Many problems arising in the physical and social sciences relate to processes which happen sequentially. Such processes are usually investigated by means of the theory of stationary stochastic processes, but there have been some attempts to develop techniques which are not subject to the conceptual difficulties inherent in the probabilistic approach. These difficulties stem from the fact that in practice one is often restricted to a single record which, from the probabilistic point of view, is only one sample from an ensemble of possible records. In some instances such a viewpoint seems artificial, and for some time series it is questionable whether any objective reality corresponds to the idea of an ensemble of possible time series. For example, as noted in Feller (1967), a theory of probability based on a frequency interpretation cannot meaningfully attach a probability to a statement such as “the sun will rise tomorrow”, because to do so one would have to set up a conceptual universe of possible worlds.

The dynamical aspects of single channel gating can be modelled by a Markov renewal process, with states aggregated into two classes corresponding to the receptor channel being open or closed, and with brief sojourns in either class not detected. This paper is concerned with the relation between the amount of time, for a given record, in which the channel appears to be open compared to the amount in which it is actually open and the difference in their proportions; this may be used to obtain information on the unobserved actual process from the observed one. Results, with extensions, on exponential families have been applied to obtain relevant generating functions and asymptotic normal distributions, including explicit forms for the parameters. Numerical results are given as illustration in special cases.

Sun and Waterman model DNA mutations during the PCR reaction by a non-canonical branching process. Mean-field approximated values fit the simulated values surprisingly well. We prove this as a theoretical result, for a wide range of the parameters. Thus, we bound explicitly the biases, in law and in the mean, that the mean-field approximation induces in the random number of mutations of a DNA molecule, as a function of the initial number of molecules, of the number of PCR cycles, of the efficiency rate and of the mutation rate. The range where we prove that the approximation is good contains the observed mutation rates in many actual PCR reactions.

We transpose the ordinary differential equation method (used for decreasing stepsize stochastic algorithms) to a dynamical system method to study dynamical systems disturbed by a noise decreasing to zero. We prove that such an algorithm does not fall into a regular trap if the noise is exciting in an unstable direction.

The almost sure convergence of the maximum of a strictly stationary sequence is studied. We show that, if a particular mixing condition, related to the asymptotic independence of maxima defined by O'Brien, is satisfied, then the maximum behaves asymptotically like the quantile F←(1-1/n) whenever the marginal tail distribution decreases quickly enough. A necessary condition for the almost sure convergence of the maximum is derived. Applications are also discussed.

For random planar stationary tessellations a parameter system of three mean values and two directional distributions of the edges is considered. To investigate the interdependences between these parameters their joint range is described, i.e. the set of values which can be realized by tessellations. The constructive proof is based on mixtures of stationary regular tessellations. To explore the range for the class of stationary ergodic tessellations a procedure is used which we refer to as iteration.

Kingman [5] proved a formula that expresses the joint distribution of the processes where b is a regular point in the state space of a Hunt process. We give an extension of this formula, as well as several interesting facts related to it, for the case when Φ is any finely perfect subset of the state space. We also establish some connections between this result and results on last-exit decompositions.

Consider the basic location problem in which k locations from among n given points X1,…,Xn are to be chosen so as to minimize the sum M(k; X1,…,Xn) of the distances of each point to the nearest location. It is assumed that no location can serve more than a fixed finite number D of points. When the Xi, i ≥ 1, are i.i.d. random variables with values in [0,1]d and when k = ⌈n/(D+1)⌉ we show that

where α := α(D,d) is a positive constant, f is the density of the absolutely continuous part of the law of X1, and c.c. denotes complete convergence.

Distance measurements are useful tools in stochastic geometry. For a Boolean model Z in ℝd, the classical contact distribution functions allow the estimation of important geometric parameters of Z. In two previous papers, several types of generalized contact distributions have been investigated and applied to stationary and nonstationary Boolean models. Here, we consider random sets Z which are generated as the union sets of Poisson processes X of k-flats, k ∈ {0, …, d-1}, and study distances from a fixed point or a fixed convex body to Z. In addition, we also consider the distances from a given flat or a flag consisting of flats to the individual members of X and investigate the associated process of nearest points in the flats of X. In particular, we discuss to which extent the directional distribution of X is determined by this point process. Some of our results are presented for more general stationary processes of flats.

