A Course on Large Deviations with an Introduction to Gibbs Measures.

4.2. Varadhan’s theorem and Bryc’s theorem 35
is the Cramér rate for the law of Sn/n; see (1.1) and use the contraction
principle. The above supremum is attained at x = 2
3 log 2 instead of E[X1] =
log √ 2. Thus, the paths that contribute the most are the ones for which
Sn/n ∼ 2
3 log 2, i.e. Wn ∼ (22/3 ) n . They do occur with an exponentially
small probability, but when they do occur, they have an exponentially large
contribution. The two balance out to produce the mean value (3/2) n . LDP
and the rate function has a unique zero at x = 2
3 log 2.
The above exercises and example point to a relationship between large
deviations and statistical mechanics where one encounters integrals of the
form e nf dµn. The distributions νn in Exercise 4.9 are examples of Gibbs
measures. We will see more of this in the second part of the course. The
next Section 4.3 provides an appetizer in the context of a mean-field model.
Varadhan’s theorem gives asymptotics of integrals as a consequence of
an LDP. Perhaps not surprisingly, knowing the asymptotics of sufficiently
many integrals is equivalent to an LDP. Let Cb(X ) denote the set of bounded
and continuous functions on X .
Bryc’s theorem. Let {µn} be a sequence of probability measures on a metric
space X . Assume {µn} is exponentially tight with normalization rn.
Suppose the limit
1
Γ(f) = lim
n→∞ rn log
e rnf dµn
exists for all bounded continuous functions f. Then, LDP(µn, rn, I) holds
with the tight rate function
(4.2)
I(x) = sup
f∈Cb(X )
{f(x) − Γ(f)}.
The above theorem reminds us again of weak convergence of probability
measures.
Of course, given the LDP, Varadhan’s theorem implies that Γ(f) =
sup(f − I). Note, however, that the relation between Γ and I is not the
convex duality we will see in Chapter 5, where the functions f are continuous
linear functions. Even though until this point we have only seen convex rate
functions, the rate function in (4.2) need not be convex; recall Definition
2.14. In fact, the next section has an LDP with a nonconvex rate function.
Also, Exercise 5.27 shows how simple it is to come up with such a situation.
Proving that the limit Γ(f) exists for all bounded continuous functions
may be too hard to achieve. However, for the LDP to hold one really
needs the limit to exist for a rich enough class of functions; see for example
Theorem 4.4.10 of [8].
If X a metric vector space, then a rich enough class of functions that
would ensure the LDP via Bryc’s theorem is the class of concave Lipschitz