Consider two sequences of bounded random variables, a value and a timing process, that satisfy the large deviation principle (LDP) with rate-function J( , ) and whose cumulative process satisfies the LDP with rate-function I(). Under mixing conditions, an LDP for estimates of I constructed by transforming an estimate of J is proved. For the case of cumulative renewal processes it is demonstrated that this approach is favorable to a more direct method as it ensures the laws of the estimates converge weakly to a Dirac measure at I .

We consider the sample paths of the order statistics of i.i.d. random variables with common distribution function F. If F is strictly increasing (but possibly having discontinuities), we prove that the sample paths of the order statistics satisfy the large deviation principle in the Skorohod (J1) topology. If F corresponds to a discrete distribution, we prove that the sample paths satisfy the large deviation principle in the topology of weak convergence. Versions of Sanov’s Theorem are deduced as a corollary to these results. A number of illustrative examples are presented, including applications to the sample paths of trimmed means and Hill Plots. 1

With great affection this paper is dedicated to Henry McKean on the occasion of his 75th birthday. ABSTRACT. Given a constrained minimization problem, under what conditions does there exist a related, unconstrained problem having the same minimum points? This basic question in global optimization motivates this paper, which answers it from the viewpoint of statistical mechanics. In this context, it reduces to the fundamental question of the equivalence and nonequivalence of ensembles, which is analyzed using the theory of large deviations and the theory of convex functions. In a 2000 paper appearing in the Journal of Statistical Physics, we gave necessary and sufficient conditions for ensemble equivalence and nonequivalence in terms of support and concavity properties of the microcanonical entropy. In later research we significantly extended those results by introducing a class of Gaussian ensembles, which are obtained from the canonical ensemble by adding an exponential factor involving a quadratic function of the Hamiltonian. The present paper is an overview of our work on this topic. Our most important discovery is that even when the microcanonical and canonical ensembles are not equivalent, one can often find a Gaussian ensemble that satisfies a strong form of equivalence with the microcanonical ensemble known as universal equivalence. When translated back into optimization theory, this implies that an unconstrained minimization problem involving a Lagrange multiplier and a quadratic penalty function has the same minimum points as the original constrained problem. The results on ensemble equivalence discussed in this paper are illustrated in the context of the Curie–Weiss–Potts lattice-spin model.

It is known that simulation of the mean position of a reflected random walk {Wn} exhibits non-standard behavior, even for light-tailed increment distributions with negative drift. The Large Deviation Principle (LDP) holds for deviations below the mean, but for deviations at the usual speed above the mean the rate function is null. This paper takes a deeper look at this phenomenon. Conditional on a large sample mean, a complete sample path LDP analysis is obtained. Let I denote the rate function for the one dimensional increment process. If I is coercive, then given a large simulated mean position, under general conditions our results imply that the most likely asymptotic behavior, ψ, of the paths n −1 W ⌊tn ⌋ is to be zero apart from on an interval [T0, T1] ⊂ [0, 1] and to satisfy the functional equation ∇I ( d dt ψ(t)) = λ ∗ (T1 − t) whenever ψ(t) ̸ = 0. If I is non-coercive, a similar, but slightly more involved, result holds. 1