Fima C. Klebaner

A limit theorem for population-size-dependent branching processes

An analogue of the Kesten-Stigum theorem, and sufficient conditions for the geometric rate of growth in the rth mean and almost surely, are obtained for population-size-dependent branching processes.

Random variation and concentration effects in PCR

Even though the efficiency of the polymerase chain reaction (PCR) reaction decreases, analyses are made in terms of Galton-Watson processes, or simple deterministic models with constant replication probability (efficiency). Recently, Schnell and Mendoza have suggested that the form of the efficiency, can be derived from enzyme kinetics. This results in the sequence of molecules numbers forming a stochastic process with the properties of a branching process with population size dependence, which is supercritical, but has a mean reproduction number that approaches one. Such processes display ultimate linear growth, after an initial exponential phase, as is the case in PCR. It is also shown that the resulting stochastic process for a large Michaelis-Menten constant behaves like the deterministic sequence x(n) arising by iterations of the function f(x)=x+x/(1+x).

Population-size-dependent and age-dependent branching processes

Supercritical branching processes are considered which are Markovian in the age structure but where reproduction parameters may depend upon population size and even the age structure of the population. Such processes generalize Bellman-Harris processes as well as customary demographic processes where individuals give birth during their lives but in a purely age-determined manner. Although the total population size of such a process is not Markovian the age chart of all individuals certainly is. We give the generator of this process, and a stochastic equation from which the asymptotic behaviour of the process is obtained, provided individuals are measured in a suitable way (with weights according to Fishers reproductive value). The approach so far is that of stochastic calculus. General supercritical asymptotics then follows from a combination of L2 arguments and Tauberian theorems. It is shown that when the reproduction and life span parameters stabilise suitably during growth, then the process exhibits exponential growth as in the classical case. Application of the approach to, say, the classical Bellman-Harris process gives an alternative way of establishing its asymptotic theory and produces a number of martingales.

On the path to extinction

Populations can die out in many ways. We investigate one basic form of extinction, stable or intrinsic extinction, caused by individuals on the average not being able to replace themselves through reproduction. The archetypical such population is a subcritical branching process, i.e., a population of independent, asexually reproducing individuals, for which the expected number of progeny per individual is less than one. The main purpose is to uncover a fundamental pattern of nature. Mathematically, this emerges in large systems, in our case subcritical populations, starting from a large number, x, of individuals. First we describe the behavior of the time to extinction T: as x grows to infinity, it behaves like the logarithm of x, divided by r, where r is the absolute value of the Malthusian parameter. We give a more precise description in terms of extreme value distributions. Then we study population size partway (or u-way) to extinction, i.e., at times uT, for 0 < u < 1, e.g., u = 1/2 gives halfway to extinction. (Note that mathematically this is no stopping time.) If the population starts from x individuals, then for large x, the proper scaling for the population size at time uT is x into the power u − 1. Normed by this factor, the population u-way to extinction approaches a process, which involves constants that are determined by life span and reproduction distributions, and a random variable that follows the classical Gumbel distribution in the continuous time case. In the Markov case, where an explicit representation can be deduced, we also find a description of the behavior immediately before extinction.

A Family of Non-Gaussian Martingales with Gaussian Marginals

We construct a family of martingales with Gaussian marginal distributions. We give a weak construction as Markov, inhomogeneous in time processes, and compute their infinitesimal generators. We give the predictable quadratic variation and show that the paths are not continuous. The construction uses distributions Gσ having a log-convolution semigroup property. Further, we categorize these processes as belonging to one of two classes, one of which is made up of piecewise deterministic pure jump processes. This class includes the case where Gσ is an inverse log-Poisson distribution. The processes in the second class include the case where Gσ is an inverse log-gamma distribution. The richness of the family has the potential to allow for the imposition of specifications other than the marginal distributions.

CONDITIONS FOR INTEGRABILITY OF MARKOV CHAINS

We give simple sufficient conditions for integrability of continuous-time Markov chains in terms of their infinitesimal parameters. Similar conditions for regularity are stated first, and a simple proof given.

Population-dependent branching processes with a threshold

A branching process model where offspring distributions depend on the threshold as well as on the population size is introduced. Behaviour of such models is related to the behaviour of the corresponding deterministic models, whose behaviour is known from the chaos theory. Asymptotic behaviour of such branching processes is obtained when the population threshold is large. If the initial population size is comparable to the threshold then the size of the nth generation relative to the threshold has a normal distribution with the mean being the nth iterate of the one-step mean function. If the initial population size is negligible when compared to the threshold and the offspring distributions converge then the size of any fixed generation approaches that of a size-dependent branching process. These results are supported by a simulation study.

Tracking Volatility

We propose an adaptive algorithm for tracking historical volatility. The algorithm borrows ideas from nonparametric statistics. In particular, we assume that the volatility is a several times differentiable function with a bounded highest derivative. We propose an adaptive algorithm with a Kalman filter structure, which guarantees the same asymptotics (well known from statistical inference) with respect to the sample size n, n → ∞. The tuning procedure for this filter is simpler than for a GARCH filter.

Autoregressive approximation in branching processes with a threshold

It is shown that population dependent branching processes for large values of threshold can be approximated by Gaussian processes centered at the iterates of the corresponding deterministic function. If the deterministic system has a stable limit cycle, then in the vicinity of the cycle points the corresponding stochastic system can be approximated by an autoregressive process. It is shown that it is possible to speed up convergence to the limit so that the processes converge weakly to the stationary autoregressive process. Similar results hold for noisy dynamical systems when random noise satisfies certain conditions and the corresponding dynamical system has stable limit cycles.

Stochasticity in the adaptive dynamics of evolution: the bare bones

First a population model with one single type of individuals is considered. Individuals reproduce asexually by splitting into two, with a population-size-dependent probability. Population extinction, growth and persistence are studied. Subsequently the results are extended to such a population with two competing morphs and are applied to a simple model, where morphs arise through mutation. The movement in the trait space of a monomorphic population and its possible branching into polymorphism are discussed. This is a first report. It purports to display the basic conceptual structure of a simple exact probabilistic formulation of adaptive dynamics.