Suppose that {Xt} is a Markov chain such as the state space model for a threshold GARCH time series. The regularity assumptions for a drift condition approach to establishing the ergodicity of {Xt} typically are ϕ-irreducibility, aperiodicity, and a minorization condition for compact sets. These can be very tedious to verify due to the discontinuous and singular nature of the Markov transition probabilities. We first demonstrate that, for Feller chains, the problem can at least be simplified to focusing on whether the process can reach some neighborhood that satisfies the minorization condition. The results are valid not just for the transition kernels of Markov chains but also for bounded positive kernels, opening the possibility for new ergodic results. More significantly, we show that threshold GARCH time series and related models of interest can often be embedded into Feller chains, allowing us to apply the conclusions above.

Customers arrive sequentially at times x1 < x2 < · · · < xn and stay for independent random times Z1, …, Zn > 0. The Z-variables all have the same distribution Q. We are interested in situations where the data are incomplete in the sense that only the order statistics associated with the departure times xi + Zi are known, or that the only available information is the order in which the customers arrive and depart. In the former case we explore possibilities for the reconstruction of the correct matching of arrival and departure times. In the latter case we propose a test for exponentiality.

Contemporaneous aggregation of N independent copies of a random-coefficient AR(1) process with random coefficient a ∈ (−1, 1) and independent and identically distributed innovations belonging to the domain of attraction of an α-stable law (0 < α < 2) is discussed. We show that, under the normalization N1/α, the limit aggregate exists, in the sense of weak convergence of finite-dimensional distributions, and is a mixed stable moving average as studied in Surgailis, Rosiński, Mandrekar and Cambanis (1993). We focus on the case where the slope coefficient a has probability density vanishing regularly at a = 1 with exponent b ∈ (0, α − 1) for α ∈ (1, 2). We show that in this case, the limit aggregate {X̅t} exhibits long memory. In particular, for {X̅t}, we investigate the decay of the codifference, the limit of partial sums, and the long-range dependence (sample Allen variance) property of Heyde and Yang (1997).

In this paper we describe methods for randomly thinning certain classes of spatial point processes. In the case of a Markov point process, the proposed method involves a dependent thinning of a spatial birth-and-death process, where clans of ancestors associated with the original points are identified, and where we simulate backwards and forwards in order to obtain the thinned process. In the case of a Cox process, a simple independent thinning technique is proposed. In both cases, the thinning results in a Poisson process if and only if the true Papangelou conditional intensity is used, and, thus, can be used as a graphical exploratory tool for inspecting the goodness-of-fit of a spatial point process model. Several examples, including clustered and inhibitive point processes, are considered.

For a time series, a plot of sample covariances is a popular way to assess its dependence properties. In this paper we give a systematic characterization of the asymptotic behavior of sample covariances of long-memory linear processes. Central and noncentral limit theorems are obtained for sample covariances with bounded as well as unbounded lags. It is shown that the limiting distribution depends in a very interesting way on the strength of dependence, the heavy-tailedness of the innovations, and the magnitude of the lags.

In a repulsive point process, points act as if they are repelling one another, leading to underdispersed configurations when compared to a standard Poisson point process. Such models are useful when competition for resources exists, as in the locations of towns and trees. Bertil Matérn introduced three models for repulsive point processes, referred to as types I, II, and III. Matérn used types I and II, and regarded type III as intractable. In this paper an algorithm is developed that allows for arbitrarily accurate approximation of the likelihood for data modeled by the Matérn type-III process. This method relies on a perfect simulation method that is shown to be fast in practice, generating samples in time that grows nearly linearly in the intensity parameter of the model, while the running times for more naive methods grow exponentially.

Let {Zt}t≥0 be a Lévy process with Lévy measure ν, and let τ(t)=∫0tr(u) d u, where {r(t)}t≥0 is a positive ergodic diffusion independent from Z. Based upon discrete observations of the time-changed Lévy process Xt≔Zτt during a time interval [0,T], we study the asymptotic properties of certain estimators of the parameters β(φ)≔∫φ(x)ν(d x), which in turn are well known to be the building blocks of several nonparametric methods such as sieve-based estimation and kernel estimation. Under uniform boundedness of the second moments of r and conditions on φ necessary for the standard short-term ergodic property limt→ 0 E φ(Zt)/t = β(φ) to hold, consistency and asymptotic normality of the proposed estimators are ensured when the time horizon T increases in such a way that the sampling frequency is high enough relative to T.

We consider a model for a time series of spatial locations, in which points are placed sequentially at random into an initially empty region of ℝd, and given the current configuration of points, the likelihood at location x for the next particle is proportional to a specified function βk of the current number (k) of points within a specified distance of x. We show that the maximum likelihood estimator of the parameters βk (assumed to be zero for k exceeding some fixed threshold) is consistent in the thermodynamic limit where the number of points grows in proportion to the size of the region.

We introduce a modification of the linear ARCH (LARCH) model (Giraitis, Robinson, and Surgailis (2000)) - a special case of Sentana's (1995) quadratic ARCH (QARCH) model - for which the conditional variance is a sum of a positive constant and the square of an inhomogeneous linear combination of past observations. Necessary and sufficient conditions for the existence of a stationary solution with finite variance are obtained. We give conditions under which the stationary solution with infinite fourth moment can exhibit long memory, the leverage effect, and a Lévy-stable limit behavior of partial sums of squares.

