Under a first order moment condition on the immigration mechanism,we show that an appropriately scaled supercritical and irreducible multi-type continuous state and continuous time branching process with immigration(C...Under a first order moment condition on the immigration mechanism,we show that an appropriately scaled supercritical and irreducible multi-type continuous state and continuous time branching process with immigration(CBI process)converges almost surely.If an x log(x)moment condition on the branching mechanism does not hold,then the limit is zero.If this x log(x)moment condition holds,then we prove L1 convergence as well.The projection of the limit on any left non-Perron eigenvector of the branching mean matrix is vanishing.If,in addition,a suitable extra power moment condition on the branching mechanism holds,then we provide the correct scaling for the projection of a CBI process on certain left non-Perron eigenvectors of the branching mean matrix in order to have almost sure and L1 limit.Moreover,under a second order moment condition on the branching and immigration mechanisms,we prove L2 convergence of an appropriately scaled process and the above-mentioned projections as well.A representation of the limits is also provided under the same moment conditions.展开更多
Motivated by sample path decomposition of the stationary continuous state branching process with immigration, a general population model is considered using the idea of immortal individual. We compute the joint distri...Motivated by sample path decomposition of the stationary continuous state branching process with immigration, a general population model is considered using the idea of immortal individual. We compute the joint distribution of the random variables： the time to the most recent common ancestor （MRCA）, the size of the current population, and the size of the population just before MRCA. We obtain the bottleneck effect as well. The distribution of the number of the oldest families is also established. These generalize the results obtained by Y. T. Chen and J. F. Delmas.展开更多
In this paper, a three dimensional matrix valued rational interpolant (TGMRI) is first constructed by making use of the generalized inverse of matrices. The interpolants are of the Thiele type branched continued fra...In this paper, a three dimensional matrix valued rational interpolant (TGMRI) is first constructed by making use of the generalized inverse of matrices. The interpolants are of the Thiele type branched continued fraction form, with matrix numerator and scalar denominator. Some properties of TGMRI are given. An efficient recursive algorithm is proposed. The results in the paper can be extend to n variable.展开更多
Bivariate vector valued rational interpolants are established by means of Thiele-type branched continued fractions and Samelson inverse over rectangular grids with holes, characterisation theorem with topologic struct...Bivariate vector valued rational interpolants are established by means of Thiele-type branched continued fractions and Samelson inverse over rectangular grids with holes, characterisation theorem with topologic structure is brought in light and uniqueness theorem in some sense is obtained.展开更多
基金supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciencessupported by the Royal Society Newton International Fellowship and the EU-funded Hungarian(Grant No.EFOP-3.6.1-16-2016-00008)。
文摘Under a first order moment condition on the immigration mechanism,we show that an appropriately scaled supercritical and irreducible multi-type continuous state and continuous time branching process with immigration(CBI process)converges almost surely.If an x log(x)moment condition on the branching mechanism does not hold,then the limit is zero.If this x log(x)moment condition holds,then we prove L1 convergence as well.The projection of the limit on any left non-Perron eigenvector of the branching mean matrix is vanishing.If,in addition,a suitable extra power moment condition on the branching mechanism holds,then we provide the correct scaling for the projection of a CBI process on certain left non-Perron eigenvectors of the branching mean matrix in order to have almost sure and L1 limit.Moreover,under a second order moment condition on the branching and immigration mechanisms,we prove L2 convergence of an appropriately scaled process and the above-mentioned projections as well.A representation of the limits is also provided under the same moment conditions.
基金Acknowledgements The author would like to express his sincere thanks to his advisor Professor Zenghu Li for his persistent encouragements and suggestions and Professor J. F. Delmas for his careful check of this work. Thanks are also given to the anonymous referees for the suggestions. This work was supported in part by the National Natural Science Foundation of China （Grant No. 11131003） and the 985 Program.
文摘Motivated by sample path decomposition of the stationary continuous state branching process with immigration, a general population model is considered using the idea of immortal individual. We compute the joint distribution of the random variables： the time to the most recent common ancestor （MRCA）, the size of the current population, and the size of the population just before MRCA. We obtain the bottleneck effect as well. The distribution of the number of the oldest families is also established. These generalize the results obtained by Y. T. Chen and J. F. Delmas.
文摘In this paper, a three dimensional matrix valued rational interpolant (TGMRI) is first constructed by making use of the generalized inverse of matrices. The interpolants are of the Thiele type branched continued fraction form, with matrix numerator and scalar denominator. Some properties of TGMRI are given. An efficient recursive algorithm is proposed. The results in the paper can be extend to n variable.
文摘Bivariate vector valued rational interpolants are established by means of Thiele-type branched continued fractions and Samelson inverse over rectangular grids with holes, characterisation theorem with topologic structure is brought in light and uniqueness theorem in some sense is obtained.
