The Kingman tree length process has infinite quadratic variation

@article{Dahmer2014TheKT,
  title={The Kingman tree length process has infinite quadratic variation},
  author={Iulia Dahmer and Robert Knobloch and A. Wakolbinger},
  journal={Electronic Communications in Probability},
  year={2014},
  volume={19},
  pages={1-12}
}
In the case of neutral populations of fixed sizes in equilibrium whose genealogies are described by the Kingman N-coalescent back from time t consider the associated processes of total tree length as t increases. We show that the (cadlag) process to which the sequence of compensated tree length processes converges as N tends to infinity is a process of infinite quadratic variation; therefore this process cannot be a semimartingale. This answers a question posed in Pfaffelhuber et al. (2011). 
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References

SHOWING 1-10 OF 28 REFERENCES
The tree length of an evolving coalescent
A well-established model for the genealogy of a large population in equilibrium is Kingman’s coalescent. For the population together with its genealogy evolving in time, this gives rise to a…
Asymptotic sampling formulae for -coalescents
We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results…
The evolving beta coalescent
In mathematical population genetics, it is well known that one can represent the genealogy of a population by a tree, which indicates how the ancestral lines of individuals in the population coalesce…
On the Two Oldest Families for the Wright-Fisher Process
We extend some of the results of Pfaffelhuber and Wakolbinger on the process of the most recent common ancestors in evolving coalescent by taking into account the size of one of the two oldest…
Dynamics of the evolving Bolthausen-Sznitman coalecent
TLDR
The limiting behavior of the evolution of the time back to the most recent common ancestor and the total length of the branches in the tree are obtained and a new result concerning the number of blocks in the Bolthausen-Sznitman coalescent is obtained.
Tree-valued resampling dynamics Martingale problems and applications
The measure-valued Fleming–Viot process is a diffusion which models the evolution of allele frequencies in a multi-type population. In the neutral setting the Kingman coalescent is known to generate…
Asymptotic results concerning the total branch length of the Bolthausen-Sznitman coalescent
BETA-COALESCENTS AND CONTINUOUS STABLE RANDOM TREES
TLDR
It is proved that Beta-coalescents can be embedded in continuous stable random trees based on a construction of the Donnelly-Kurtz lookdown process using continuous random trees, which is of independent interest.
ASYMPTOTIC RESULTS ABOUT THE TOTAL BRANCH LENGTH OF THE BOLTHAU-SEN-SZNITMAN COALESCENT
Abstract We study the total branch length Ln of the Bolthausen-Sznitman coalescent as the sample size n tends to infinity. Asymptotic expansions for the moments of Ln are presented. It is shown that…
The process of most recent common ancestors in an evolving coalescent
...
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