1.1. GARTNER-ELLIS
¨
THEOREM 3
(2) Λ(θ) is differentiable in
DΛ.o
(3) Λ(·) is steep:
lim
θ→a+
Λ (θ) = lim
θ→b−
Λ (θ) = ∞.
Theorem 1.1.4. Let Assumption 1.1.1 hold. For any closed set F ⊂ R,
(1.1.4) lim sup
n→∞
1
bn
log P{Yn ∈ F } ≤ − inf
λ∈F
Λ∗(λ).
If we further assume that the logarithmic moment function Λ(θ) is essentially
smooth, then for any open set G ⊂ R,
(1.1.5) lim inf
n→∞
1
bn
log P{Yn ∈ G} ≥ − inf
λ∈G
Λ∗(λ).
Proof. We first prove the upper bound (1.1.4). Without loss generality we may
assume that inf
λ∈F
Λ∗(λ)
0. Let 0 l inf
λ∈F
Λ∗(λ)
be fixed but arbitrary. By the
convexity, lower semi-continuity and goodness of
Λ∗(·)
there are real numbers α β
such that {λ ∈ R;
Λ∗(λ)
≤ l} = [α, β]. Clearly, F ∩ [α, β] = ∅. Therefore, there are
α α and β β such that F ∩ (α , β ) = ∅. Obviously,
Λ∗(α
),
Λ∗(β
) l. We
claim that
Λ∗(α
) = sup
θ0
θα − Λ(θ) ,
Λ∗(β
) = sup
θ0
θβ − Λ(θ) .
In fact, the relation
Λ∗(α
) = sup
θ≥0
θα − Λ(θ)
would lead to
Λ∗(α
) ≤
Λ∗(λ)
for any λ α . This is impossible when λ ∈ [α, β].
In addition,
P{Yn ∈ F } ≤ P{Yn ∈ (α , β )} = P{Yn ≤ α } + P{Yn ≥ β }.
Consequently,
lim sup
n→∞
1
bn
log P{Yn ∈ F }
≤ max lim sup
n→∞
1
bn
log P{Yn ≤ α }, lim sup
n→∞
1
bn
log P{Yn ≥ β } .
For any θ 0,
P{Yn ≥ β } ≤
e−θβ bn
E exp θbnYn .
Hence,
lim sup
n→∞
1
bn
log P{Yn ≥ β } ≤ −θβ + Λ(θ).
Taking infimum over θ 0 on the right hand side gives that
lim sup
n→∞
1
bn
log P{Yn ≥ β } ≤
−Λ∗(β
) ≤ −l.
