The assumption of linear costs of observation usually leads to optimal stopping boundaries which are straight lines. For non-linear costs of observation, the question arises of how the shape of cost functions influences the shape of optimal stopping boundaries. In Irle (1987), (1988) it was shown that, under suitable assumptions on c, for the problem of optimal stopping (Wt + x)+ - c(s + t), t ∊ [0,∞), the optimal stopping boundary h(t) can be enscribed between k1/c'(t) and k2/c'(t) for some constants k1, k2. In this paper we find the exact asymptotic expansion h(t) = 1/(4c'(t))(1 + o(1)).

We study an example of a hit-and-run random walk on the symmetric group $\mathbf S_n$ . Our starting point is the well-understood top-to-random shuffle. In the hit-and-run version, at each single step, after picking the point of insertion j uniformly at random in $\{1,\ldots,n\}$ , the top card is inserted in the jth position k times in a row, where k is uniform in $\{0,1,\ldots,j-1\}$ . The question is, does this accelerate mixing significantly or not? We show that, in $L^2$ and sup-norm, this accelerates mixing at most by a constant factor (independent of n). Analyzing this problem in total variation is an interesting open question. We show that, in general, hit-and-run random walks on finite groups have non-negative spectrum.

The stochastic version of the simple epidemic is considered in this paper. A method which has computational advantages is developed for computing the probability distribution of the process. The method is based on the Jordan canonical form of the matrix representing the system of differential-difference equations for the simple epidemic. It is also shown that the method can be applied to the general epidemic as well as any other right-shift process.

This paper addresses random packing of equal-sized disks in a manner such that no disk has a gap on its circumference large enough to accommodate an extra touching neighbour. This structure generalises the deterministic packing models discussed in classical geometry (Coxeter (1961), Hilbert and Cohn-Vossen (1952)). Relationships with the dual mosaic formed by joining the centres of touching disks are established. Constraints on the neighbourhood of disks and on the packing density are established.

The Edgeworth expansion gives an indication of the rate of convergence of the distribution function of the sum of a fixed number of random variables to the normal distribution. A similar expansion is given here for the distribution function of the sum of a random number N of random variables, when the probability generating function of N takes a special form.

Let $(Z_n)_{n\geq0}$ be a supercritical Galton–Watson process. Consider the Lotka–Nagaev estimator for the offspring mean. In this paper we establish self-normalized Cramér-type moderate deviations and Berry–Esseen bounds for the Lotka–Nagaev estimator. The results are believed to be optimal or near-optimal.

We define a class of anticipative flows on Poisson space and compute its Radon-Nikodym derivative. This result is applied to statistical testing in an anticipative queueing problem.

In this paper we investigate the limiting behavior of the failure rate for the convolution of two or more life distributions. In a previous paper on mixtures, Block, Mi and Savits (1993) showed that the failure rate behaves like the limiting behavior of the strongest component. We show a similar result here for convolutions. We also show by example that unlike a mixture population, the ultimate direction of monotonicity does not necessarily follow that of the strongest component.

Let {Zt, t ≧ 0} be an irreducible regular semi-Markov process with transition probabilities Pij (t). Let f(t) be non-negative and non-decreasing to infinity, and let λ ≧ 0. This paper identifies a large set of functions f(t) with the solidarity property that convergence of the integral ≧ eλtf(t)Pij(t) dt for a specific pair of states i and j implies convergence of the integral for all pairs of states. Similar results are derived for the Markov renewal functions Mij (t). Among others it is shown that f(t) can be taken regularly varying.

We consider a two-sided exit problem for a Lévy process with no positive jumps. The Laplace transform of the time when the process first exits an interval from above is obtained. It is expressed in terms of another Laplace transform for the one-sided exit problem. Applications of this result are discussed. In particular, a new expression for the solution to the two-sided exit problem is obtained. The joint distribution of the minimum and the maximum values of such a Lévy process is also studied.

We present a very short derivation of the integral representation of the two-sided Skorokhod reflection Z of a continuous function X of bounded variation, which is a generalization of the integral representation of the one-sided map featured in Anantharam and Konstantopoulos (2011) and Konstantopoulos et al. (1996). We also show that Z satisfies a simpler integral representation when additional conditions are imposed on X.

Direct and simple proofs are given of the equivalence of partial balance over a set of states and the fact that the corresponding equilibrium distribution is insensitive to nominal sojourn time in that set and independent of its past and remaining duration. By a counter-example it is shown that solely insensitivity is not sufficient for partial balance.

Miyazawa and Taylor (1997) introduced a class of assemble-transfer batch service queueing networks which do not have tractable stationary distribution. However by assuming a certain additional arrival process at each node when it is empty, they obtain a geometric product-form stationary distribution which is a stochastic upper bound for the stationary distribution of the original network. In this paper we develop a stochastic lower bound for the original network by introducing an additional departure process at each node which tends to remove all the customers present in it. This model in combination with the aforementioned upper bound model gives a better sense for the properties of the original network.

The classical technique of uniformization (or randomization) for bounded continuous-time Markov chains and Markov reward structures is extended to dynamic systems generated by arbitrary non-negative generators. Most notably, these include so-called input-output models in economic analysis. The results are of practical interest for both computational and theoretical purposes. Particularly, the recursive computation and the limiting behaviour of cumulative reward structures for non-negative dynamic systems is concluded as a special application. Two numerical examples are included to illustrate the conditions and the results.

A method used by electrical engineers to analyse polyphase alternating current systems suggests a generalisation to arbitrary plane polygons of a theorem on triangles nowadays known, for obscure reasons, as ‘Napoleon's Theorem': the centroids of equilateral triangles erected on the sides of an arbitrary triangle form the vertices of an equilateral triangle. The generalisation to other polygons uses a construction first studied by C.-A. Laisant in 1877; results of Jesse Douglas (1940) and the author (1941) are re-derived by means of the elementary algebra of finite-dimensional vector spaces over the field of complex numbers.

The age and type distributions (conditioned on non-extinction) in a critical Bellman-Harris process are shown to converge in probability to a deterministic distribution under a finite first-moment hypothesis on the offspring distribution and a tightness condition on the lifetime distributions.

Double-stranded DNA molecules can be damaged by enzymic action or radiation, in a manner which creates randomly-located single-stranded breaks (nicks). The accumulation of these leads eventually to the double-stranded breakage of the molecule, because two opposite-strand nicks within a critical distance of each other establish conditions for breakage. We study the random variable N, defined as the number of nicks needed for double-stranded breakage to occur. We develop an asymptotic theory which is needed for practical computations.

We investigate some effects that the ‘light' trimming of a sum Sn = X1 + X2 + · ·· + Xn of independent and identically distributed random variables has on behaviour of iterated logarithm type. Light trimming is defined as removing a constant number of summands from Sn. We consider two versions: (r)Sn, which is obtained by deleting the r largest Xi from Sn, and , which is obtained by deleting the r variables Xi which are largest in absolute value from Sn. We summarise some relevant results from Rogozin (1968), Heyde (1969), and later writers concerning the untrimmed sum, and add some new results concerning trimmed sums. Among other things we show that a general form of the law of the iterated logarithm holds for but not (completely) for .

Starting from a Markov chain with a finite or a countable infinite alphabet, we consider the chain obtained when all but one symbol are indistinguishable for the practitioner. We study conditions on the transition matrix of the Markov chain ensuring that the image chain has continuous or discontinuous transition probabilities with respect to the past.