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 \zeta, we construct a ``strong memoryless time'' \hat \zeta such that \zeta-t is exponentially distributed conditional on {\hat \zeta\leq t, \zeta>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.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000005

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.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ513 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Refined distributional approximations for the uncovered set in the Johnson-Mehl model

Let [Phi]z be the uncovered set (i.e., the complement of the union of intervals) at time z in the one-dimensional Johnson-Mehl model. We derive a bound for the total variation distance between the distribution of the number of components of [Phi]z[intersection](0,t] and a compound Poisson-geometric distribution, which is sharper and simpler than an earlier bound obtained by Erhardsson. We also derive a previously unavailable bound for the total variation distance between the distribution of the Lebesgue measure of [Phi]z[intersection](0,t] and a compound Poisson-exponential distribution. Both bounds are O(z[beta](t)/t) as t-->[infinity], where z[beta](t) is defined so that the expected number of components of [Phi]z[beta](t)[intersection](0,t] converges to [beta]>0 as t-->[infinity], and the parameters of the approximating distributions are explicitly calculated.Johnson-Mehl model Uncovered set Compound Poisson approximation Error bound Markov process Renewal reward process

Reciprocal properties of random fields on undirected graphs

We clarify and refine the definition of a reciprocal random field on an undirected graph, with the reciprocal chain as a special case, by introducing four new properties: the factorizing, global, local, and pairwise reciprocal properties, in decreasing order of strength, with respect to a set of nodes delta. They reduce to the better-known Markov properties if 8 is the empty set, or, with the exception of the local property, if delta is a complete set. Conditions for each reciprocal property to imply the next stronger property are derived, and it is shown that, conditionally on the values at a set of nodes delta(0), all four properties are preserved for the subgraph induced by the remaining nodes, with respect to the node set delta \ delta(0). We note that many of the above results are new even for reciprocal chains

On the number of high excursions of linear growth processes

Uni- and bidirectional linear growth processes arise from a Poisson point process on R x [0, [infinity]) with intensity measure l x [Lambda] (where l = Lebesgue measure) in a certain way. They can be used to model how "growth" is initiated at random times from random points on the line and from each point proceeds at constant speed in one or both directions. Each finite interval will be completely "overgrown" after an a.s. finite time, a time which equals the maximum of a linear growth process on the interval. We give here, using the coupling version of the Stein Chen method, an upper bound for the total variation distance between the distribution of the number of excursions above a threshold z for a linear growth process in an interval of finite length L, and a Poisson distribution. The bound tends to 0 as L and z grow to x in a proper fashion. The general results are then applied to two specific examples.Poisson point process Linear growth process Number of excursions Stein Chen method Coupling Total variation distance Poisson convergence