Richards, Jordan and Wadsworth, Jennifer and Tawn, Jonathan and Brown, Simon
(2021)
*Extremes of Aggregated Random Variables and Spatial Processes.*
PhD thesis, Lancaster University.

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## Abstract

This thesis aims to develop on literature for modelling the extremal behaviour of aggregates of random variables or spatial processes, where here the aggregate refers to the arithmetic mean, or sum, of a collection of random variables, or the integral of a stochastic process. The tail behaviour of aggregates is of interest to practitioners in industries such as financial trading, where extreme returns or losses of a portfolio, i.e., a weighted aggregate of financial derivatives, are of interest. Another area where the literature is applicable is in risk management for river flooding, which typically occur with heavy rain- or snowfall over a catchment area; this problem can be formulated as an extreme value analysis of the total volume of rain or snow that falls within a specified spatio-temporal region. Aggregation acts as a smoothing operation, meaning that all information about the underlying process that feeds the aggregate variable is lost; this can potentially lead to unreliable inference when only the sample aggregate data are available for modelling. However, given that data for the underlying process are available, we can exploit the relationship between the statistical properties of this process and the extremal behaviour of the aggregate to improve on inference; we provide some approaches for establishing such a relationship. We derive the first-order behaviour of the survival function of the weighted sum of random variables, as this aggregate variable tends to its upper-endpoint. We do this first for a bivariate sum with dependence within the set of underlying variables modelled using two widely applied limiting characterisations of extremal dependence. We then extend these results to a d−variate sum for finite d, and with dependence modelled fully using certain copulae. In both cases, we establish links between the extremal behaviour of the underlying random variables and the aggregate variable. We further detail a data-driven approach for modelling the extremes of spatial aggregates. Here we propose a fully spatial model for the extremal behaviour of the underlying process, which relies on conditional methods; we then draw replications from this model to approximate the distribution of the spatial aggregate. Whilst this approach can be applied to any spatial process, we apply it to precipitation and detail considerations that must be taken to make this feasible. A method for accommodating spatial non-stationarity in the extremal dependence structure of data is also proposed. This relies on transformation of the original coordinate system to a new latent space where stationarity can reasonably be assumed.