Boundedly finite measures: separation and convergence by an algebra of functions

Löhr, Wolfgang; Rippl, Thomas

doi:10.1214/16-ecp17

Cited by 6 publications

(10 citation statements)

References 21 publications

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“…Their approach rests on the Stone-Weierstrass theorem. Using Proposition 4.1 we are able to extend the results from Löhr and Rippl (2016) to the case of random measures. Proof.…”

Section: Lipschitz Functions Determine Convergence In Distributionmentioning

confidence: 84%

“…In Kallenberg (2017), a family C ⊆ CB + b (X) satisfying condition (ii) is called an approximating class of I and it was shown that such families determine vague convergence in M(X), see (Kallenberg, 2017, Lemma 4.1). Löhr and Rippl (2016) obtained a very general result which gives sufficient conditions for a family of functions to be convergence determining for vague convergence of measures in M(X), see (Löhr and Rippl, 2016, Theorem 2.3). Their approach rests on the Stone-Weierstrass theorem.…”

Section: Lipschitz Functions Determine Convergence In Distributionmentioning

confidence: 99%

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A note on vague convergence of measures

Basrak

Planinić

2019

Statistics & Probability Letters

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We propose a new approach to vague convergence of measures based on the general theory of boundedness due to Hu (1966). The article explains how this connects and unifies several frequently used types of vague convergence from the literature. Such an approach allows one to translate already developed results from one type of vague convergence to another. We further analyze the corresponding notion of vague topology and give a new and useful characterization of convergence in distribution of random measures in this topology.

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“…Their approach rests on the Stone-Weierstrass theorem. Using Proposition 4.1 we are able to extend the results from Löhr and Rippl (2016) to the case of random measures. Proof.…”

Section: Lipschitz Functions Determine Convergence In Distributionmentioning

confidence: 84%

Section: Lipschitz Functions Determine Convergence In Distributionmentioning

confidence: 99%

A note on vague convergence of measures

Basrak

Planinić

2019

Statistics & Probability Letters

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“…If, in addition, Π X is convergence determining and Y is Feller, E x [H(X t , y)] t→0 − −− → H(x, y) by (12) for all y, which implies…”

Section: Assume the Following Three Propertiesmentioning

confidence: 99%

“…For the measurability (6), we use Remark 2.4: We need to show that Π X is convergence determining and (t, x) → E y [ x AE , Y t ] is continuous. For the former, recall that by Le Cam's theorem [10] (see also [12]), the set of functions Π X ⊆ C b (E X ) on a completely regular Hausdorff space E X is convergence determining for Radon probability measures, if it is multiplicatively closed and induces the topology of E X . In our case,…”

Section: Examplesmentioning

confidence: 99%

Duality and the well-posedness of a martingale problem

Depperschmidt¹,

Greven²,

Pfaffelhuber³

2019

Preprint

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For two Polish state spaces E X and E Y , and an operator G X , we obtain existence and uniqueness of a G X -martingale problem provided there is a dual process Y on E Y solving a G Y -martingale problem. Duality here means the existence of a rich function H and transition kernels (µ t ) t≥0 on E X such that E y [H(x, Y t )] = µ t (x, dx ′ )H(x ′ , y) for all (x, y) ∈ E X ×E Y and t ≥ 0. While duality is well-known to imply uniqueness of the G X -martingale problem, we give here a set of conditions under which duality also implies existence without using approximation techniques. As examples, we treat branching and resampling models, such as Feller's branching diffusion and the Fleming-Viot superprocess.

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“…We here consider the open ε-neighborhood of 0 and the closed complements (which are Polish spaces), where we want to have the restriction of our measures to converge weakly as finite measures. This is usually formalized as follows (see [DVJ08,LR16] for this object).…”

Section: Excursion Law Entrance Law and Process Conditioned On Survivalmentioning

confidence: 99%

Genealogy-valued Feller diffusion

Depperschmidt¹,

Greven²

2019

Preprint

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We consider the evolution of the genealogy of the population currently alive in a Feller branching diffusion model. In contrast to the approach via labeled trees in the continuum random tree world, the genealogies are modeled as equivalence classes of ultrametric measure spaces, elements of the space Í. This space is Polish and has a rich semigroup structure for the genealogy. We focus on the evolution of the genealogy in time and the large time asymptotics conditioned both on survival up to present time and on survival forever. We develop the calculus in such a way that it can be applied in the future to more complicated systems as logistic branching or state dependent branching.We prove existence, uniqueness and Feller property of solutions of the martingale problem for this genealogy valued, i.e. Í-valued Feller diffusion. We give the precise relation to the time-inhomogeneous Í 1 -valued Fleming-Viot process. The uniqueness is shown via Feynman-Kac duality with the distance matrix augmented Kingman coalescent. Via the Feynman-Kac duality we deduce the generalized branching property of the Í-valued Feller diffusion. Using a semigroup operation on Í together with the branching property we obtain a Lévy-Khintchine formula for Í-valued Feller diffusion and determine explicitly the Lévy measure.For h > 0 we obtain a decomposition into depth-h subfamilies which leads to a representation of the in terms of a Cox point process of genealogies of single ancestor subfamilies. Furthermore, correcting a result from the Ê + -valued literature, we will identify the Í-valued process conditioned to survive until a finite time T . This is the key ingredient of the excursion law of the Í-valued Feller diffusion.We study long time asymptotics of the Í-valued Feller diffusion conditioned to survive forever, its generalized quasi-equilibrium and Kolmogorov-Yaglom limit law and show the limit processes solve well-posed Í-valued martingale problems. We also obtain various representations of the long time limits: backbone construction of the Palm distribution, the Í-valued version of the Kallenberg tree, the Í-valued version of Feller's branching diffusion with immigration from an immortal line à la Evans. This requires considering conditioned martingale problems which are of different form that those in the unconditioned case. On the level of Ívalued processes we still have equality (in law) of the Q-process, i.e., the process conditioned to survive up to time T in the limit T → ∞, the size-biased process and Evans' branching process with immigration from an immortal line. The Í-valued generalized quasi-equilibrium is a size-biased version of the Kolmogorov-Yaglom limit law.The above results are also key tools for analyzing genealogies in spatial branching populations. We construct the genealogy of the spatial version of the Feller diffusion on a countable group (super random walk). We give results on a martingale problem characterization, duality, generalized branching property and the long time behavior for this object. As an appl...

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Boundedly finite measures: separation and convergence by an algebra of functions

Cited by 6 publications

References 21 publications

A note on vague convergence of measures

A note on vague convergence of measures

Duality and the well-posedness of a martingale problem

Genealogy-valued Feller diffusion

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