While max-stable processes are typically written as pointwise maxima over an infinite number of stochastic processes, in this paper, we consider a family of representations based on ℓp-norms. This family includes both the construction of the Reich–Shaby model and the classical spectral representation by de Haan (1984) as special cases. As the representation of a max-stable process is not unique, we present formulae to switch between different equivalent representations. We further provide a necessary and sufficient condition for the existence of an ℓp-norm-based representation in terms of the stable tail dependence function of a max-stable process. Finally, we discuss several properties of the represented processes such as ergodicity or mixing.

This paper considers the expression of expectations and density function of the extinction time for a continuous-time Markov branching process.

This paper contains a simple proof of Steck's recurrence equation for determining the power of the standard one-sided Kolmogorov test for a distribution function.

A three-stage real diffusion process is used as a model of the T-cell count of an HIV-positive individual who is to receive antiviral therapy such as AZT. The ‘quality of life' of such a person is identified as the sojourn time of the diffusion process above a certain critical T-cell level c. The time of introducing therapy is defined as the first-passage time of the diffusion to a prescribed level z > c. The distribution of the sojourn time of the diffusion above the level c depends on the level z at which therapy is initiated. The expected sojourn time is explicitly computed as a function of z for the particular diffusion process defining the model. There is a simple criterion for determining when to start therapy as early as possible.

Events that occur consecutively or simultaneously cause some other event as effect. The latter can be observed with noise, and the problem is to estimate the weights of the causes in the realization of the effect.

We consider a continuum percolation model consisting of two types of nodes, namely legitimate and eavesdropper nodes, distributed according to independent Poisson point processes in R2 of intensities λ and λE, respectively. A directed edge from one legitimate node A to another legitimate node B exists provided that the strength of the signal transmitted from node A that is received at node B is higher than that received at any eavesdropper node. The strength of the signal received at a node from a legitimate node depends not only on the distance between these nodes, but also on the location of the other legitimate nodes and an interference suppression parameter γ. The graph is said to percolate when there exists an infinitely connected component. We show that for any finite intensity λE of eavesdropper nodes, there exists a critical intensity λc < ∞ such that for all λ > λc the graph percolates for sufficiently small values of the interference parameter. Furthermore, for the subcritical regime, we show that there exists a λ0 such that for all λ < λ0 ≤ λc a suitable graph defined over eavesdropper node connections percolates that precludes percolation in the graphs formed by the legitimate nodes.

Invariant stochastic properties are studied for circuit processes, i.e. processes whose sample paths can be represented by a denumerable set of overlapping directed circuits c in a denumerable set S and a set of positive weights , when the family of the weights varies.

∗ A shorter version of this account has appeared in the Australian Journal of Statistics 36 (1994) 1–8, with major differences in Section 5; in that journal Ted's influence on Australian statistics is stressed, while in this journal we concentrate on his contributions to the Applied Probability Trust.

A family $\{Q_{\beta}\}_{\beta \geq 0}$ of Markov chains is said to exhibit metastable mixing with modes$S_{\beta}^{(1)},\ldots,S_{\beta}^{(k)}$ if its spectral gap (or some other mixing property) is very close to the worst conductance $\min\!\big(\Phi_{\beta}\big(S_{\beta}^{(1)}\big), \ldots, \Phi_{\beta}\big(S_{\beta}^{(k)}\big)\big)$ of its modes for all large values of $\beta$. We give simple sufficient conditions for a family of Markov chains to exhibit metastability in this sense, and verify that these conditions hold for a prototypical Metropolis–Hastings chain targeting a mixture distribution. The existing metastability literature is large, and our present work is aimed at filling the following small gap: finding sufficient conditions for metastability that are easy to verify for typical examples from statistics using well-studied methods, while at the same time giving an asymptotically exact formula for the spectral gap (rather than a bound that can be very far from sharp). Our bounds from this paper are used in a companion paper (O. Mangoubi, N. S. Pillai, and A. Smith, arXiv:1808.03230) to compare the mixing times of the Hamiltonian Monte Carlo algorithm and a random walk algorithm for multimodal target distributions.

Suppose k balls are dropped into n boxes independently with uniform probability, where n, k are large with ratio approximately equal to some positive real $\lambda$ . The maximum box count has a counterintuitive behavior: first of all, with high probability it takes at most two values $m_n$ or $m_n+1$ , where $m_n$ is roughly $\frac{\ln n}{\ln \ln n}$ . Moreover, it oscillates between these two values with an unusual periodicity. In order to prove this statement and various generalizations, it is first shown that for $X_1,\ldots,X_n$ independent and identically distributed discrete random variables with common distribution F, under mild conditions, the limiting distribution of their maximum oscillates in three possible families, depending on the tail of the distribution. The result stated at the beginning follows from the ensemble equivalence for the order statistics in various allocations problems, obtained via conditioning limit theory. Results about the number of ties for the maximum, as well as applications, are also provided.

