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
