Computational methods for sums of random variables

Abstract

A typical problem in the field of rare-event estimation is to find the probability P(S > γ) where S := X 1 + · · · + X d for a fixed d ∈ N + and where the γ ∈ R is large or increasing. In applications we often wish to understand the behaviour of a combination of random factors. Hence the random variable S is ubiquitous in real-world modeling problems. It can model, for example, aggregate risk or portfolio value for holding d risky assets [124,152], the aggregate losses for d insurance policy claims [14,107], and the combined signal interference from d wireless transmission sources [73]. Probabilities of this form are used to understand how a system would behave under extreme scenarios such as a market crash, a power surge, or a natural disaster. One is typically interested in, not just the quantity P(S > γ), but the behaviour of the summands when the extreme event {S > γ} occurs.