Stan Zachary

A Note on Veraverbeke's Theorem

We give an elementary probabilistic proof of Veraverbekes theorem for the asymptotic distribution of the maximum of a random walk with negative drift and heavy-tailed increments. The proof gives insight into the principle that the maximum is in general attained through a single large jump.

Multivariate extrapolation in the offshore environment

We consider the estimation of the extremes of the metocean climate, in particular those of the univariate and joint distributions of wave height, wave period and wind speed. This is of importance in the design of oil rigs and other marine structures which must be able to withstand extreme environmental loadings. Such loadings are often functions of two or more metocean variables and the problem is to estimate the extremes of their joint distribution, typically beyond the range of the observed data. The statistical methodology involves both univariate and multivariate extreme value theory. Multivariate theory which avoids (often very inappropriate) prior assumptions about the nature of the statistical association between the variables is a fairly recent development. We review and adapt this theory, presenting simpler descriptions and proofs of the key results. We study in detail an application to data collected over a nine-year period at the Alwyn North platform in the northern North Sea. We consider the many problems arising in the analysis of such data, including those of seasonality and short-term dependence, and we show that multivariate extreme value theory may indeed be used to estimate probabilities and return periods associated with extreme events. We consider also the confidence intervals associated with such estimates and the implications for future data collection and analysis. Finally we review further both the statistical and engineering issues raised by our analysis.

The probability of exceeding a high boundary on a random time interval for a heavy-tailed random walk

We study the asymptotic probability that a random walk with heavy-tailed increments crosses a high boundary on a random time interval. We use new techniques to extend results of Asmussen (1998) to completely general stopping times, uniformity of convergence over all stopping times, and a wide class of nonlinear boundaries. We give also some examples and counterexamples.

The Performance of Single Resource Loss Systems in Multiservice Networks

Recent developments in communication networks have led to much interest in systems where traac of widely diiering characteristics is integrated together. In earlier work the authors develop an analysis of single resource loss systems under the assumption of heavy traac. In this paper we discuss the analysis with special emphasis on its practical implementation for solving real world examples that arise in the study of multiservice networks. The assumption of heavy traac also holds in this paper, but there is good reason to expect that results are also accurate when the resource is near to critical loading.

Probability theory of capacity value of additional generation

This paper describes a new probability theory of the capacity value of additional generation in electrical power systems. A closed-form expression for the effective load carrying capability or equivalent firm capacity of a small additional capacity is derived. This depends on the mean and variance of the distribution of available additional generation capacity, and the shape of the distribution of the difference between available existing capacity and demand, near zero margin. The theory extends naturally to the case where the pre-existing background and additional resource are not statistically independent. The theory may be used to explain and confirm the generality of various well-known properties of capacity value results, as is illustrated using Great Britain examples. Of particular note is the common observation that if the distribution for demand is shifted so as to increase the calculated risk, then the capacity value of additional generation increases. The new theory demonstrates that this is not true in general, but rather is a consequence of the shape near zero margin of the probability distribution of the margin of existing generating capacity over demand.

Optimal control of storage for arbitrage, with applications to energy systems

We study the optimal control of storage which is used for arbitrage, i.e. for buying a commodity when it is cheap and selling it when it is expensive. Our particular concern is with the management of energy systems, although the results are generally applicable. We consider a model which may account for nonlinear cost functions, market impact, input and output rate constraints and inefficiencies or losses in the storage process. We develop an algorithm which is maximally efficient in the sense that it incorporates the result that, at each point in time, the optimal management decision depends only a finite, and typically short, time horizon. We give examples related to the management of a real-world system.

Heavy-Tailed and Long-Tailed Distributions

In this chapter we are interested in (right-) tail properties of distributions, i.e. in properties of a distribution which, for any x, depend only on the restriction of the distribution to (x, ∞). More generally it is helpful to consider tail properties of functions.

Densities and Local Probabilities

This chapter is devoted to local long-tailedness and to local subexponentiality. First we consider densities with respect to either Lebesgue measure on \(\mathbb{R}\) or counting measure on \(\mathbb{Z}\). Next we study the asymptotic behaviour of the probabilities to belong to an interval of a fixed length. We give the analogues of the basic properties of the tail probabilities including two analogues of Kesten’s estimate, and provide sufficient conditions for probability distributions to have these local properties.

Maximum of Random Walk

In this chapter, we study a random walk whose increments have a (right) heavy-tailed distribution with a negative mean. The maximum of such a random walk is almost surely finite, and our interest is in the tail asymptotics of the distribution of this maximum, for both infinite and finite time horizons; we are further interested in the local asymptotics for the maximum in the case of an infinite time horizon. We use direct probabilistic techniques and show that, under the appropriate subexponentiality conditions, the main reason for the maximum to be far away from zero is again that a single increment of the walk is similarly large.

Deviations from the Delft Method in Quality Control for Streamer Shaping

The Delft method has been widely accepted for quality control in geodetic positioning. It was originally developed for onshore surveys, where the survey networks are fully connected. In the offshore seismic surveying industry, the use of streamers originally required a change in this approach. Only one streamer was used, so that it was impossible to create a fully connected survey network. To solve that problem, compasses were used to supplement the network. This required the use of streamer shape models.