Abstract
We consider the weak convergence of iterates of so-called centred quadratic stochastic operators. These iterations allow us to study the discrete time evolution of probability distributions of vector-valued traits in populations of inbreeding or hermaphroditic species, whenever the offspring’s trait is equal to an additively perturbed arithmetic mean of the parents’ traits. It is shown that for the existence of a weak limit, it is sufficient that the distributions of the trait and the perturbation have a finite variance or have tails controlled by a suitable power function. In particular, probability distributions from the domain of attraction of stable distributions have found an application, although in general the limit is not stable.
Highlights
The theory of quadratic stochastic operators (QSOs) is rooted in the work of Bernstein [6]. Such operators are applied there to model the evolution of a discrete probability distribution of a finite number of biotypes in a process of inheritance
Since the seventies of the twentieth century, the field is steadily evolving in many directions, for a detailed review of mathematical results and open problems, see [15]
In the infinite dimensional case, QSOs were first considered on the 1 space, containing the discrete probability distributions
Summary
The theory of quadratic stochastic operators (QSOs) is rooted in the work of Bernstein [6]. Many interesting models were considered in [16] which is interesting due to the presented extensions and indicated possibilities of studying limit behaviours of infinite dimensional quadratic stochastic operators through finite dimensional ones. One needs more restrictive, appropriate for L1 spaces, assumptions on the QSO, e.g. in [4] a kernel form (cf Definition 1) was assumed Even in this subclass, it is not readily possible to prove convergence of a trajectory of a QSO. The operators are built into models of continuous time evolution of the trait’s distribution and the size of the population With these (and additional technical assumptions, like bounds on moment growth), they obtained a convergence slightly stronger than weak convergence.
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