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A continuous-state branching process with immigration having branching mechanism
and immigration mechanism
, a CBI
process for short, may have either of two different asymptotic regimes, depending on whether
, the CBI process has either a limit distribution or a growth rate dictated by the branching dynamics. When
, immigration overwhelms branching dynamics. Asymptotics in the latter case are studied via a nonlinear time-dependent renormalization in law. Three regimes of weak convergence are exhibited. Processes with critical branching mechanisms subject to a regular variation assumption are studied. This article proves and extends results stated by M. Pinsky in ‘Limit theorems for continuous state branching processes with immigration’ (Bull. Amer. Math. Soc.78, 1972).
For a one-locus haploid infinite population with discrete generations, the celebrated model of Kingman describes the evolution of fitness distributions under the competition of selection and mutation, with a constant mutation probability. This paper generalises Kingman’s model by using independent and identically distributed random mutation probabilities, to reflect the influence of a random environment. The weak convergence of fitness distributions to the globally stable equilibrium is proved. Condensation occurs when almost surely a positive proportion of the population travels to and condenses at the largest fitness value. Condensation may occur when selection is favoured over mutation. A criterion for the occurrence of condensation is given.
In this paper we consider the beta(2 − α, α)-coalescents with 1 < α < 2 and study the moments of external branches, in particular, the total external branch length of an initial sample of n individuals. For this class of coalescents, it has been proved that nα-1T(n) →DT, where T(n) is the length of an external branch chosen at random and T is a known nonnegative random variable. For beta(2 − α, α)-coalescents with 1 < α < 2, we obtain limn→+∞n3α-5 𝔼(Lext(n) − n2-α𝔼T)2 = ((α − 1)Γ(α + 1))2Γ(4 − α) / ((3 − α)Γ(4 − 2α)).
A dual-axis rotational Inertial Navigation System (INS) has received wide attention in recent years because of high performance and low cost. However, some errors of inertial sensors such as stochastic errors are not averaged out automatically during navigation. Therefore a Twice Position-fix Reset (TPR) method is provided to enhance accuracy of a dual-axis rotational INS by compensating stochastic errors. According to characteristics of an azimuth error introduced by stochastic errors of an inertial sensor in the dual-axis rotational INS, both an azimuth error and a radial-position error are much better corrected by the TPR method based on an optimised error propagation equation. As a result, accuracy of the dual-axis rotational INS is prominently enhanced by the TPR method, as is verified by simulations and field tests.
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