1.2. LDP FOR NON-NEGATIVE RANDOM VARIABLES 15

We now prove “(1.2.12) =⇒ (1.2.11)”. We may assume that the right hand side

of (1.2.11) is positive. For any k ≥ 0,

θ[p−1k]+1

([p−1k] + 1)!

bn

[p−1k]+1 EYnp−1k]+1

[

1/p

≤

∞

m=0

θm

m!

bn

m

EYn

m

1/p

.

By Jensen inequality, Stirling formula and an argument similar to the one used for

(1.2.15), we can prove that for any 0 δ there is C 0 independent of n such

that

E exp

pθ

1 +

bnYn

1/p

(1.2.17)

≤ exp

pθ

1 +

+ C

1 + δ

δ

∞

m=0

θm

m!

bn

m

EYn

m

1/p

p

.

By (1.2.12) (with θ being replaced by

pθ

1 +

), we get

pΨ

θ

1 +

≤ max 0, p lim inf

n→∞

1

bn

log

∞

m=0

θm

m!

bn

m

EYn

m

1/p

.

Since the left hand side is positive, we have

lim inf

n→∞

1

bn

log

∞

m=0

θm

m!

bn

m

EYn

m

1/p

≥ Ψ

θ

1 +

.

By the lower semi-continuity of Ψ(·), letting →

0+

on the right hand side gives

(1.2.11).

Finally, “(1.2.13) =⇒ (1.2.14)” also follows from the estimate given in (1.2.17).

An immediate application of Lemma 1.2.6 is the following G¨artener-Ellis-type

theorem.

Theorem 1.2.7. Assume that for each θ ≥ 0, the limit

(1.2.18) Ψ(θ) ≡ lim

n→∞

1

bn

log

∞

m=0

θm

m!

bn

m EYnm

1/p

exists as an extended real number. Assume that the function Ψ(θ) is essentially

smooth on R+ (Definition 1.2.3). For each λ 0,

(1.2.19) lim

n→∞

1

bn

log P{Yn ≥ λ} = −IΨ(λ)

where the rate function IΨ(·) is defined by

(1.2.20) IΨ(λ) = p sup

θ0

θλ1/p

− Ψ(θ) λ 0.

Proof. By Lemma 1.2.6, the condition posed in Corollary 1.2.5 is satisfied by

Λp(θ) = pΨ(θ/p).

We now consider the case of a single random variable.