arXiv subjects

Torkel Erhardsson

Publications and source records attributed to Torkel Erhardsson.

3 recordsLinked to original sources

arXiv2020

Normal approximation for mixtures of normal distributions and the evolution of phenotypic traits

An explicit bound is given for the Kolmogorov distance between a mixture of normal distributions and a normal distribution with properly chosen parameter values. A random variable X has a mixture of normal distributions if its conditional distribution given some sigma-algebra is normal. The bound depends only on the first two moments of the first two conditional moments of X given this sigma-algebra. As an application, the Yule-Ornstein-Uhlenbeck model, used in the field of phylogenetic comparative methods, is considered. A bound is derived for the Kolmogorov distance between the distribution of the average value of a phenotypic trait over n related species and a normal distribution. The bound goes to 0 as n goes to infinity, extending earlier limit theorems by Bartoszek and Sagitov.

math.PR↗

arXiv2014

Conditions for convergence of random coefficient AR(1) processes and perpetuities in higher dimensions

A $d$-dimensional RCA(1) process is a generalization of the $d$-dimensional AR(1) process, such that the coefficients $\{M_t;t=1,2,\ldots\}$ are i.i.d. random matrices. In the case $d=1$, under a nondegeneracy condition, Goldie and Maller gave necessary and sufficient conditions for the convergence in distribution of an RCA(1) process, and for the almost sure convergence of a closely related sum of random variables called a perpetuity. We here prove that under the condition $\Vert {\prod_{t=1}^nM_t}\Vert \stackrel{\mathrm{a.s.}}{\longrightarrow}0$ as $n\to\infty$, most of the results of Goldie and Maller can be extended to the case $d>1$. If this condition does not hold, some of their results cannot be extended.

math.ST↗

arXiv2004

Strong memoryless times and rare events in Markov renewal point processes

Let W be the number of points in (0,t] of a stationary finite-state Markov renewal point process. We derive a bound for the total variation distance between the distribution of W and a compound Poisson distribution. For any nonnegative random variable ζ, we construct a ``strong memoryless time'' \hat ζsuch that ζ-t is exponentially distributed conditional on {\hat ζ\leq t, ζ>t}, for each t. This is used to embed the Markov renewal point process into another such process whose state space contains a frequently observed state which represents loss of memory in the original process. We then write W as the accumulated reward of an embedded renewal reward process, and use a compound Poisson approximation error bound for this quantity by Erhardsson. For a renewal process, the bound depends in a simple way on the first two moments of the interrenewal time distribution, and on two constants obtained from the Radon-Nikodym derivative of the interrenewal time distribution with respect to an exponential distribution. For a Poisson process, the bound is 0.

math.PR↗