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14
Limit theorems for triangular urn schemes
 PROB. THEORY RELATED FIELDS
, 2005
"... We study a generalized Pólya urn with balls of two colours and a triangular replacement matrix; the urn is not required to be balanced. We prove limit theorems describing the asymptotic distribution of the composition of the urn after a long time. Several different types of asymptotics appear, depen ..."
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Cited by 57 (3 self)
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We study a generalized Pólya urn with balls of two colours and a triangular replacement matrix; the urn is not required to be balanced. We prove limit theorems describing the asymptotic distribution of the composition of the urn after a long time. Several different types of asymptotics appear, depending on the ratio of the diagonal elements in the replacement matrix; the limit laws include normal, stable and MittagLeffler distributions as well as some less familiar ones. The results are in some cases similar to, but in other cases strikingly different from, the results for irreducible replacement matrices.
Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
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Cited by 48 (3 self)
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Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
Branching processes with immigration and related topics
 Front. Math. China
"... Abstract: This is a survey on recent progresses in the study of branching processes with immigration, generalized OrnsteinUhlenbeck processes and affine Markov processes. We mainly focus on the applications of skew convolution semigroups and the connections in those processes. ..."
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Cited by 6 (0 self)
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Abstract: This is a survey on recent progresses in the study of branching processes with immigration, generalized OrnsteinUhlenbeck processes and affine Markov processes. We mainly focus on the applications of skew convolution semigroups and the connections in those processes.
Growth of GaltonWatson trees: immigration and lifetimes
, 2010
"... We study certain consistent families (Fλ)λ≥0 of GaltonWatson forests with lifetimes as edge lengths and/or immigrants as progenitors of the trees in Fλ. Specifically, consistency here refers to the property that for each µ ≤ λ, the forest Fµ has the same distribution as the subforest of Fλ spanned ..."
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Cited by 3 (0 self)
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We study certain consistent families (Fλ)λ≥0 of GaltonWatson forests with lifetimes as edge lengths and/or immigrants as progenitors of the trees in Fλ. Specifically, consistency here refers to the property that for each µ ≤ λ, the forest Fµ has the same distribution as the subforest of Fλ spanned by the black leaves in a Bernoulli leaf colouring, where each leaf of Fλ is coloured in black independently with probability µ/λ. The case of exponentially distributed lifetimes and no immigration was studied by Duquesne and Winkel and related to the genealogy of Markovian continuousstate branching processes. We characterise here such families in the framework of arbitrary lifetime distributions and immigration according to a renewal process, related to Sagitov’s (nonMarkovian)generalisation of continuousstate branching renewal processes, and similar processes with immigration.
Proof(s) of the Lamperti representation of continuous state branching processes
, 2008
"... Abstract. The representation of continuousstate branching processes (CSBPs) as timechanged Lévy processes with no negative jumps was discovered by John Lamperti in 1967 but was never proved. The goal of this paper is to provide a proof, and we actually provide two. The first one relies on studying ..."
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Cited by 2 (1 self)
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Abstract. The representation of continuousstate branching processes (CSBPs) as timechanged Lévy processes with no negative jumps was discovered by John Lamperti in 1967 but was never proved. The goal of this paper is to provide a proof, and we actually provide two. The first one relies on studying the timechange, using martingales and the LévyItô representation of Lévy processes. It gives insight into a stochastic differential equation satisfied by CSBPs and on its relevance to the branching property. The other method studies the timechange in a discrete model, where an analogous Lamperti representation is evident, and provides functional approximations to Lamperti transforms by introducing a new topology on Skorohod space. Some classical arguments used to study CSBPs are reconsidered and simplified. 1.
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, 908
"... Asymptotic regimes for the partition into colonies of a branching process with emigration ..."
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Asymptotic regimes for the partition into colonies of a branching process with emigration
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, 904
"... A limit theorem for trees of alleles in branching processes with rare neutral mutations ..."
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A limit theorem for trees of alleles in branching processes with rare neutral mutations