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Large Deviations 2 – LDPs, Rate Functions and Lower Semi-Continuity | Eventually Almost Everywhere

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So in the previous post we discussed Cramer’s theorem on large deviations for means of i.i.d. random variables. It’s worth stepping back and thinking more abstractly about what we showed. Each has some law, which we think of as a measure on , though this could equally well be some other space, depending on where the random variables are supported. The law of large numbers asserts that as , these measures are increasingly concentrated at a single point in , which in this case is . Cramer’s theorem then asserts that the measure of certain sets not containing this point of concentration decays exponentially in n, and quantifies the exponent, a so-called rate function, via a Legendre transform of the log moment generating function of the underlying distribution. One key point is that we considered only certain sets , though we could equally well have considered . What would happen if we wanted to consider an interval, say ? Well, , and we might as well assume that is sufficiently continu

Remarks from Cramer’s Theorem So in the previous post we discussed Cramer’s theorem on large deviations for means of i.i.d. random variables. It’s worth stepping back and thinking more abstractly about what we showed. Each has some law, which we think of as a measure on , though this could equally well be some other space, depending on where the random variables are supported. The law of large numbers asserts that as , these measures are increasingly concentrated at a single point in , which in this case is . Cramer’s theorem then asserts that the measure of certain set

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