In this paper,we consider numerical and trigonometric series with a very general monotonicity condition.First,a fundamental decomposition is established from which the sufficient parts of many classical results in Fou...In this paper,we consider numerical and trigonometric series with a very general monotonicity condition.First,a fundamental decomposition is established from which the sufficient parts of many classical results in Fourier analysis can be derived in this general setting.In the second part of the paper a necessary and sufficient condition for the uniform convergence of sine series is proved generalizing a classical theorem of Chaundy and Jolliffe.展开更多
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
基金Supported by the European Research Council Advanced Grant(Grant No.267055)
文摘In this paper,we consider numerical and trigonometric series with a very general monotonicity condition.First,a fundamental decomposition is established from which the sufficient parts of many classical results in Fourier analysis can be derived in this general setting.In the second part of the paper a necessary and sufficient condition for the uniform convergence of sine series is proved generalizing a classical theorem of Chaundy and Jolliffe.
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.
