1.2. LDP FOR NON-NEGATIVE RANDOM VARIABLES 11
A basic assumption of the artner-Ellis theorem (Theorem 1.2.4 below) for non-
negative random variables is the existence of the logarithmic moment generating
function on
R+.
Correspondent to Definition 1.1.3, we adopt the following notion
of essential smoothness for the functions defined on
R+.
Definition 1.2.3. A convex function Λ(θ):
R+
−→ [0, ∞] is said to be essen-
tially smooth on
R+,
if
(1) There is a θ0 0 such that Λ(θ) for every 0 θ θ0.
(2) Λ(θ) is differentiable in the interior
o
= (0,a) (0 a ∞) of the
domain =
R+;
Λ(θ) ∞}.
(3) The function Λ(θ) is steep at the right end (of the domain) and is flat at
the left end, i.e.,
lim
θ→a−
Λ (θ) = and Λ
(0+)
lim
θ→0+
Λ(θ) Λ(0)
θ
= 0.
The following theorem appears as a version of Ga¨ rtner-Ellis large deviation.
Theorem 1.2.4. Assume that for all θ 0, the limit
(1.2.4) Λ(θ) = lim
n→∞
1
bn
log E exp θbnYn
exists as an extended real number, and that the function Λ(θ) is essentially smooth
on R+.
(1) The function
(1.2.5) I(λ) = sup
θ0
θλ Λ(θ) λ 0
is strictly increasing and continuous on R+. Consequently, the LDP given
defined by (1.2.1) and (1.2.2) and the LDP defined by (1.2.3) are equiva-
lent.
(2) The equivalent forms (1.2.1), (1.2.2) and (1.2.3) hold.
Proof. Let ξ be an independent random variable with distribution
P{ξ = −1} = P{ξ = 1} =
1
2
.
We have
E exp θbnξYn =
1
2
E exp θbnYn + E exp θbnYn θ R.
Consequently,
lim
n→∞
1
bn
log E exp θbnξYn = Λ(|θ|) θ R.
Notice that Λ(|θ|) is essentially smooth on R according to Definition 1.1.3. Applying
Theorem 1.1.4 to the sequence {ξYn} we have the large deviation principle given
in (1.2.1) and (1.2.2) with the rate function given by
I(λ) = sup
θ∈R
θλ Λ(|θ|) = sup
θ0
θλ Λ(θ) ∀λ 0. (1.2.6)
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