1.2. LDP FOR NON-NEGATIVE RANDOM VARIABLES 11

A basic assumption of the G¨ 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 DΛ

o

= (0,a) (0 a ≤ ∞) of the

domain DΛ = {θ ∈

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)