1.2. LDP FOR NON-NEGATIVE RANDOM VARIABLES 13
where
(1.2.9) Ip(λ) = sup
θ0
θλ1/p
− Λp(θ) λ 0.
Proof. Replacing Yn by Yn
1/p
in Theorem 1.2.4 completes the proof.
In view of the Taylor expansion,
(1.2.10) E exp θbnYn
1/p
=
∞
m=0
θm
m!
bn
mEYn m/p,
one may attempt to estimate EYn
m/p
when establishing (1.2.7) by “standard” ap-
proach becomes technically difficult. When 1/p is not integer, however, it is not
very pleasant to deal with the (possibly) fractional power m/p. To resolve this
problem, we introduce the following lemma.
Lemma 1.2.6. Let p 0 be fixed and let Ψ: [0, ∞) −→ [0, ∞] be a non-decreasing
lower semi-continuous function. Assume that the domain of Ψ has the form
DΨ ≡ {θ; Ψ(θ) ∞} = [0,a)
where 0 a ≤ ∞, and that Ψ(θ) is continuous on DΨ.
(1) The lower bound
(1.2.11) lim inf
n→∞
1
bn
log
∞
m=0
θm
m!
bn
m
EYn
m
1/p
≥ Ψ(θ) θ 0
holds if and only if
(1.2.12) lim inf
n→∞
1
bn
log E exp θbnYn
1/p
≥ pΨ
θ
p
θ 0.
(2) The upper bound
(1.2.13) lim sup
n→∞
1
bn
log
∞
m=0
θm
m!
bn
m
EYn
m
1/p
≤ Ψ(θ) θ 0
holds if and only if
(1.2.14) lim sup
n→∞
1
bn
log E exp θbnYn
1/p
≤ pΨ
θ
p
θ 0.
Proof. We first prove “(1.2.11) =⇒ (1.2.12)”. We may assume that in (1.2.12),
the right hand side is positive. By the expansion (1.2.10) we have
E exp θbnYn
1/p
≥
θ[pm]+1
([pm] + 1)!
bn
[pm]+1EYn
[pm]+1
p
m = 0, 1, · · · .
By Jensen inequality
bn
[pm]+1
EYn
[pm]+1
p ≥ bn
pm
EYn
m
[pm]+1
pm
.
Therefore, as bn
pmEYn m
≥ 1 we have
bn
[pm]+1
EYn
[pm]+1
p ≥ bn
pm
EYn
m
.
