%0 Journal Article %1 watkins2000consistency %A Watkins, Joseph C. %D 2000 %J J. Math. Biol. %K CLT branching_processes deterministic_limit exchangeability large_populations urn_models %N 3 %P 253--271 %R 10.1007/s002850000041 %T Consistency and fluctuation theorems for discrete time structured population models having demographic stochasticity %U http://dx.doi.org/10.1007/s002850000041 %V 41 %X Abstract. In this paper we prove a consistency theorem (law of large numbers) and a fluctuation theorem (central limit theorem) for structured population processes. The basic assumptions for these theorems are that the individuals have no statistically distinguishing features beyond their class and that the interaction between any two individuals is not too high. We apply these results to density dependent models of Leslie type and to a model for flour beetle dynamics. MR A sequence of stochastic models is considered for the growth in discrete time of a population that is divided into classes. The sequence of models is indexed by $n$n, where $n$n is some measure of habitat size. Under a number of conditions, it is shown that the sequence of models has the consistency property that, for every time $t 0$t≥0, as $n ınfty$n→∞, the sequence of normalised vectors of class sizes $X^n(t)$X−−n(t) converges in mean square to $x(t)$x(t), the solution of a corresponding deterministic model. A fluctuation theorem is proved, according to which $n(X^n(t) - x(t))$n−√(X−−n(t)−x(t)) converges in distribution to $Z(t)$Z(t), where $Z$Z is a mean zero Gaussian process. The proof makes use of a central limit theorem for exchangeable random variables, which is proved in an appendix. The results are applied to density dependent models of Leslie type and to a model for flour beetle dynamics.

@article{watkins2000consistency, abstract = {Abstract. In this paper we prove a consistency theorem (law of large numbers) and a fluctuation theorem (central limit theorem) for structured population processes. The basic assumptions for these theorems are that the individuals have no statistically distinguishing features beyond their class and that the interaction between any two individuals is not too high. We apply these results to density dependent models of Leslie type and to a model for flour beetle dynamics. [MR] A sequence of stochastic models is considered for the growth in discrete time of a population that is divided into classes. The sequence of models is indexed by $n$n, where $n$n is some measure of habitat size. Under a number of conditions, it is shown that the sequence of models has the consistency property that, for every time $t \ge 0$t≥0, as $n \rightarrow \infty$n→∞, the sequence of normalised vectors of class sizes $\overline{X}^{n}(t)$X−−n(t) converges in mean square to $x(t)$x(t), the solution of a corresponding deterministic model. A fluctuation theorem is proved, according to which $\sqrt{n}(\overline{X}^{n}(t) - x(t))$n−√(X−−n(t)−x(t)) converges in distribution to $Z(t)$Z(t), where $Z$Z is a mean zero Gaussian process. The proof makes use of a central limit theorem for exchangeable random variables, which is proved in an appendix. The results are applied to density dependent models of Leslie type and to a model for flour beetle dynamics. }, added-at = {2011-04-21T23:47:58.000+0200}, author = {Watkins, Joseph C.}, biburl = {https://www.bibsonomy.org/bibtex/26abff905bd97448663c68cb415582cab/peter.ralph}, coden = {JMBLAJ}, description = {MR: Publications results for "MR Number=(1792676)"}, doi = {10.1007/s002850000041}, fjournal = {Journal of Mathematical Biology}, interhash = {4ea63da2d9c0940d7b9c7e67bdc1f58a}, intrahash = {6abff905bd97448663c68cb415582cab}, issn = {0303-6812}, journal = {J. Math. Biol.}, keywords = {CLT branching_processes deterministic_limit exchangeability large_populations urn_models}, mrclass = {92D25 (60F05)}, mrnumber = {1792676 (2001k:92060)}, mrreviewer = {Andris Abakuks}, number = 3, pages = {253--271}, timestamp = {2013-01-20T14:26:25.000+0100}, title = {Consistency and fluctuation theorems for discrete time structured population models having demographic stochasticity}, url = {http://dx.doi.org/10.1007/s002850000041}, volume = 41, year = 2000 }