2 1. BASICS ON LARGE DEVIATIONS

exists as an extended real number. Further, the origin belongs to the interior DΛ

o

of

the domain DΛ = {θ ∈ R; Λ(θ) ∞} of the function Λ(θ).

By H¨ older inequality, Λ(θ) is a convex function. Consequently, DΛ o = (a, b) for

some −∞ ≤ a 0 b ≤ ∞ and Λ(θ) is continuous in its domain DΛ. Define the

Fenchel-Legendre transform Λ∗(λ) of Λ(θ) as

(1.1.2)

Λ∗(λ)

= sup

θ∈R

θλ − Λ(θ) λ ∈ R.

To discuss the role played by the function

Λ∗(·),

we introduce the following

definition.

Definition 1.1.2. A function I: R −→ [0, ∞] is called a rate function if it is

lower semi-continuous: For any l 0, the level set

Il = {λ ∈ R; I(λ) ≤ l}

is a closed set. Further, a rate function is said to be good if every level set is compact

in R.

We point out an equivalent statement of lower semi-continuity: For any λn,λ ∈

R with λn → λ,

(1.1.3) lim inf

n→∞

I(λn) ≥ I(λ).

We now claim that under Assumption 1.1.1, the function

Λ∗(·)

is a good rate

function. Indeed, by definition

Λ∗(λn)

≥ θλn − Λ(θ) θ ∈ R n = 1, 2, · · · ,

which leads to

lim inf

n→∞

Λ∗(λn)

≥ θλ − Λ(θ) θ ∈ R.

Taking the supremum over θ ∈ R on the right-hand side proves (1.1.3).

Let l 0 be fixed. By Assumption 1.1.1 and by the continuity of Λ(θ) in its

domain there is δ 0 such that

c ≡ sup

|θ|≤δ

Λ(θ) ∞.

Consequently, for any λ with

Λ∗(λ)

≤ l,

l ≥ sup

|θ|≤δ

θλ − Λ(θ) ≥ δ|λ| − c.

Therefore, the level set

Λa

∗

= {λ ∈ R,

Λ∗(λ)

≤ l}

is compact.

In addition, the fact that

Λ∗(λ)

is the conjugate of the convex function Λ(θ)

makes

Λ∗(λ)

a convex function.

Definition 1.1.3. A convex function Λ: R −→ (−∞.∞] is essentially smooth

if:

(1) DΛ

o

= (a, b) is non-empty.