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.
In recent years there has been a resurgence of interest in generation adequacy risk assessment, due to the need to include variable generation renewables within such calculations. This paper will describe new statistical approaches to estimating the
joint distribution of demand and available VG capacity; this is required for the LOLE calculations used in many statutory adequacy studies, for example those of GB and PJM. The most popular estimation technique in the VG-integration literature is `hindcast, in which the historic joint distribution of demand and available VG is used as a predictive distribution. Through the use of bootstrap statistical analysis, this paper will show that due to extreme sparsity of data on times of high demand and low VG, hindcast results can suffer from sampling uncertainty to the extent that they have little practical meaning. An alternative estimation approach, in which a marginal distribution of available VG is rescaled according to demand level, is thus proposed. This reduces sampling uncertainty at the expense of the additional model structure assumption, and further provides a means of assessing the sensitivity of model outputs to the VG-demand relationship by varying the function of demand by which the marginal VG distribution is rescaled.
