Consider a process of identically-shaped (but not necessarily equal-sized) figures (e.g. points, clusters of points, lines, spheres) embedded at random in n-dimensional space. A simple technique is derived for finding the distribution of the distance from a fixed point, chosen independently of the process of figures, to the k th nearest figure. The technique also shows that the distribution is independent of the distribution of the orientations of the figures. It is noted that the distribution obtained above (for equal-sized figures) is identical to the distribution of the distance from a fixed figure to the k th nearest of a random process of points.

The Poisson shot-noise process discussed here takes the form f:oo H(t, s)N(ds), where N(· is the counting measure of a Poisson process and the H(·, s) are independent stochastic processes. Necessary and sufficient conditions are obtained for convergence in distribution, as t ∼ OC, to any infinitely divisible distribution. The main interest is in the explosive transient one-sided shot-noise, Y(t) = f:1 H(t, s)N(ds) where Var Y(t)∼ oc, Here conditions for asymptotic normality are discussed in detail. Important examples include the Poisson cluster point process and the integrated stationary shotnoise.

In this paper, we study ruin probabilities in two generalized risk models. The effects of timing of payments and interest on the ruin probabilities in the models are considered. The rates of interest are assumed to have a dependent autoregressive structure. Generalized Lundberg inequalities for the ruin probabilities are derived by a renewal recursive technique. An illustrative application is given to the compound binomial risk process.

Let be the scan statistic of window size r for a sequence of n bistate trials . The scan statistic Sn(r) has been successfully used in various fields of applied probability and statistics, and its distribution has been studied extensively in the literature. Currently, all existing formulae for the distribution of Sn(r) are rather complex, and they can only be numerically implemented when is a sequence of Bernoulli trials, the window size r is less than 20 and the length of the sequence n is not too large. Hence, these formulae have been limiting the practical applications of the scan statistic. In this article, we derive a simple and effective formula for the distribution of Sn(r) via the finite Markov chain embedding technique to overcome some of the limitations of the existing complex formulae. This new formula can be applied when is either a sequence of Bernoulli trials or a sequence of Markov dependent bistate trials. Selected numerical examples are given to illustrate our results.

A multitype branching process, the n-family community mating process, is introduced for the purpose of comparing extinction probabilities with those of bisexual Galton–Watson branching processes. Consideration of known properties of standard multitype branching processes leads to conditions which are both necessary and sufficient for extinction in a bisexual Galton–Watson branching process. An application is then made to the counterexample of the author's earlier paper.

We prove that the quasi-invariant measures associated to a Brownian motion with negative drift X form a one-parameter family. The minimal one is a probability measure inducing the transition density of a three-dimensional Bessel process, and it is shown that it is the density of the limit distribution limt→∞Px(X A | τ > t). It is also shown that the minimal quasi-invariant measure of infinite mass induces the density of the limit distribution ) which is the law of a Bessel process with drift.

Suppose that we are given a set of n elements which are to be arranged in some order. At each unit of time a request is made to retrieve one of these elements — the ith being requested with probability Pi. We show that the rule which always moves the requested element one closer to the front of the line minimizes the average position of the element requested among a wide class of rules for all probability vectors of the form P1 = p, P2= · ·· = Pn = (1 – p)/(n − 1). We also consider the above problem when the decision-maker is allowed to utilize such rules as ‘only make a change if the same element has been requested k times in a row', and show that as k approaches infinity we can do as well as if we knew the values of the Pi.

Assume that there are k types of insurance contracts in an insurance company. The ith related claims are denoted by {Xij, j ≥ 1}, i = 1,…,k. In this paper we investigate large deviations for both partial sums S(k; n1,…,nk) = ∑i=1k ∑j=1niXij and random sums S(k; t) = ∑i=1k ∑j=1Ni (t)Xij, where Ni(t), i = 1,…,k, are counting processes for the claim number. The obtained results extend some related classical results.

Let p denote the class of all probability measures defined on the Borel subsets of the unit interval I = [0, 1]. For each positive integer n, take Mn is convex, closed, bounded, and n-dimensional; the convex hull of the space curve {(t,t2, …, tn): 0 ≦ t ≦ 1}; e.g., see Theorems 7.2, 7.3 of [1]. At each point (c1, C2, …, cn) of Mn, define Note that v−, v+ depend only on C1, C2, …, Cn− 1; Vm only on cn; We shall as notational convenience dictates and as will be apparent from the context regard v±n as functions on Mn− 1 or on higher order moment spaces and also regard Vn as a function on moment spaces of order higher than n.

The model considered in this note has been referred to by Haight (1958), Kingman (1961) and Flatto and McKean (1977) as two queues in parallel. Customers choose the shorter of the two queues which are otherwise independent. This system is known to be inferior to a single queue feeding the two servers, but how much? Some elementary considerations provide a fresh perspective on this awkward boundary-value problem. A procedure is proposed for the solution in the context of finite waiting-room size and some comparisons are made with the single-queue system and an independent two-queue system.

