Large deviations and idempotent probability (Q2735643)

This monograph presents a complete theory of idempotent probability, with special emphasis on large deviation theory as a statement about weak convergence. One of the starting points of this theory is the observation that a large deviation principle for a sequence \((P_n)_n\) of probability measures with scale \((r_n)_n\) and rate function \(I\) may be rephrased as a weak convergence statement of the set functions \(P_n(\cdot)^{r_n}\) toward the set function \(\Pi(\cdot)=\sup_{z\in\cdot}e^{-I(z)}\). The analogies between large deviations and weak convergence indeed go deeper and are reflected by the use of a number of common techniques (characteristic functionals, projective limits, continuous mappings etc.) in both fields. Note that one of the most basic properties of \(\Pi\) is that it is maxitive in the sense that \(\Pi(A\cup B)=\Pi(A)\vee \Pi(B)\). Maxitive set functions are known in possibility theory and max-plus calculus, which have first been introduced in the seventies using various notions (possibility measures, \(A\)-measures, performance measures, etc.). NEWLINENEWLINENEWLINEThe two main purposes of the idempotent probability theory developed in the monograph under review are to systematically explore the analogy between large-deviation theory and the notion of weak convergence, and to develop a full stochastic calculus for idempotent measures and idempotent processes. Special emphasis is put on \(\tau\)-smoothness, a certain continuity `from above\'\ for increasing nets of probability measures. (All convergence assertions in this monograph are formulated for nets, rather than for sequences, of measures.) Furthermore, the martingales are replaced by their idempotent analog, the maxingales. NEWLINENEWLINENEWLINEIn Part I, the foundations of idempotent probability theory are led, and an idempotent stochastic calculus is developed. The definition of an idempotent measure differs from the one of a usual measure in the appearance of maxitivity instead of additivity. Nevertheless, idempotent versions of almost all well-known tools and basic facts from `usual\'\ probability theory are presented and proven in Chapter 1: measure theory, modes of convergence, product spaces, independence, conditioning and so on. The theory of maxingales (the idempotent version of martingales) is formulated in analogy to the theory of martingales in Chapter 2. NEWLINENEWLINENEWLINEPart II deals with the large deviation theory as a special case of weak convergence of idempotent probability measures. The author mainly considers large deviation convergence of the distributions of semimaxingales for the Skorokhod topology on the space of càdlàg-functions \([0,\infty)\to\mathbb R^d\). Here the main tools in weak convergence of semimaxingales derived in Chapter 2 (like characterisations of limits in terms of finite-dimensional distributions in Chapter 4, and as solutions to maxingale problems in Chapter 5) are employed for proving and interpreting the large-deviation results obtained. In Chapter 6, an application to networks of queueing processes is given in the frame of idempotent diffusions. Finally, in two appendices, some auxiliary lemmas on convex analysis are presented, and extensive literature remarks are given. NEWLINENEWLINENEWLINEThe monograph is written in a rather accessible, careful and organized way. The language is descriptive and clear, and the number of typoes is extremely small. This book is expected to become a standard reference on idempotent probability theory.