We look at joint regular variation properties of MA(∞) processes of the form X = (Xk, k ∈ Z), where Xk = ∑j=0∞ψjZk-j and the sequence of random variables (Zi, i ∈ Z) are independent and identically distributed with regularly varying tails. We use the setup of MO-convergence and obtain hidden regular variation properties for X under summability conditions on the constant coefficients (ψj: j ≥ 0). Our approach emphasizes continuity properties of mappings and produces regular variation in sequence space.

We consider estimation of the spot volatility in a stochastic boundary model with one-sided microstructure noise for high-frequency limit order prices. Based on discrete, noisy observations of an Itô semimartingale with jumps and general stochastic volatility, we present a simple and explicit estimator using local order statistics. We establish consistency and stable central limit theorems as asymptotic properties. The asymptotic analysis builds upon an expansion of tail probabilities for the order statistics based on a generalized arcsine law. In order to use the involved distribution of local order statistics for a bias correction, an efficient numerical algorithm is developed. We demonstrate the finite-sample performance of the estimation in a Monte Carlo simulation.

In this paper, a moderate deviation theorem for one-dimensional stable random walks in random scenery is proved. The proof relies on the analysis of maximum local times of stable random walks, and the comparison of moments between random walks in random scenery and self-intersection local times of the underlying random walks.

In this paper we extend our previous work. We consider the local-time process L of a strong Markov process X, add negative drift to L, and reflect it à la Skorokhod to obtain a process Q. The reflection of X, together with Q, is, in some sense, a macroscopic model for a service system with two priorities. We derive an expression for the joint law of the duration of an excursion, the maximum value of the process on it, and the time between successive excursions. We work with a properly constructed stationary version of the process. Examples are also given in the paper.

In a discrete-time single-type Galton--Watson branching random walk {Zn, ζn}n≤ 0, where Zn is the population of the nth generation and ζn is a collection of the positions on ℝ of the Zn individuals in the nth generation, let Yn be the position of a randomly chosen individual from the nth generation and Zn(x) be the number of points in ζn that are less than or equal to x for x∈ℝ. In this paper we show in the explosive case (i.e. m=E(Z1∣ Z0=1)=∞) when the offspring distribution is in the domain of attraction of a stable law of order α,0 <α<1, that the sequence of random functions {Zn(x)/Zn:−∞<x<∞} converges in the finite-dimensional sense to {δx:−∞<x<∞}, where δx≡ 1{N≤ x} and N is an N(0,1) random variable.

We construct an integrated probabilistic model to capture interactions between atoms of a nanocomponent. We then use this model to assess reliabilities of nanocomponents with different structures. Several properties of our proposed model are also described under a sparseness condition. The model is an extension of our previous model based on Markovian random field theory. The proposed integrated model is flexible in that pairwise relationship information among atoms as well as features of individual atoms can be easily incorporated. An important feature that distinguishes the integrated probabilistic model from our previous model is that the integrated approach uses all available sources of information with different weights for different types of interaction. In this paper we consider the nanocomponent at a fixed moment of time, say the present moment, and we assume that the present state of the nanocomponent depends only on the present states of its atoms.

We consider epidemics in populations that are partitioned into small groups known as households. Whilst infectious, a typical infective makes global and local contact with individuals chosen independently and uniformly from the whole population or their own household, as appropriate. Previously, the classical Poisson approximation for the number of survivors of a severe epidemic has been extended to the household model. However, in the current work we exploit a Sellke-type construction of the epidemic process, which enables the derivation of sufficient conditions for the existence of a compound Poisson limit theorem for the survivors of the epidemic. The results are specialised to the Reed-Frost and general stochastic epidemic models.

Consider a random walk Sn = ∑i=0nXi with negative drift. This paper deals with upper bounds for the maximum M = maxn≥1Sn of this random walk in different settings of power moment existences. As is usual for deriving upper bounds, we truncate summands. Therefore, we use an approach of splitting the time axis by stopping times into intervals of random but finite length and then choose a level of truncation on each interval. Hereby, we can reduce the problem of finding upper bounds for M to the problem of finding upper bounds for Mτ = maxn≤τSn. In addition we test our inequalities in the heavy traffic regime in the case of regularly varying tails.

