2 1. BASICS ON LARGE DEVIATIONS
exists as an extended real number. Further, the origin belongs to the interior
o
of
the domain = R; Λ(θ) ∞} of the function Λ(θ).
By older inequality, Λ(θ) is a convex function. Consequently, 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 = 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

= 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)
o
= (a, b) is non-empty.
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