Most metrics between finite point measures currently used in the literature have the flaw that they do not treat differing total masses in an adequate manner for applications. This paper introduces a new metric d̅1 that combines positional differences of points under a closest match with the relative difference in total mass in a way that fixes this flaw. A comprehensive collection of theoretical results about d̅1 and its induced Wasserstein metric d̅2 for point process distributions are given, including examples of useful d̅1-Lipschitz continuous functions, d̅2 upper bounds for the Poisson process approximation, and d̅2 upper and lower bounds between distributions of point processes of independent and identically distributed points. Furthermore, we present a statistical test for multiple point pattern data that demonstrates the potential of d̅1 in applications.

We study a flexible class of finite-disc process models with interaction between the discs. We let 𝒰 denote the random set given by the union of discs, and use for the disc process an exponential family density with the canonical sufficient statistic depending only on geometric properties of 𝒰 such as the area, perimeter, Euler-Poincaré characteristic, and the number of holes. This includes the quermass-interaction process and the continuum random-cluster model as special cases. Viewing our model as a connected component Markov point process, and thereby establishing local and spatial Markov properties, becomes useful for handling the problem of edge effects when only 𝒰 is observed within a bounded observation window. The power tessellation and its dual graph become major tools when establishing inclusion-exclusion formulae, formulae for computing geometric characteristics of 𝒰, and stability properties of the underlying disc process density. Algorithms for constructing the power tessellation of 𝒰 and for simulating the disc process are discussed, and the software is made public available.

Start with a necklace consisting of one white bead and one black bead, and add new beads one at a time by inserting each new bead between a randomly chosen adjacent pair of old beads, with the proviso that the new bead will be white if and only if both beads of the adjacent pair are black. Let Wn denote the number of white beads when the total number of beads is n. We show that EWn = n/3 and, with c2 = 2/45, that (Wn − n/3) / c√n is asymptotically standard normal. We find that, for all r ≥ 1 and n > 2r, the rth cumulant of the distribution of Wn is of the form nhr. We find the expected numbers of gaps of given length between white beads, and examine the asymptotics of the longest gaps.

The class of processes formed as the aggregation of Ornstein-Uhlenbeck processes has proved useful in modeling time series from a number of areas and includes several interesting special cases. This paper examines the second-order properties of this class. Bounds on the one-step prediction error variance are proved and consistency of the minimum contrast estimation is demonstrated.

Continuous-time autoregressive moving average (CARMA) processes with a nonnegative kernel and driven by a nondecreasing Lévy process constitute a very general class of stationary, nonnegative continuous-time processes. In financial econometrics a stationary Ornstein-Uhlenbeck (or CAR(1)) process, driven by a nondecreasing Lévy process, was introduced by Barndorff-Nielsen and Shephard (2001) as a model for stochastic volatility to allow for a wide variety of possible marginal distributions and the possibility of jumps. For such processes, we take advantage of the nonnegativity of the increments of the driving Lévy process to study the properties of a highly efficient estimation procedure for the parameters when observations are available of the CAR(1) process at uniformly spaced times 0,h,…,Nh. We also show how to reconstruct the background driving Lévy process from a continuously observed realization of the process and use this result to estimate the increments of the Lévy process itself when h is small. Asymptotic properties of the coefficient estimator are derived and the results illustrated using a simulated gamma-driven Ornstein-Uhlenbeck process.

We characterize the Lyapunov exponent and ergodicity of nonlinear stochastic recursion models, including nonlinear AR-GARCH models, in terms of an easily defined, uniformly ergodic process. Properties of this latter process, known as the collapsed process, also determine the existence of moments for the stochastic recursion when it is stationary. As a result, both the stability of a given model and the existence of its moments may be evaluated with relative ease. The method of proof involves piggybacking a Foster-Lyapunov drift condition on certain characteristic behavior of the collapsed process.

Based on the concept of multipower variation we establish a class of easily computable and robust estimators for the integrated volatility, especially including the squared integrated volatility, in Lévy-type stochastic volatility models. We derive consistency and feasible distributional results for the estimators. Furthermore, we discuss the applications to time-changed CGMY, normal inverse Gaussian, and hyperbolic models with and without leverage, where the time-changes are based on integrated Cox-Ingersoll-Ross or Ornstein-Uhlenbeck-type processes. We deduce which type of market microstructure does not affect the estimates.

In this paper, we examine the rate of convergence of moving block bootstrap (MBB) approximations to the distributions of normalized sample quantiles based on strongly mixing observations. Under suitable smoothness and regularity conditions on the one-dimensional marginal distribution function, the rate of convergence of the MBB approximations to distributions of centered and scaled sample quantiles is of order O(n−1¼ log logn).

In this paper our object is to show that a certain class of nonstationary random processes can locally be approximated by stationary processes. The class of processes we are considering includes the time-varying autoregressive conditional heteroscedastic and generalised autoregressive conditional heteroscedastic processes, amongst others. The measure of deviation from stationarity can be expressed as a function of a derivative random process. This derivative process inherits many properties common to stationary processes. We also show that the derivative processes obtained here have alpha-mixing properties.

The existence and uniqueness of maximum likelihood estimators for the time and range parameters in random sequential adsorption models are investigated.