We consider a strictly substochastic matrix or a stochastic matrix with absorbing states. By using quasi-stationary distributions we show that there is an associated canonical Markov chain that is built from the resurrected chain, the absorbing states, and the hitting times, together with a random walk on the absorbing states, which is necessary for achieving time stationarity. Based upon the 2-stringing representation of the resurrected chain, we supply a stationary representation of the killed and the absorbed chains. The entropies of these representations have a clear meaning when one identifies the probability measure of natural factors. The balance between the entropies of these representations and the entropy of the canonical chain serves to check the correctness of the whole construction.

In this paper we study the asymptotic behaviour near extinction of (sub-)critical continuous-state branching processes. In particular, we establish an analogue of Khintchine's law of the iterated logarithm near extinction time for a continuous-state branching process whose branching mechanism satisfies a given condition.

The aim of the present note is to give an alternative simpler proof to a result of Belyaev [1], namely that in a loss system of n servers with recurrent input and negative exponential service times the intervals between losses, suitably scaled to have constant mean, tend to a negative exponential distribution as n tends to infinity.

A Poisson limit theorem for sums of dissociated 0–1 random variables is refined by deriving the first terms in an asymptotic expansion. The most natural refinement does not remove all the first-order error in a number of applications to tests of clustering, and a further approximation is derived which gives excellent results in practice. The proofs are based on the technique of Stein and Chen.

Martingales constitute a basic tool in stochastic analysis; this paper considers their application to counting processes. We use this tool to revisit a renewal theorem and give extensions for various counting processes. We first consider a renewal process as a pilot example, deriving a new semimartingale representation that differs from the standard decomposition via the stochastic intensity function. We then revisit Blackwell’s renewal theorem, its refinements and extensions. Based on these observations, we extend the semimartingale representation to a general counting process, and give conditions under which asymptotic behaviour similar to Blackwell’s renewal theorem holds.

The existence of a weak limit birth-and-death process on the natural integers for the simple birth-and-death process conditional on non-extinction up to time t as t→∞ is proved. Starting from the latter a new weak limiting procedure yields a diffusion Markov process on the positive infinite semi-axis.

We consider the distribution of the age of an individual picked uniformly at random at some fixed time in a linear birth-and-death process. By exploiting a bijection between the birth-and-death tree and a contour process, we derive the cumulative distribution function for this distribution. In the critical and supercritical cases, we also give rates for the convergence in terms of the total variation and other metrics towards the appropriate exponential distribution.

We study a not necessarily symmetric random walk with interactions on ℤ, which is an extension of the one-dimensional discrete version of the sausage Wiener path measure. We prove the existence of a repulsion/attraction phase transition for the critical value λc ≡ −μ of the repulsion coefficient λ, where μ is a drift parameter. In the self-repellent case, we determine the escape speed, as a function of λ and μ, and we prove a law of large numbers for the end-point.

In this note we consider branching processes whose behavior depends on a dynamic random environment, in the sense that we assume that the offspring distributions of individuals are parametrized, over time, by the realizations of a process describing the environmental evolution. We study how the variability in time of the environment modifies the variability of total population by considering two branching processes of this kind (but subjected to different environments). We also provide conditions on the random environments in order to stochastically compare their marginal distributions in the increasing convex sense. Weaker conditions are also provided for comparisons at every fixed time of the expected values of the two populations.

Nonspatial stochastic models of populations subject to catastrophic events result in the common conclusion that the survival probability of the population is nondecreasing with respect to the random number of individuals removed at each catastrophe. The purpose of this paper is to prove that such a monotonic relationship is not true for simple spatial models based on Harris' contact processes, whose dynamics are described by hypergraph structures rather than traditional graph structures. More precisely, it is proved that, for a wide range of parameters, the destruction of (infinite) hyperplanes does not affect the existence of a nontrivial invariant measure, whereas the destruction of large (finite) cubes drives the population to extinction, a result that we depict by using the biological picture: forest fires are more devastating than tornadoes. This indicates that the geometry of the subsets struck by catastrophes is somewhat more important than their area, thus the need to consider spatial rather than nonspatial models in this context.