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Weak Stability of Centred Quadratic Stochastic Operators

Krzysztof BartoszekDepartment of Mathematics, Uppsala University, 751 06, Uppsala, SwedenJoachim DomstaThe Krzysztof Brzeski Institute of Applied Informatics, The State University of Applied Sciences in Elbla̧g, ul. Wojska Polskiego 1, 82–300, Elbląg, PolandMałgorzata PułkaDepartment of Probability and Biomathematics, Gdańsk University of Technology, ul. Narutowicza 11/12, 80–233, Gdańsk, Poland

2017en

ABI

Annotatsiya

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.

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DOI: 10.1007/s40840-017-0575-8

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