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

.