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

2.2. Formal large deviations 13
We are interested in the weight µn assigns to an outcome x ∈ X . In
Example 1.1 these weights decayed like e −cn for atypical points. This is the
kind of situation we want to study, and in particular we wish to compute the
constant c exactly. It could happen that c is infinite. This either happens
because x can never take place or because the probability actually decays
faster than exponentially. On the other hand, c could also be 0 which means
that x turns out to be “less rare” than we thought and the probability decays
slower than exponentially. Looking for the correct rate of decay is thus part
of the problem. Let us say then that, at scale rn ↗ ∞, the weight µn assigns
to x decays like e −crn . We still would like to compute the constant c. In
fact, this is a function of x that we will call the rate function and denote by
I(x).
This is of course not precise. Often we cannot talk about the weights
assigned to individual elements x of the space. Instead we must talk about
weights assigned to events, or subsets, of the space. But, thinking informally
for just a bit longer, on an exponential scale it makes sense to regard an event
A as rare as its least rare outcome. That is, the weight assigned to a set A
should be the maximal weight µn assigns to an element of A, i.e. e −rn infA I .
On a technical level, it is too much to expect actual convergence for all sets
A on account of boundary effects. A more reasonable formulation would be
to say that for all measurable sets A:
(2.1)
− inf I(x) ≤ lim
x∈A◦ n→∞
1
rn
log µn(A) ≤ lim
n→∞
1
rn
log µn(A) ≤ − inf I(x),
x∈A
where A ◦ and A are, respectively, the topological interior and closure of A.
Remark 2.2. The limsup for closed sets and liminf for open sets remind
us of weak convergence of probability measures where the same boundary
issue arises; see Appendix A.1 for the definition of weak convergence.
Example 2.3. Let us revisit the Bernoulli i.i.d. sequence {Xn} that we
considered in Example 1.1. If we let µn(A) = P {Sn/n ∈ A} and recall the
function Ip in (1.1), then {µn} satisfy (2.1) with normalization n and rate Ip.
Indeed, take first an open set G. For any point s ∈ G ∩ [0, 1] taking n large
enough implies that [ns]/n ∈ G and thus P {Sn/n ∈ G} ≥ P {Sn = [ns]}.
This implies that
1
lim
n→∞ n log P {Sn/n ∈ G} ≥ lim
n→∞ P {Sn = [ns]} = −Ip(s).
This inequality also holds for s ∈ G [0, 1], since Ip(s) is infinite then. Now
we can take sup over points s ∈ G and the lower bound in (2.1) follows by
taking G = A ◦ .