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.