On coupling of discrete renewal processes

Lindvall, Torgny

doi:10.1007/bf00534882

Cited by 55 publications

(21 citation statements)

References 5 publications

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“…The question of existence of a common ancestor and the time to the MRCA can therefore be investigated via classical results of Lindvall [8] on coupling times of discrete renewal processes, which are controlled in the power-law case via applications of Karamata's Tauberian theorem for power series; see, e.g. .…”

Section: Model and Main Resultsmentioning

confidence: 99%

The Ancestral Process of Long-Range Seed Bank Models

Blath¹,

Casanova²,

Kurt³

et al. 2013

J. Appl. Probab.

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We present a new model for seed banks, where direct ancestors of individuals may have lived in the near as well as the very far past. The classical Wright-Fisher model, as well as a seed bank model with bounded age distribution considered in Kaj, Krone and Lascoux (2001) are special cases of our model. We discern three parameter regimes of the seed bank age distribution, which lead to substantially different behaviour in terms of genetic variability, in particular with respect to fixation of types and time to the most recent common ancestor. We prove that, for age distributions with finite mean, the ancestral process converges to a time-changed Kingman coalescent, while in the case of infinite mean, ancestral lineages might not merge at all with positive probability. Furthermore, we present a construction of the forward-in-time process in equilibrium. The mathematical methods are based on renewal theory, the urn process introduced in Kaj, Krone and Lascoux (2001) as well as on a paper by Hammond and Sheffield (2013).

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Section: Model and Main Resultsmentioning

confidence: 99%

The Ancestral Process of Long-Range Seed Bank Models

Blath¹,

Casanova²,

Kurt³

et al. 2013

J. Appl. Probab.

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“…The recurrence distribution generating the renewal sequence {UJ/UQ} has a finite moment of order i>, whence Proposition 2 of Lindvall [6] yields Uoc -Uj = o(j~u) and since J^ffj = 1 w e conclude that the above sum is o(j~"). The remaining component is In view of these, the dominated convergence theorem can be used to show that the above sum is o(j~"), and the proof is complete.…”

Section: 5) Is O{j~v)mentioning

confidence: 85%

“…Jo Let y -F t (6) and use the backward equation (1.1) to obtain the simpler expression terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700030445 150 Anthony G. Pakes [5] Letting 6 -0 in this integral gives the well-known formula for E{(T) and in particular we see t h a t these expectations are either both finite or both infinite.…”

Section: The Time Of Last Mutationmentioning

confidence: 99%

An infinite alleles version of the Markov branching process

Pakes

1989

J Aust Math Soc A

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Individuals in a population which grows according to the rules defining the Markov branching process can mutate into novel allelic forms. We obtain some results about the time of the last mutation and the limiting frequency spectrum. In the present context these results refine certain results obtained in the discrete time case and they answer some conjectures still unresolved for the discrete time case.

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“…:n and similarly '1~+ =(ij~)k? :n· Use of Lindvall (1979), Proposition 1 gives the assertion (i) = (ii). -The result (ii) ==> (i) is known in case p ~ 2 by Davies and Grubel (1983) where a similar stronger result is given for p ~ 2.…”

Section: P(iif4=ai For Some J)~m/3mn·mentioning

confidence: 94%

“…So also by using that P*(X 0 =0)=1, we have to a stationary process (fi,,)::'::-ro and it is well known that this process has an imbedded Markov structure in the sense that z.=inf{k~n: ifk=l} is a Markov sequence and so (fin).~0 is independent of (i7.)n? :o given X 0 =Z 0 • As in Lindvall (1979) -one constructs also a second renewal sequence (if~) coupled with (if,,) such that if~=ii. for n~Twhere Tisa random time.…”

Section: P(iif4=ai For Some J)~m/3mn·mentioning

confidence: 99%

Convergence rates in the strong law for bounded mixing sequences

Berbee¹

1987

Probab. Th. Rel. Fields

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Summary. Speed of convergence is studied for a Marcinkiewicz-Zygmund strong law for partial sums of bounded dependent random variables under conditions on their mixing rate. Though o:-mixing is also considered, the most interesting result concerns absolutely regular sequences. The results are applied to renewal theory to show that some of the estimates obtained by other authors on coupling are best possible. Another application sharpens a result for averaging a function along a random walk.

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On coupling of discrete renewal processes

Cited by 55 publications

References 5 publications

The Ancestral Process of Long-Range Seed Bank Models

The Ancestral Process of Long-Range Seed Bank Models

An infinite alleles version of the Markov branching process

Convergence rates in the strong law for bounded mixing sequences

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