A population of susceptible individuals partitioned into m groups and exposed to a contagious disease is considered.

The progress of this simple epidemic is modeled by an m -dimensional stochastic process, whose components are the number of infective individuals in the respective groups at time t. Exact and approximate formulas for the joint and marginal state probabilities are obtained.

The focus of our attention is the limit distribution of the sum of independent and identically distributed random vectors from which all the extreme summands are removed. The problem is rather trivial if the summands are ordered by their norms. It is of much more interest when the vertices of the convex hull generated by the vectors are taken as the extremes.

Let {Z(t)}t0 be an age-dependent branching process with immigration. For a general class of functions Φ(x), a necessary and sufficient condition is given for when E{Φ (Z(t))} <∞. This result is a direct generalization of a theorem proven for the branching process without immigration.

Let (Yn, Nn)n≥1 be independent and identically distributed bivariate random variables such that the Nn are positive with finite mean ν and the Yn have a common heavy-tailed distribution F. We consider the process (Zn)n≥1 defined by Zn = Yn - Σn-1, where It is shown that the probability that the maximum M = maxn≥1Zn exceeds x is approximately as x → ∞, where F' := 1 - F. Then we study the integrated tail of the maximum of a random walk with long-tailed increments and negative drift over the interval [0, σ], defined by some stopping time σ, in the case in which the randomly stopped sum is negative. Finally, an application to risk theory is considered.

We study a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = ℝ+ d , with drift r 0 ∈ ℝ d and Hurst parameter H ∈ (½, 1). Under a natural stability condition on the drift vector r 0 and reflection directions, we establish a geometric drift towards a compact set for the 1-skeleton chain Ž̆ of the RFBM process Z; that is, there exist β, b ∈ (0, ∞) and a compact set C ⊂ S such that ΔV(x):= E x [V(Ž̆(1))] − V(x) ≤ −βV(x) + b 1 C (x), x ∈ S, for an exponentially growing Lyapunov function V : S → [1, ∞). For a wide class of Markov processes, such a drift inequality is known as a necessary and sufficient condition for exponential ergodicity. Indeed, similar drift inequalities have been established for reflected processes driven by standard Brownian motions, and our result can be viewed as their fractional Brownian motion counterpart. We also establish that the return times to the set C itself are geometrically bounded. Motivation for this study is that RFBM appears as a limiting workload process for fluid queueing network models fed by a large number of heavy-tailed ON/OFF sources in heavy traffic.

We provide an algorithm to find the value and an optimal strategy of the Ten Thousand dice game solitaire variant in the framework of Markov control processes. Once an optimal critical threshold is found, the set of nonstopping states of the game becomes finite and the solution is found by a backwards algorithm that gives the values for each one of these states of the game. The algorithm is finite and exact. The strategy to find the critical threshold comes from the continuous pasting condition used in optimal stopping problems for continuous-time processes with jumps.

Results similar to those in [3] are obtained for one-dimensional diffusion processes with discounted cost. The stronger assumption that at least one of the inaccessible boundaries is natural enables us to identify the solution of a differential equation corresponding to the future expected cost function.

A time-series consisting of white noise plus Brownian motion sampled at equal intervals of time is exactly orthogonalized by a discrete cosine transform (DCT-II). This paper explores the properties of a version of spectral analysis based on the discrete cosine transform and its use in distinguishing between a stationary time-series and an integrated (unit root) time-series.

The explosion probability before time t of a branching diffusion satisfies a nonlinear parabolic partial differential equation. This equation, along with the natural boundary and initial conditions, has only the trivial solution, i.e. explosion in finite time does not occur, provided the creation rate does not grow faster than the square power at ∞.

We find a ‘natural' measure on the set of convex infinite cylinders touching a convex body K in En. Using the curvature measures, we can calculate the probability that a convex infinite cylinder and a convex body K are tangent in some previously determined regions. As a particular case we obtain a measure on all q-planes touching K.

For many years the modified Mercalli (MM) scale has been used to describe earthquake damage and effects observed at scattered locations. In the next stage of an analysis involving MM data, isoseismal lines based on the observations have been added to maps by hand, i.e. subjectively. However a few objective methods have been proposed (by e.g. De Rubeis et al., Brillinger, Wald et al. and Pettenati et al.). The work presented here develops objective methods further. In particular the ordinal character of the MM scale is specifically taken into account. Numerical smoothing is basic to the approach and methods involving splines, local polynomial regression and wavelets are illustrated. The approach also allows the inclusion of explanatory variables, for example site effects. The procedure is implemented for data from the 17 October 1989 Loma Prieta earthquake.

We present sufficient conditions for the weak convergence of the distributions of the kth order statistics from a conditionally mixing array of random variables to limit laws which are represented in terms of a mixed compound Poisson distribution.

A new representation for the characteristic function of the multivariate strictly geo-stable distribution is presented. The representation is appealing from a parametric viewpoint: its parameters have an intuitive probabilistic interpretation; and it is particularly useful for estimating the parameters of the geo-stable distribution.

We give some explicit formulae for the prices of two path-dependent options which combine Brownian averages and penalizations. Because these options are based on both the maximum and local time of Brownian motion, obtaining their prices necessitates some involved study of homogeneous Brownian functionals, which may be of interest in their own right.

Gorostiza and Wakolbinger (1991), and Dawson and Perkins (1991) established the same persistence criterion for a class of critical branching particle systems and for a class of superprocesses respectively. In this note we take another approach to the criterion and present a simpler proof of it.

Consider a fork-join queue, where each job upon arrival splits into k tasks and each joins a separate queue that is attended by a single server. Service times are independent, exponentially distributed random variables. Server i works at rate , where μ is constant. We prove that the departure process becomes stochastically faster as the service rates become more homogeneous in the sense of stochastic majorization. Consequently, when all k servers work with equal rates the departure process is stochastically maximized.

We follow the terminology of Khintchine's monograph [4].

Let us consider a random stream of calls entering a service system which consists of n lines. The service durations are assumed to be identically distributed random variables which are independent both of each other and of the course of the incoming stream.

To optimize the performance of an M/M/s queue, Moder and Phillips [1] have studied the problem in which there are always n1 open counters ready to offer service. The actual number of open counters increases by one, up to a maximum n2, every time the number of waiting customers exceeds a certain critical number; it decreases by one every time the queue length becomes smaller than another critical number.

Some properties of the Voronoi polygon associated with a moving particle in a regular tessellation are studied. In particular, the area of the variable polygon has played a relevant role in previous studies on plant competition, animal predation, adjustment of the sex ratio in bisexual populations and the gregarious behaviour of animal species.