Let X1, …, Xn be independent random points drawn from an absolutely continuous probability measure with density f in ℝd. Under mild conditions on f, wederive a Poisson limit theorem for the number of large probability nearest-neighbour balls. Denoting by Pn the maximum probability measure of nearest-neighbour balls, this limit theorem implies a Gumbel extreme value distribution for nPn − ln n as n → ∞. Moreover, we derive a tight upper bound on the upper tail of the distribution of nPn − ln n, which does not depend on f.

The spatial distribution of canopy gaps was analysed on three sites (total 71 ha) in the tropical moist forest of Tai National Park, Ivory Coast. Pattern analysis revealed a clustered distribution of gaps for two of the three sites. Catena dependent gap formation processes might explain local differences in the occurrence and distribution of gaps. Gap densities, sizes and percentage forest area in gap phase are higher on the upper and middle slope than on the crest or lower slope. As a consequence, regeneration of gap dependent tree species might be directed to the catena positions with the highest disturbance regime. The spatial distribution of gap dependent species can be clumped, not only due to the regeneration within gaps, but also due to the clustered nature of gap distribution on its own.

After some introductory general remarks on recent investigations involving population models, two broad classes of stochastic model are discussed further, viz. spatial nearest-neighbour lattice models and doubly stochastic models.

In Section 1 of the paper, the first type of model is considered primarily for its relevance to recent work by the author and others affecting the practical design and analysis of replicated field experiments.

In Section 2, doubly stochastic processes are discussed more theoretically, particularly models investigated recently by the author involving infinitesimal transition operators in continuous time linear in the (variable) parameters.

Some new numerical results on extinction and other ‘absorption' probabilities are presented; these are intended to throw further light on the extent to which the assumption of ‘white-noise' variability of the parameters can be a useful approximation to more realistic models.

Motivated by problems in the analysis of spatial data, we prove some general Poisson limit theorems for the U-statistics of Hoeffding (1948). The theorems are applied to tests of clustering or collinearities in plane data; nearest neighbour distances are also considered.

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.