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 R;
Λ∗(λ)
l} = [α, β]. Clearly, F [α, β] = ∅. Therefore, there are
α α and β β such that F , 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 , 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.
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