Large Deviations 3 – Gartner-Ellis Theorem: Where do the all terms come from? | Eventually Almost Everywhere

We want to drop the i.i.d. assumption from Cramer’s theorem, to get a criterion for a general LDP as defined in the previous post to hold. Preliminaries For general random variables on with laws , we will continue to have an upper bound like in Cramer’s theorem, provided the moment generating functions of converge as required. For analogy with Cramer, take . The Gartner-Ellis theorem gives conditions for the existence of a suitable lower bound and, in particular, when this is the same as the upper bound. We define the logarithmic moment generating function and assume that the limit exists for all . We also assume that , where . We also define the Fenchel-Legendre transform as before: We say is an exposed point of if for some , Such a is then called an exposing hyperplane. One way of thinking about this definition is that is convex, but is strictly convex in any direction at an exposed point. Alternatively, at an exposed point y, there is a vector such that has a global minimum

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• Mike Liang