Varadhan's Lemma

Varadhan's Lemma is a result in probability theory that helps evaluate the asymptotic behavior of certain exponential averages of random variables. Specifically, it provides a way to approximate the growth rate of expectations of exponential functions, which is essential in large deviations theory. Essentially, it states that as the sample size grows large, these expectations can be approximated by optimizing a related function called the rate function. This lemma simplifies complex probability calculations, allowing us to understand the unlikely but significant events' probabilities in stochastic systems as they become large.