1.2. LDP FOR NON-NEGATIVE RANDOM VARIABLES 17
(1) Assume that
(1.2.23) lim inf
n→∞
1
bn
log P{Yn ≥ λ} ≥ −I(λ) (λ 0).
Then for every θ 0
(1.2.24) lim inf
n→∞
1
bn
log
∞
m=0
θm
m!
bn
m
EYn
m
1/p
≥ sup
λ0
θλ1/p
−
p−1I(λ)
.
(2) Assume that
(1.2.25) lim sup
n→∞
1
bn
log P{Yn ≥ λ} ≤ −I(λ) (λ 0).
Then for every θ 0 satisfying
(1.2.26) lim
l→∞
lim sup
n→∞
1
bn
log
∞
m=0
θm
m!
bn
m
E
(
Yn
m1{Yn≥l}
)
1/p
= 0
we have
(1.2.27) lim sup
n→∞
1
bn
log
∞
m=0
θm
m!
bn
m
EYn
m
1/p
≤ sup
λ0
θλ1/p
−
p−1I(λ)
.
Proof. For any λ 0,
EYn
m
≥
λmP{Yn
≥ λ}.
Consequently,
∞
m=0
θm
m!
bn
m
EYn
m
1/p
≥ exp
bnθλ1/p
P{Yn ≥ λ}
1/p
.
By (1.2.23)
lim inf
n→∞
1
bn
log
∞
m=0
θm
m!
bn
m
EYn
m
1/p
≥
θλ1/p
−
p−1I(λ).
Taking supremum over λ 0 on the right hand side gives (1.2.24).
To prove (1.2.27), write
∞
m=0
θm
m!
bn
m EYnm
1/p
(1.2.28)
=
∞
m=0
θm
m!
bn
m
E
(
Yn
m1{Ynl}
)
1/p
+
∞
m=0
θm
m!
bn
m
E
(
Yn
m1{Yn≥l}
)
1/p
.
Given 0, partition the interval [0,l] into 0 = λ0 · · · λN = l such that
for each 1 ≤ i ≤ l the length of the sub-interval Ai = [λi−1,λi] is less than . So
