Some Useful Functions for Functional Large Deviations

Abstract

Useful descriptions of stochastic models are often provided when they are represented as functions of well understood stochastic models. Properties of the well understood model can be preserved by the representation. For example, the large deviation principle (LDP) is preserved by continuous maps via the contraction principle and weak convergence is preserved by continuous maps via the continuous mapping theorem. In a recent paper, Ganesh and O'Connell demonstrate the value of particular topological space of functions indexed by the positive real line by showing: (1) a large class of partial sums process satisfy the functional LDP in the space; and (2) the supremum map is continuous when restricted to an appropriate subspace. This makes the space useful, for example, in proving logarithmic asymptotics for the stationary waiting-time distribution at a single server queue. This paper facilitates further deductions from the space by considering other useful maps: inversion, composition and first passage time. It is shown that composition and inversion are continuous when restricted to an appropriate subspace. First passage time is not continuous, but with additional rate-function assumptions the LDP can be deduced by explicit calculation. A number of examples of these results are presented.