Quote:
Originally Posted by Max Raker
Can you explain a little about the work of Srinivasa Varadhan? I think of all the Abel Prize winners, I know the least about what he achieved
Varadhan received the Abel Prize for his work in
large deviations. I have heard it said that large deviations is the next level in the hierarchy of approximation tools consisting of: law of large numbers, central limit theorem, and large deviations.
Let {X
n} be a sequence of mean zero, iid random variables, and let
Xn = (X
1 + ... + X
n)/n. Given a > 0, we wish to understand the asymptotic behavior of P(
Xn > a). A naive application of the central limit theorem would lead us to replace this with P(N > n
1/2a), where N has a normal distribution. In the case where each X
n has unit variance, so that N is a standard normal, we have
Or, in terms of a formal limit theorem,
We might speculate, then, that
But we would be wrong. As it turns out, the central limit theorem is not sharp enough to handle this situation, which involves such extremely rare events.
One of the characterizing features of the central limit theorem is that it is an
invariance principle. The limiting distribution (i.e. the normal) does not depend on the distribution of the X
n's. The same is not true for large deviations. In our setting, we have
where the function
γ depends on the distribution of the X
n's.
The basics of large deviations for iid sequences are discussed in Section 1.9 of
Probability: Theory and Examples. The theory of large deviations, however, extends well beyond iid sequences. There are applications, for example, to the modeling and study of metastable behavior, as discussed in
Random perturbations of dynamical systems and
Large deviations and metastability. Another good reference is
Large Deviations for Stochastic Processes.