Graphical models with heavy-tailed factors can be used to model extremal dependence or causality between extreme events. In a Bayesian network, variables are recursively defined in terms of their parents according to a directed acyclic graph (DAG). We focus on max-linear graphical models with respect to a special type of graph, which we call a tree of transitive tournaments. The latter is a block graph combining in a tree-like structure a finite number of transitive tournaments, each of which is a DAG in which every two nodes are connected. We study the limit of the joint tails of the max-linear model conditionally on the event that a given variable exceeds a high threshold. Under a suitable condition, the limiting distribution involves the factorization into independent increments along the shortest trail between two variables, thereby imitating the behaviour of a Markov random field.

We are also interested in the identifiability of the model parameters in the case when some variables are latent and only a subvector is observed. It turns out that the parameters are identifiable under a criterion on the nodes carrying the latent variables which is easy and quick to check.

A Markov tree is a random vector indexed by the nodes of a tree whose distribution is determined by the distributions of pairs of neighbouring variables and a list of conditional independence relations. Upon an assumption on the tails of the Markov kernels associated to these pairs, the conditional distribution of the self-normalized random vector when the variable at the root of the tree tends to infinity converges weakly to a random vector of coupled random walks called a tail tree. If, in addition, the conditioning variable has a regularly varying tail, the Markov tree satisfies a form of one-component regular variation. Changing the location of the root, that is, changing the conditioning variable, yields a different tail tree. When the tails of the marginal distributions of the conditioning variables are balanced, these tail trees are connected by a formula that generalizes the time change formula for regularly varying stationary time series. The formula is most easily understood when the various one-component regular variation statements are tied up into a single multi-component statement. The theory of multi-component regular variation is worked out for general random vectors, not necessarily Markov trees, with an eye towards other models, graphical or otherwise.

It is well known that Monte Carlo integration with variance reduction by means of control variates can be implemented by the ordinary least squares estimator for the intercept in a multiple linear regression model. A central limit theorem is established for the integration error if the number of control variates tends to infinity. The integration error is scaled by the standard deviation of the error term in the regression model. If the linear span of the control variates is dense in a function space that contains the integrand, the integration error tends to zero at a rate which is faster than the square root of the number of Monte Carlo replicates. Depending on the situation, increasing the number of control variates may or may not be computationally more efficient than increasing the Monte Carlo sample size.

Extreme value theory for random vectors and stochastic processes with continuous trajectories is usually formulated for random objects where the univariate marginal distributions are identical. In the spirit of Sklar's theorem from copula theory, such marginal standardization is carried out by the pointwise probability integral transform. Certain situations, however, call for stochastic models whose trajectories are not continuous but merely upper semicontinuous (USC). Unfortunately, the pointwise application of the probability integral transform to a USC process does not, in general, preserve the upper semicontinuity of the trajectories. In this paper we give sufficient conditions to enable marginal standardization of USC processes and we state a partial extension of Sklar's theorem for USC processes. We specialize the results to max-stable processes whose marginal distributions and normalizing sequences are allowed to vary with the coordinate.

Classical extreme value theory for stationary sequences of random variables can to a large extent be paraphrased as the study of exceedances over a high threshold. A special role within the description of the temporal dependence between such exceedances is played by the extremal index. Parts of this theory can be generalized not only to random variables on an arbitrary state space hitting certain failure sets, but even to a triangular array of rare events on an abstract probability space. In the case of M4 (maxima of multivariate moving maxima) processes, the arguments take a simple and direct form.

For arbitrary stationary sequences of random variables satisfying a mild mixing condition, distributional approximations are established for functionals of clusters of exceedances over a high threshold. The approximations are in terms of the distribution of the process conditionally on the event that the first variable exceeds the threshold. This conditional distribution is shown to converge to a nontrivial limit if the finite-dimensional distributions of the process are in the domain of attraction of a multivariate extreme-value distribution. In this case, therefore, limit distributions are obtained for functionals of clusters of extremes, thereby generalizing results for higher-order stationary Markov chains by Yun (2000).

The formation of Ni germanosilicides during solid-state interaction between Ni and heavily B-doped strained epitaxial Si1-xGex films with x=0.18, 0.32 and 0.37 is studied. No NiSi2 is found in these samples even after annealing at 850 °C, which can be compared to the formation of NiSi2 at 750 °C on Si(100). Resistance and diffraction studies for the Si0.82Ge0.18 sample indicate that NiSi0.82Ge0.18 forms and the NiSi0.82Ge0.18/Si0.82Ge0.18 structure is stable from 400 to 700 °C. For the NiSi1-uGeu formed in all Si1-xGex samples, where u can be different from x, a strong film texturing is observed. When the Ge fraction is increased from 18 at.% to 32–37 at.%, the morphological stability of the film is degraded and a substantial increase in sheet resistance occurs already at 600 °C. The contact resistivity for the NiSi0.8Ge0.2/Si0.8Ge0.2 interface formed at 550 °C is determined as 1.2×10-7 Ωcm2, which satisfies the RS contact resistivity requirement for the 70 nm technology node.

The reactions of Ni with polycrystalline Si, Si0.82Ge0.18 and Si0.818Ge0.18C0.002 films in two different configurations during rapid thermal processing were studied. For the usually studied planar configuration with 20 nm thick Ni on 130–290 nm thick Si1-x-yGexCy, NiSi1-xGex(C) forms at 450°C on either Si0.82Ge0.18 or Si0.818Ge0.18C0.002, comparable to NiSi formed on Si. However, the agglomeration of NiSi1-xGex(C) on Si0.818Ge0.18C0.002 occurs at 625°C, about 50°C higher than that of NiSi1-xGex on Si0.82Ge0.18. For thin-film lateral diffusion couples, a 200-nm thick Ni film was in contact with 80–130 nm thick Si1-x-yGexCy through 1–10 μm sized contact openings in a 170 nm thick SiO2 isolation. While the Ni3Si phase was formed for both the Si0.82Ge0.18 and Si0.818Ge0.18C0.002 samples, the presence of 0.2 at.% C caused a slightly slower lateral growth.