DonskerVaradhan

DonskerVaradhan refers to a fundamental result in probability theory concerning the large deviations of empirical measures. Specifically, the Donsker-Varadhan theorem provides a precise asymptotic formula for the probability that a sequence of empirical measures deviates significantly from the true underlying distribution. It is named after Monroe D. Donsker and G. R. Varadhan, who developed this theory in the 1970s.

The theorem establishes that for a sequence of independent and identically distributed random variables, the logarithm of the probability of observing an empirical measure far from the true distribution is proportional to the sample size, and the proportionality constant is given by the rate function (or entropy function) evaluated at the deviating measure. This rate function quantifies the "cost" of observing such a deviation.

The Donsker-Varadhan theorem is a powerful tool in large deviation theory and has found applications in various fields, including statistical mechanics, information theory, and machine learning. It allows for the analysis of rare events and provides insights into the asymptotic behavior of complex probabilistic systems. The theorem has been extended and generalized in many ways, becoming a cornerstone of modern probability and statistical physics. Its impact lies in its ability to provide quantitative estimates for the likelihood of extreme outcomes.