In this paper, we are concerned with preservation properties of first and second order by an operator L representable in terms of a stochastic process Z with non-decreasing right-continuous paths. We introduce the derived operator D of L and the derived process V of Z in order to characterize the preservation of absolute continuity and convexity. To obtain different characterizations of the preservation of convexity, we introduce two kinds of duality, the first referring to the process Z and the second to the derived process V. We illustrate the preceding results by considering some examples of interest both in probability and in approximation theory - namely, mixtures, centred subordinators, Bernstein polynomials and beta operators. In most of them, we find bidensities to describe the duality between the derived processes. A unified approach based on stochastic orders is given.

For autoregressive time series with positive innovations which either have heavy right or left tails, linear programming parameter estimates of the autoregressive coefficients have good rates of convergence. However, the asymptotic distribution of the estimators depends heavily on the distribution of the process and thus cannot be used for inference. A bootstrap procedure circumvents this difficulty. We verify the validity of the bootstrap and also give some general comments on the bootstrapping of heavy tailed phenomena.

Batches of immigrants arrive in a region at event times of a renewal process and individuals grow according to a Bellman-Harris branching process. Tribal emigration allows the possibility that all descendants of a group of immigrants collectively leave the region at some instant.

A number of results are derived giving conditions for the existence of a limiting distribution for the population size. These conditions can be given either in terms of the immigration distribution or in terms of the distribution of emigration times. Some limit theorems are obtained when the latter conditions are not fulfilled.

In this paper, we study a maintenance model with general repair and two types of replacement: failure and preventive replacement. When the system fails a decision is made whether to replace or repair it. The repair degree that affects the virtual age of the system is assumed to be a random function of the repair-cost and the virtual age at failure time. The system can be preventively replaced at any time before failure. The objective is to find the repair/replacement policy minimizing the long-run expected average cost per unit time. It is shown that a generalized repair-cost-limit policy is optimal and the preventive replacement time depends on the virtual age of the system and on the length of the operating time since the last repair. Computational procedures for finding the optimal repair-cost limit and the optimal average cost are developed. This model includes many well-known models as special cases and the approach provides a unified treatment of a wide class of maintenance models.

The relationship between the Wiener-Hopf factorisation of matrices and the solution of systems of certain operator equations is discussed in an algebraic setting. It is shown that the study of the waiting time process of the nth arriving group of the general single server bulk queue leads to equations of that type. This system of equations may be considered as an extension of Lindley's waiting-time equation.

We consider the portfolio optimization problem of maximizing the asymptotic growth rate under a combination of fixed and proportional costs. Expressing the asymptotic growth rate in terms of the risky fraction process, the problem can be transformed to that of controlling a diffusion in one dimension. Then we use the corresponding quasivariational inequalities to obtain the explicit shape together with the existence of an optimal impulse control strategy. This optimal strategy is given by only four parameters: two for the stopping boundaries and two for the new risky fractions the investor chooses at these times.

We use the fact that the Palm measure of a stationary random measure is invariant to phase space change to generalize the light traffic formula initially obtained for stationary processes on a line to general spaces. This formula gives a first-order expansion for the expectation of a functional of the random measure when its intensity vanishes. This generalization leads to new algorithms for estimating gradients of functionals of geometrical random processes.

For n independent, identically distributed uniform points in [0, 1]d, d ≥ 2, let Ln be the total distance from the origin to all the minimal points under the coordinatewise partial order (this is also the total length of the rooted edges of a minimal directed spanning tree on the given random points). For d ≥ 3, we establish the asymptotics of the mean and the variance of Ln, and show that Ln satisfies a central limit theorem, unlike in the case d = 2.

In this paper we introduce multitype branching processes with inhomogeneous Poisson immigration, and consider in detail the critical Markov case when the local intensity r(t) of the Poisson random measure is a regularly varying function. Various multitype limit distributions (conditional and unconditional) are obtained depending on the rate at which r(t) changes with time. The asymptotic behaviour of the first and second moments, and the probability of nonextinction are investigated.

We derive asymptotic theory for the extremogram and cross-extremogram of a bivariate GARCH(1,1) process. We show that the tails of the components of a bivariate GARCH(1,1) process may exhibit power-law behavior but, depending on the choice of the parameters, the tail indices of the components may differ. We apply the theory to five-minute return data of stock prices and foreign-exchange rates. We judge the fit of a bivariate GARCH(1,1) model by considering the sample extremogram and cross-extremogram of the residuals. The results are in agreement with the independent and identically distributed hypothesis of the two-dimensional innovations sequence. The cross-extremograms at lag zero have a value significantly distinct from zero. This fact points at some strong extremal dependence of the components of the innovations.

