12. Random Walks and Factorisation Identities

In this chapter, several remarkable and rather useful relations establishing interconnections between different characteristics of random walks (the so-called boundary functionals) are derived, and the arising problems are related to the simplest boundary problems of Complex Analysis. Section 12.1 introduces the concept of factorisation identity and derives two fundamental identities of that kind. Some consequences of these identities, including the trichotomy theorem on the oscillatory behaviour of random walks and a one-sided version of the Strong Law of Large Numbers are presented in Sect. 12.2. Pollaczek–Spitzer’s identity and an identity for the global maximum of the random walk are derived in Sect. 12.3, followed by illustrating these results by examples from the ruin theory and the theory of queueing systems in Sect. 12.4. Sections 12.5 and 12.6 are devoted to studying the cases where factorisation components can be obtained in explicit form and so closed form expressions are available for the distributions of a number of important boundary functionals. Sections 12.7 and 12.8 employ factorisation identities to derive the asymptotic properties of the distribution of the excess of a random walk of a high level and that of the global maximum of the walk, and also to analyse the distribution of the first passage time.