The Kingman tree length process has infinite quadratic variation

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 (c\`adl\`ag) 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).Comment: 13 pages, 3 figure

Total length of the genealogical tree for quadratic stationary continuous-state branching processes

“…(See Janson and Kersting [14] for the external length asymptotics.) The corresponding limiting process has been studied in Pfaffelhuber, Wakolbinger and Weisshaupt [20] as well as Dahmer, Knobloch and Wakolbinger [5] where it is proved the limiting process is not a semi-martingale. Extension has been provided for other Λ-coalescents, see Kersting, Schweinsberg and Wakolbinger [15] for Beta-coalescent and Schweinsberg [24] for the Bolthausen-Sznitman coalescent.…”

Probabilistic aspects of $Λ$-coalescents in equilibrium and in evolution

“…Writing J n,s (f ) for the random variable (10) with (X k ) = (X 0 k ) replaced by (X s k ), we thus obtain…”

“…Theorem 5.1. For f ∈ F and s ∈ R, let J n,s (f ) be as in (10), but now evaluated at the coalescent tree T n (n 1−α s) instead of T n (0). Then the sequence of stationary processes (J n,s (f )) −∞<s<∞ , n ≥ 1, converges as n → ∞ in finite-dimensional distributions to the moving average process…”

A representation for exchangeable coalescent trees and generalized tree-valued Fleming-Viot processes

“…In [16] and in the present article, the level is the rank among the individuals at the respective time according to the time of the latest descendant. Although the levels in finite restrictions of the lookdown model differ from the labels in the Moran model, the processes of the unlabeled genealogical trees coincide which is used to study the length of the genealogical trees in Pfaffelhuber, Wakolbinger, and Weisshaupt [43] and Dahmer, Knobloch, and Wakolbinger [11].…”