We consider an L1 analogue of the least squares estimator for the parameters of stationary, finite-order autoregressions. This estimator, the least absolute deviation (LAD), is shown to be strongly consistent via a result that may have independent interest. The striking feature is that the conditions are so mild as to include processes with infinite variance, notably the stationary, finite autoregressions driven by stable increments in Lα, α > 1. Finally, sampling properties of LAD are compared to those of least squares. Together with a known convergence rate result for least squares, the Monte Carlo study provides evidence for a conjecture on the convergence rate of LAD.

Introduced recently in mathematical finance by Bayer et al. (2016), the rough Bergomi model has proved particularly efficient to calibrate option markets. We investigate some of its probabilistic properties, in particular proving a pathwise large deviations principle for a small-noise version of the model. The exponential function (continuous but superlinear) as well as the drift appearing in the volatility process fall beyond the scope of existing results, and a dedicated analysis is needed.

The purpose of this note is to extend results on critical Galton-Watson branching processes, due to Feller and Lamperti: if the number of individuals at time zero tends to infinity, the process converges, with two different normalizations, to certain diffusion processes in the functional form, discussed in, e.g., Billingsley.

A formal approach to produce a model for the data-generating distribution based on partial knowledge is the well-known maximum entropy method. In this approach, partial knowledge about the data-generating distribution is formulated in terms of some information constraints and the model is obtained by maximizing the Shannon entropy under these constraints. Frequently, in reliability analysis the problem of interest is the lifetime beyond an age t. In such cases, the distribution of interest for computing uncertainty and information is the residual distribution. The information functions involving a residual life distribution depend on t, and hence are dynamic. The maximum dynamic entropy (MDE) model is the distribution with the density that maximizes the dynamic entropy for all t. We provide a result that relates the orderings of dynamic entropy and the hazard function for distributions with monotone densities. Applications include dynamic entropy ordering within some parametric families of distributions, orderings of distributions of lifetimes of systems and their components connected in series and parallel, record values, and formulation of constraints for the MDE model in terms of the evolution paths of the hazard function and mean residual lifetime function. In particular, we identify classes of distributions in which some well-known distributions, including the mixture of two exponential distributions and the mixture of two Pareto distributions, are the MDE models.

The block replacement policy, wherein items are replaced at regular intervals of time and on failure, is rather wasteful because sometimes almost new items are also removed. As an alternative a policy of replacement by new items at regular intervals of time and by used items on failure, is suggested. The consequences of this policy, called used item replacement policy, are studied for Erlangian and sub-exponential life-time distributions. The latter distribution which is the difference of two negative exponential distributions, does not seem to have received much attention in the literature so far.

For the queueing system M/G/∞ some distributions connected with the associated busy periods are derived.

We give a unified way of obtaining approximation formulas for the steady-state distribution of the queue length in the M/GI/s queue. The approximations of Hokstad (1978) and Case A of Tijms et al. (1981) are derived again. The main interest of this paper is in considering the theoretical meaning of the assumptions given in the literature. Having done this, we derive new approximation formulas. Our discussion is based on one version of the steady-state equations, called the basic equations in this paper. The basic equations are derived for M/GI/s/k with finite and infinite k. Similar approximations are possible for M/GI/s/k (k < +∞).

For the directed edge preferential attachment network growth model studied by Bollobás et al. (2003) and Krapivsky and Redner (2001), we prove that the joint distribution of in-degree and out-degree has jointly regularly varying tails. Typically, the marginal tails of the in-degree distribution and the out-degree distribution have different regular variation indices and so the joint regular variation is nonstandard. Only marginal regular variation has been previously established for this distribution in the cases where the marginal tail indices are different.

We provide a probabilistic proof of the characterization of point processes (on the real line) with the order statistic property. The characterization is used to investigate the homogeneity of such processes and is also related to the martingale theory associated with point processes.

The annual earthquake predictions of the China Seismological Bureau (CSB) are evaluated by means of an R score (an R score is approximately 0 for completely random guesses, and approximately 1 for completely successful predictions). The average R score of the annual predictions in China in the period 1990–1998 is about 0.184, significantly larger than 0.0. However, background seismicity is higher in seismically active regions. If a ‘random guess' prediction is chosen to be proportional to the background seismicity, the expected R score is 0.123, and the nine-year mean R score of 0.184 as observed is only marginally higher than this background value. Monte Carlo tests indicate that the probability of attaining an R score of actual prediction by background seismicity based on random guess is about . It is concluded that earthquake prediction in China is still in a very preliminary stage, barely above a pure chance level.