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.