Abstract
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 (càdlàg) 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).
Highlights
Introduction and main resultThe Kingman coalescent is a classical model in mathematical population genetics used for describing the genealogies for a wide class of population models.The population models in question are neutral, exchangeable and with an offspring distribution of finite variation
We show that the process to which the sequence of compensated tree length processes converges as N tends to infinity is a process of infinite quadratic variation; this process cannot be a semimartingale
This is a process with values in the set of partitions of {1, . . . , N } which starts in the partition in singletons and has the following dynamics: given the process is in state πk, it jumps at rate k 2 to a state πk−1 which is obtained by merging two randomly chosen elements of πk
Summary
The Kingman coalescent is a classical model in mathematical population genetics used for describing the genealogies for a wide class of population models (see e.g [21]). The jumps of the (compensated) tree length process Lld,N happen at the times lines exist level N in the N -look-down graph and they have sizes equal to the life-lengths up to N of these lines. In order to understand the jumps of the limiting process Lld that occur in In the infinite look-down graph, for every k ∈ N, k ≥ 2 consider the process ηk of time points at which the lines that were born at level k reach level ∞. The size h of this jump is equal to the life-length T G of the line G that dies at this time point (see the proof of Theorem 2).
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