By use of a method of dissection, a formula is derived for the variance in the number of lattice points covered by a rectangle of arbitrary size, lying on a square lattice of unit spacing and inclined to the lattice at an angle of which the tangent is a rational fraction.

The ray solution of Felsenstein's n-allele random selection diffusion process is given for small values of the selection parameters. This solution holds away from, but not near, the boundary of frequency space. The solution is possible only because the coefficients of the associated Jacobi field equations agree uniformly with those for the case of zero selection up to fourth powers in the selection parameters, whilst the covariance of the diffusion has only quadratic dependence.

In this paper, we refine the concept of Γ-variation up to second order, and we give a characterization of this type of asymptotic behaviour. We apply our results to obtain uniform rates of convergence in the weak convergence of renormalised sample maxima to the double exponential distribution. In a second application we derive a rate of convergence result for the Hill estimator.

We are interested in estimating the intensity parameter of a Boolean model of discs (the bombing model) from a single realization. To do so, we derive the conditional distribution of the points (germs) of the underlying Poisson process. We demonstrate how to apply coupling from the past to generate samples from this distribution, and use the samples thus obtained to approximate the maximum likelihood estimator of the intensity. We discuss and compare two methods: one based on a Monte Carlo approximation of the likelihood function, the other a stochastic version of the EM algorithm.

We consider a class of impulse control problems for general underlying strong Markov processes on the real line, which allows for an explicit solution. The optimal impulse times are shown to be of a threshold type and the optimal threshold is characterised as a solution of a (typically nonlinear) equation. The main ingredient we use is a representation result for excessive functions in terms of expected suprema.

In this paper we study the random walk on the hypercube (ℤ / 2ℤ)n which at each step flips k randomly chosen coordinates. We prove that the mixing time for this walk is of the order (n / k)logn. We also prove that if k = o(n) then the walk exhibits cutoff at (n / 2k)logn with window n / 2k.

We study the asymptotic behavior of a new particle filter approach for the estimation of hidden Markov models. In particular, we develop an algorithm where the latent-state sequence is segmented into multiple shorter portions, with an estimation technique based upon a separate particle filter in each portion. The partitioning facilitates the use of parallel processing, which reduces the wall-clock computational time. Based upon this approach, we introduce new estimators of the latent states and likelihood which have similar or better variance properties compared to estimators derived from standard particle filters. We show that the likelihood function estimator is unbiased, and show asymptotic normality of the underlying estimators.

An rth-order extremal process Δ(r) = (Δ(r)t)t≥0 is a continuous-time analogue of the rth partial maximum sequence of a sequence of independent and identically distributed random variables. Studying maxima in continuous time gives rise to the notion of limiting properties of Δt(r) as t ↓ 0. Here we describe aspects of the small-time behaviour of Δ(r) by characterising its upper and lower classes relative to a nonstochastic nondecreasing function bt > 0 with limt↓bt = 0. We are then able to give an integral criterion for the almost sure relative stability of Δt(r) as t ↓ 0, r = 1, 2, . . ., or, equivalently, as it turns out, for the almost sure relative stability of Δt(1) as t ↓ 0.

We consider a discrete-time branching random walk in the boundary case, where the associated random walk is in the domain of attraction of an α-stable law with 1 < α < 2. We prove that the derivative martingale Dn converges to a nontrivial limit D∞ under some regular conditions. We also study the additive martingale Wn and prove that n1/αWn converges in probability to a constant multiple of D∞.

In this paper we compute the absorbing time Tn of an n-dimensional discrete-time Markov chain comprising n components, each with an absorbing state and evolving in mutual exclusion. We show that the random absorbing time Tn is well approximated by a deterministic time tn that is the first time when a fluid approximation of the chain approaches the absorbing state at a distance 1 / n. We provide an asymptotic expansion of tn that uses the spectral decomposition of the kernel of the chain as well as the asymptotic distribution of Tn, relying on extreme values theory. We show the applicability of this approach with three different problems: the coupon collector, the erasure channel lifetime, and the coupling times of random walks in high-dimensional spaces.