6 1. BASICS ON LARGE DEVIATIONS
variables deviate away from their equilibrium state, or on the event that the random
variables take large values. In the following we consider a historically important
example. Let {Xk}k≥1 be a real independent and identically distributed (i.i.d.)
sequence such that there is a c 0 such that
(1.1.8) E exp c|X1| ∞.
If we take Yn to be the sample average
(1.1.9) Xn =
1
n
X1 + · · · + Xn n = 1, 2 · · ·
bn = n, then Assumption 1.1.1 holds with
Λ(θ) = log E exp θX1 .
Unfortunately, Λ(θ) is not essentially smooth in general. Indeed, it is straightfor-
ward to check that Λ(θ) fails to be steep as X1 is bounded. Nevertheless, the large
deviation principle known as the Cram´ er’s large deviation principle (Theorem 2.3.6,
p. 44, [47]) claims that (1.1.4) and (1.1.5) (with bn = n) hold under (1.1.8).
Further,
log E exp θX1 ≥ θEX1
using Jensen inequality. Hence,
Λ∗(EX1)
= 0. On the other hand, assume that
λ ∈ R satisfies
Λ∗(λ)
= 0. We have that
λθ ≤ log E exp θX1 θ ∈ R.
In view of the fact that
lim
θ→0
θ−1
log E exp θX1 = EX1,
letting θ →
0+
and letting θ →
0−
give, respectively, λ ≤ EX1 and λ ≥ EX1.
Summarizing our argument, Λ∗(λ) = 0 if and only if λ = EX1. By the goodness
of the rate function Λ∗(·), therefore, for given 0,
inf
|λ−EX1|≥
Λ∗(λ)
0.
This observation shows that the probability that the sample average deviates away
from the sample average EX1 has a genuine exponential decay. We summarize our
discussion in the following theorem (Cram´ er’s large deviation principle).
Theorem 1.1.5. Under the assumptions (1.1.8),
lim sup
n→∞
1
n
log P{Xn ∈ F } ≤ − inf
x∈F
Λ∗(x),
lim inf
n→∞
1
n
log P{Xn ∈ G} ≥ − inf
x∈G
Λ∗(x)
for any closed set F ⊂ R and any open set G ⊂ R.
In particular, for any 0,
lim sup
n→∞
1
n
log P |Xn − EX1| ≥ 0.
For the inverse of the G¨ artner-Ellis theorem, we state the following Varadhan’s
integral lemma.
