A Delayed Yule Process
نویسندگان
چکیده
In now classic work, David Kendall (1966) recognized that the Yule process and Poisson process could be related by a (random) time change. Furthermore, he showed that the Yule population size rescaled by its mean has an almost sure exponentially distributed limit as t → ∞. In this note we introduce a class of coupled delayed continuous time Yule processes parameterized by 0 < α ≤ 1 and find a representation of the Poisson process as a delayed Yule process at delay rate α = 1/2. Moreover we extend Kendall’s limit theorem to include a larger class of positive martingales derived from functionals that gauge the population genealogy. Specifically, the latter is exploited to uniquely characterize the moment generating functions of distributions of the limit martingales, generalizing Kendall’s mean one exponential limit. A connection with fixed points of the Holley-Liggett smoothing transformation also emerges in this context, about which much is known from general theory in terms of moments, tail decay, and so on.
منابع مشابه
Connecting Yule Process, Bisection and Binary Search Tree via Martingales
We present new links between some remarkable martingales found in the study of the Binary Search Tree or of the bisection problem, looking at them on the probability space of a continuous time binary branching process.
متن کاملThe Yule-Walker Equations as a Least Squares Problem and the Need for Tapering
The most commonly used method for estimating the time domain parameters of an autoregressive process is to use the Yule-Walker equations. The Yule-Walker estimates of the parameters of an autoregressive process are known to often be highly biased. There is a Fourier transform relationship between the autocovariance sequence for an autoregressive process (the estimates of which are used in the Y...
متن کاملMultitapering for Estimating Time Domain Parameters of Autoregressive Processes
The most commonly used method for estimating the time domain parameters of an autoregressive process is to use the Yule-Walker equations. The Yule-Walker estimates of the parameters of an autoregressive process of order p, or AR(p), are known to often be highly biased. This can lead to inappropriate order selection and very poor forecasting. There is a Fourier transform relationship between the...
متن کاملTime to a single hybridization event in a group of species with unknown ancestral history.
We consider a stochastic process for the generation of species which combines a Yule process with a simple model for hybridization between pairs of co-existent species. We assume that the origin of the process, when there was one species, occurred at an unknown time in the past, and we condition the process on producing n species via the Yule process and a single hybridization event. We prove r...
متن کاملMultipath mitigation in spectrum estimation using ℓ1 minimization
We consider the problem of spectrum estimation of an AutoRegressive (AR) process in a sparse multipath environment. The presence of even a small number of delayed and attenuated replica of the source signal in the received signal may severely degrade the performance of classical AR spectrum estimation methods. Dwelling on the sparsity of the multipath reflections, we propose an approach which l...
متن کامل