$$U$$ U -Statistics of Ornstein–Uhlenbeck Branching Particle System

Abstract

We consider a branching particle system consisting of particles moving according to the Ornstein-Uhlenbeck process in R d and undergoing a binary, supercritical branching with a constant rate λ > 0.This system is known to fulfil a law of large numbers (under exponential scaling). Recently the question of the corresponding central limit theorem has been addressed. It turns out that the normalization and form of the limit in the CLT fall into three qualitatively different regimes, depending on the relation between the branching intensity and the parameters of the Orstein-Uhlenbeck process. In the present paper we extend those results to U -statistics of the system proving a law of large numbers and a central limit theorem. MSC: primary 60F05; 60J80 secondary 60G20

A Central Limit Theorem for Punctuated Equilibrium

“…However, these studies use a very heavy functional analysis apparatus, which unlike the direct one here, is completely inaccessible to the applied reader. We can observe the same competition between the tree's speciation and OU's adaptation (drift) rates, resulting in a phase transition when the latter is half the former (the same as in the no jumps case Adamczak and Miłoś [1,2], Bartoszek and Sagitov [6]). We show here that if large jumps are rare enough, then the contemporary sample mean will be asymptotically normally distributed.…”

“…Branching Ornstein-Uhlenbeck models commonly have three asymptotic regimes (Adamczak and Miłoś [1,2], Ané et al [3], Bartoszek [5], Bartoszek and Sagitov [6], Ren et al [29,30]). The dependency between the adaptation rate α and branching rate λ = 1 governs in which regime the process is.…”

“…sample, in the critical case, α = 1/2, we can, after appropriate rescaling, still recover the "near" i.i.d. behaviour and if 0 < α < 1/2, then the process has "long memory" ("local correlations dominate over the OU's ergodic properties", Adamczak and Miłoś [1,2]). In the OU process with jumps setup the same asymptotic regimes can be observed, even though Adamczak and Miłoś [1,2], Ren et al [29,30]) assume that the tree is observed at a given time point, t, with n t being random.…”

Phylogenetic confidence intervals for the optimal trait value

“…Our main result, Theorem 1, should be compared with the limit theorems obtained in [1] and [2]. They also revealed three asymptotic regimes in a related, though different setting, dealing with a branching Omstein-Uhlenbeck process.…”

“…The classical Brownian motion model [14] can be viewed as a special case with a = 0 and e being irrelevant. Using a regression framework one can apply standard regression theory methods to compute confidence intervals for «(), Xo) conditionally 1116 K. BARTOSZEKAND S. SAGITOV on (a, [0][1][2] [15], [20], [28], [33]. However, confidence intervals are often not mentioned in phylogenetic comparative studies [8].…”