A non-exponential extension of Sanov’s theorem via convex duality

Abstract

This work is devoted to a vast extension of Sanov's theorem, in Laplace principle form, based on alternatives to the classical convex dual pair of relative entropy and cumulant generating functional. The abstract results give rise to a number of probabilistic limit theorems and asymptotics. For instance, widely applicable non-exponential large deviation upper bounds are derived for empirical distributions and averages of i.i.d. samples under minimal integrability assumptions, notably accommodating heavy-tailed distributions. Other interesting manifestations of the abstract results include uniform large deviation bounds, variational problems involving optimal transport costs, and constrained super-hedging problems, as well as an application to error estimates for approximate solutions of stochastic optimization problems. The proofs build on the Dupuis-Ellis weak convergence approach to large deviations as well as the duality theory for convex risk measures.

Extended Laplace principle for empirical measures of a Markov chain

“…The original idea for extensions of Laplace principles of this form is due to Lacker [34], who pursued this in the context of independent and identically distributed (i.i.d.) sequences of random variables instead of Markov chains.…”

“…The initial goal was to provide a setting to study more than just exponential tail behavior of random variables, as is given by large deviations theory. The extension of Sanov’s theorem he proved [34, Theorem 3.1] can be used to derive many interesting results, such as polynomial large deviation upper bounds, robust large deviation bounds, robust laws of large numbers, asymptotics of optimal transport problems, and more, while several possibilities remain unexplored.…”

“…Intuition, applicability, and difficulties in dealing with the above result are very similar to the i.i.d. case and are described in detail in the introduction of [34]. The main differences for Markov chains are conditions (B1) and (B2).…”

“…case, e.g. [34, Chapter 4 and 6] appear more difficult to obtain for Markov chains. A thorough analysis of the range of applications of Theorem 1.1 remains incomplete for now, as the goal of this work is to give a detailed account of one application of Theorem 1.1 to robust Markov chains.…”

“…The large deviations principle for empirical measures of a Markov chain can equivalently be stated in Laplace principle form, which builds on the convex dual pair of relative entropy (or Kullback– Leibler divergence) and cumulant generating functional f ↦ ln ʃ exp ( f ). Following the approach by Lacker (2016) in the independent and identically distributed case, we generalize the Laplace principle to a greater class of convex dual pairs. We present in depth one application arising from this extension, which includes large deviation results and a weak law of large numbers for certain robust Markov chains—similar to Markov set chains—where we model robustness via the first Wasserstein distance.…”