Exact simulation approaches for a class of diffusion bridges have recently been proposed based on rejection sampling techniques. The existing rejection sampling methods may not be practical owing to small acceptance probabilities. In this paper we propose an adaptive approach that improves the existing methods significantly under certain scenarios. The idea of the new method is based on a layered process, which can be simulated from a layered Brownian motion with reweighted layer probabilities. We will show that the new exact simulation method is more efficient than existing methods theoretically and via simulation.

Known results on the moments of the distribution generated by the two-locus Wright–Fisher diffusion model, and the duality between the diffusion process and the ancestral process with recombination are briefly summarized. A numerical method for computing moments using a Markov chain Monte Carlo simulation and a method to compute closed-form expressions of the moments are presented. By applying the duality argument, the properties of the ancestral recombination graph are studied in terms of the moments.

Identifiability of evolutionary tree models has been a recent topic of discussion and some models have been shown to be nonidentifiable. A coalescent-based rooted population tree model, originally proposed by Nielsen et al. (1998), has been used by many authors in the last few years and is a simple tool to accurately model the changes in allele frequencies in the tree. However, the identifiability of this model has never been proven. Here we prove this model to be identifiable by showing that the model parameters can be expressed as functions of the probability distributions of subsamples, assuming that there are at least two (haploid) individuals sampled from each population. This a step toward proving the consistency of the maximum likelihood estimator of the population tree based on this model.

Under mild assumptions, we show that the exact convergence rate in total variation is also exact in weaker Wasserstein distances for the Metropolis–Hastings independence sampler. We develop a new upper and lower bound on the worst-case Wasserstein distance when initialized from points. For an arbitrary point initialization, we show that the convergence rate is the same and matches the convergence rate in total variation. We derive exact convergence expressions for more general Wasserstein distances when initialization is at a specific point. Using optimization, we construct a novel centered independent proposal to develop exact convergence rates in Bayesian quantile regression and many generalized linear model settings. We show that the exact convergence rate can be upper bounded in Bayesian binary response regression (e.g. logistic and probit) when the sample size and dimension grow together.

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

We describe the random meeting motion of a finite number of investors in markets with friction as a Markov pure-jump process with interactions. Using a sequence of these, we prove a functional law of large numbers relating the large motions with the finite market of the so-called continuum of agents.

We consider an insurance model, where the underlying point process is a Cox process. Using a martingale approach applied to diffusion processes, finite-time Lundberg inequalities are obtained. By change-of-measure techniques, Cramér–Lundberg approximations are derived.

This paper builds a mixture representation of the reliability function of the conditional residual lifetime of a coherent system in terms of the reliability functions of conditional residual lifetimes of order statistics. Some stochastic ordering properties for the conditional residual lifetime of a coherent system with independent and identically distributed components are obtained, based on the stochastically ordered coefficient vectors.

The aim of this paper is to provide the conditions necessary to reduce the complexity of state filtering for finite stochastic systems (FSSs). A concept of lumpability for FSSs is introduced. In this paper we assert that the unnormalised filter for a lumped FSS has linear dynamics. Two sufficient conditions for such a lumpability property to hold are discussed. We show that the first condition is also necessary for the lumped FSS to have linear dynamics. Next, we prove that the second condition allows the filter of the original FSS to be obtained directly from the filter for the lumped FSS. Finally, we generalise an earlier published result for the approximation of a general FSS by a lumpable FSS.

In this paper we present a method to recover a time-homogeneous piecewise constant volatility from a finite set of perpetual put option prices. The whole calculation process of the volatility is decomposed into easy computations in many fixed disjoint intervals. In each interval, the volatility is obtained by solving a system of nonlinear equations.

We consider the asymmetric simple exclusion process which starts from a product measure such that all the sites to the left of zero (including zero) are occupied and the right of 0 (excluding 0) are empty. We label the particle initially at 0 as the leading particle. We study the long-term behaviour of this process near large sites when the leading particle's holding time is different from that of the other particles. In particular, we assume that the leading particle moves at a slower rate than the other particles. We call this modified asymmetric simple exclusion process the road-hog process. Coupling and stochastic ordering techniques are used to derive the density profile of this process. Road-hog processes are useful in modelling series of exponential queues with Poisson and non-Poisson input process. The density profiles dramatically illustrate the flow of customers through the queues.