Multi-type branching processes with time-dependent branching rates

Under mild nondegeneracy assumptions on branching rates in each generation, we provide a criterion for almost sure extinction of a multi-type branching process with time-dependent branching rates. We also provide a criterion for the total number of particles (conditioned on survival and divided by the expectation of the resulting random variable) to approach an exponential random variable as time goes to ∞.

Entry time statistics to different shrinking sets

We consider [Formula: see text]-mixing dynamical systems [Formula: see text] and we find conditions on families of sets [Formula: see text] so that [Formula: see text] tends in law to an exponential random variable, where [Formula: see text] is the entry time to [Formula: see text]

A generalized spatial-temporal model of rainfall based on a clustered point process

The objective in this paper is to present and fit a relatively simple stochastic spatial-temporal model of rainfall in which the arrival times of rain cells occur in a clustered point process. In the x - y plane, rain cells are represented as discs; each disc having a random radius; the locations of the disc centres being given by a two-dimensional Poisson process. The intensity of each cell is a random variable that remains constant over the area of the disc and throughout the lifetime of the cell, the lifetime being an exponential random variable. The cells are randomly classified from 1 to n with different parameters for the different cell types, so that the random variables of an arbitrary cell, e. g. radius and intensity, are correlated. Multi-site second-order properties are derived and used to fit the model to hourly rainfall data taken from six sites in the Thames basin, UK.

A Metastable Result for the Finite Multidimensional Contact Process

We prove that for a contact process restricted to the cube [1,n]d and initially fully occupied, the time to die out, when it is suitably normalized, converges to an exponential random variable as n tends to infinity.

On single-lane roads

A road which narrows at a bottleneck from an ∞-lane road to a one-lane road is studied with the aid of two stochastic processes. Special attention is given to headways and gaps. At the bottleneck an equilibrium headway can be viewed as the maximum of a shifted exponential random variable and a minimum headway. After the bottleneck the situation becomes far more complicated. However, limiting results are obtained for headways and gaps at a large distance from the bottleneck. The asymptotic behavior of headways and gaps is largely determined by the behavior of the desired speed distribution at the lower extreme of its support.

On single-lane roads

A road which narrows at a bottleneck from an ∞-lane road to a one-lane road is studied with the aid of two stochastic processes. Special attention is given to headways and gaps. At the bottleneck an equilibrium headway can be viewed as the maximum of a shifted exponential random variable and a minimum headway. After the bottleneck the situation becomes far more complicated. However, limiting results are obtained for headways and gaps at a large distance from the bottleneck. The asymptotic behavior of headways and gaps is largely determined by the behavior of the desired speed distribution at the lower extreme of its support.