throughout the session and independent of the session length. The second type of model assumes that the workload arrives in discrete entities, packets, according to a Poisson process throughout the session, and such that the size of each packet is chosen independently from a given packet size distribution. The duration and the continuous or discrete rate of traffic in one session is independent of the traffic in any other session, although in general the sessions overlap. One novelty in this work is that we point out how these two types of models differ in their asymptotic behavior and that we explain the origin of the qualitative differencies.We are going to introduce the workload models using directly an integral representation with respect to Poisson measures, as in Kurtz (1996) andÇ aglar (2004), rather than working with a more traditional Poisson shot...

We study the genealogical structure of samples from a population for which any given generation is made up of direct descendents from several previous generations. These occur in nature when there are seed banks or egg banks allowing an individual to leave offspring several generations in the future. We show how this temporal structure in the reproduction mechanism causes a decrease in the coalescence rate. We also investigate the effects of age-dependent neutral mutations.Our main result gives weak convergence of the scaled ancestral process, with the usual diffusion scaling, to a coalescent process which is equivalent to a time-changed version of Kingman's coalescent. Running head: Seed Bank Coalescents