We investigate the degree profile and total weight in Apollonian networks. We study the distribution of the degrees of vertices as they age in the evolutionary process. Asymptotically, the (suitably-scaled) degree of a node with a fixed label has a Mittag-Leffler-like limit distribution. The degrees of nodes of later ages have different asymptotic distributions, influenced by the time of their appearance. The very late arrivals have a degenerate distribution. The result is obtained via triangular Pólya urns. Also, via the Bagchi–Pal urn, we show that the number of terminal nodes asymptotically follows a Gaussian law. We prove that the total weight of the network asymptotically follows a Gaussian law, obtained via martingale methods. Similar results carry over to the sister structure of the k-trees, with minor modification in the proof methods, done mutatis mutandis.

Cross-covariances between the Bernoulli thinned processes of an arbitrary point process are determined. When the point process is renewal it is shown that zero correlation implies independence. An example is given to show that zero covariance between intervals does not imply zero covariance between counts. Mark-dependent thinning of Markov renewal processes is discussed and the results are applied to the overflow queue. Here we give an example of two uncorrelated but dependent renewal processes, neither of which is Poisson, which yield a Poisson process when superposed. Finally, we study Markov-chain thinning of renewal processes.

We consider the same circuit switching problem as in Mitra [1]. The calculation of the blocking probabilities is reduced to finding the partition function for a closed exponential pseudo-network with L−1 customers. This pseudo-network differs from that in [1] in one respect only: service rates at nodes 1, 2, …, p depend on the queue length. The asymptotic expansion developed in [1] follows from our exact expression for the partition function.

Intrinsic volumes are key functionals in convex geometry and have also appeared in several stochastic settings. Here we relate them to questions of regularity in Gaussian processes with regard to Itô–Nisio oscillation and metrization of GB/GC indexing sets. Various bounds and estimates are presented, and questions for further investigation are suggested. From alternate points of view, much of the discussion can be interpreted in terms of (i) random sets and (ii) properties of (deterministic) infinite-dimensional convex bodies.

In this paper we develop the theory of the W and Z scale functions for right-continuous (upwards skip-free) discrete-time, discrete-space random walks, along the lines of the analogous theory for spectrally negative Lévy processes. Notably, we introduce for the first time in this context the one- and two-parameter scale functions Z, which appear for example in the joint deficit at ruin and time of ruin problems of actuarial science. Comparisons are made between the various theories of scale functions as one makes time and/or space continuous.

A classical result about Markov jump processes states that a certain class of dynamical systems given by ordinary differential equations are obtained as the limit of a sequence of scaled Markov jump processes. This approach fails if the scaling cannot be carried out equally across all entities. In the present paper we present a convergence theorem for such an unequal scaling. In contrast to an equal scaling the limit process is not purely deterministic but still possesses randomness. We show that these processes constitute a rich subclass of piecewise-deterministic processes. Such processes apply in molecular biology where entities often occur in different scales of numbers.

Statistical inference for Markov processes is commonly based on the maximum likelihood method of estimation and the likelihood ratio criterion for testing hypotheses. Construction of estimators and test statistics by these methods require that a model be chosen in the form of a family of transition density functions. In this paper, asymptotic properties of the maximum likelihood estimator and of the likelihood ratio statistic λn are examined when the model chosen for their construction is incorrect—that is, when no density in the model is a density for the transition probability distribution of the Markov process. It is shown that if and λn are constructed from a ‘regular’ incorrect model, then is consistent and asymptotically normally distributed and the asymptotic null distribution of −2 log λn is that of a linear combination of independent chi-squared random variables. These results are applied to propose measures of the performance of the test based on λn when the statistic is constructed from an incorrect model.

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.

Seeds are randomly scattered in ℝd according to an m-dependent point process. Each seed has its own potential germination time. From each seed that succeeds in germinating, a spherical inhibited region grows to prohibit germination of any seed with later potential germination time. We show that under certain conditions on the distribution of the potential germination time, the number of germinated seeds in a large region has an asymptotic normal distribution.

In this paper, a reflected stochastic differential equation (SDE) with jumps is studied for the case where the constraint acts on the law of the solution rather than on its paths. These reflected SDEs have been approximated by Briand et al. (2016) using a numerical scheme based on particles systems, when no jumps occur. The main contribution of this paper is to prove the existence and the uniqueness of the solutions to this kind of reflected SDE with jumps and to generalize the results obtained by Briand et al. (2016) to this context.