1.1. GARTNER-ELLIS

¨

THEOREM 5

According to the assumption there is θ θ such that Λ(θ ) ∞. Using Chebyshev

inequality and (1.1.4) one can show that

lim

M→∞

lim sup

n→∞

1

bn

log E exp θbnYn 1{Yn≥M} = −∞.

Letting b → ∞ on the right hand side of the previous display leads to (1.1.6).

We now come to the proof of (1.1.5). It suﬃces to show that for any λ0 ∈ G,

lim inf

n→∞

1

bn

log P{Yn ∈ G} ≥

−Λ∗(λ0).

(1.1.7)

We may assume that Λ∗(λ0) ∞. By essential smoothness of Λ(·) and by

the definition of Λ∗(·) there is a θ0 ∈ DΛ o such that Λ∗(θ) = λ0θ0 − Λ(θ0) and

λ0 = Λ (θ0).

Let δ 0 be so small that (x0 − δ, x0 + δ) ⊂ G.

P{Yn ∈ G} ≥ P{Yn ∈ (x0 − δ, x0 + δ)}

≥

e−(θ0λ0+|θ0|δ)bn

E exp θ0bnYn 1{Yn∈(x0−δ,

x0+δ)}

.

Given λ = λ0,

Λ∗(λ)

λθ0 − Λ(θ0), for otherwise λ = Λ (θ0) = λ0. The

goodness and the lower semi-continuity of

Λ∗(·)

imply that

Λ(θ0) sup

λ∈(x0−δ, x0+δ)

λθ0 −

Λ∗(λ)

.

Write

E exp θ0bnYn = E exp θ0bnYn 1{Yn∈(x0−δ,

x0+δ)}

+ E exp θ0bnYn 1{Yn∈(x0−δ,

x0+δ)}

.

By Assumption 1.1.1 and by (1.1.6), we get

Λ(θ0) ≤ max lim inf

n→∞

1

bn

log E exp θ0bnYn 1{Yn∈(x0−δ,

x0+δ)}

,

sup

λ∈(x0−δ,x0+δ)

λθ0 −

Λ∗(λ)

.

Consequently,

lim inf

n→∞

1

bn

log E exp θ0bnYn 1{Yn∈(x0−δ,

x0+δ)}

≥ Λ(θ0).

Thus,

lim inf

n→∞

1

bn

log P{Yn ∈ G} ≥ −θ0λ0 − |θ0|δ + Λ(θ0) =

−Λ∗(θ0)

− |θ0|δ.

Finally, letting δ →

0+

on the right hand side proves (1.1.7).

The limit form described in (1.1.4) and (1.1.5) is called large deviation principle

(LDP) in the literature. Theorem 1.1.4 is known as the G¨ artner-Ellis theorem on

large deviations. In application, the main focus is on the event that the random