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

1 +
bnYn
1/p
(1.2.17)
exp

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

1 +
), we get

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.
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