1.1. GARTNER-ELLIS

¨

THEOREM 7

Theorem 1.1.6. Let

Λ∗(·)

be a good rate function and let Q(λ) be a continuous

function on R.

(1) Assume that (1.1.5) holds for every open set G. Then

lim inf

n→∞

1

bn

log E exp bnQ(Yn) ≥ sup

λ∈R

Q(λ) −

Λ∗(λ)

.

(2) Assume that (1.1.4) holds for every closed set F and that

lim sup

n→∞

1

bn

log E exp (1 + )bnQ(Yn) ∞

for some 0. Then for every θ ∈ R,

lim sup

n→∞

1

bn

log E exp bnQ(Yn) ≤ sup

λ∈R

Q(λ) −

Λ∗(λ)

.

Proof. We omit the proof of part (2) since it is similar to the proof of (1.1.6).

To prove part (1), let λ0 ∈ R be fixed but arbitrary. By continuity, for given

0 there is a δ 0 such that |Q(λ) − Q(λ0)| if |λ − λ0| δ. Hence,

E exp bnQ(Yn) ≥ exp bn(Q(λ0) − ) P{|Yn − λ0| δ}.

Consequently,

lim inf

n→∞

1

bn

log E exp bnQ(Yn) ≥ Q(λ0) − − inf

|λ−λ0|δ

Λ∗(λ)

≥ Q(λ0) − −

Λ∗(λ0).

Letting →

0+,

and then taking supremum over λ0 ∈ R on the right hand side, we

have established the desired lower bound.

The general theory of large deviations has been extended to the random variables

taking values in abstract topological spaces. Let Yn be a sequence of random

variables taking values in a separable Banach space B. To extend the G¨artner-Ellis

theorem, we assume the existence of the limit

Λ(f) ≡ lim

n→∞

1

bn

log E exp bnf(Yn) f ∈

B∗

instead of (1.1.1), where B∗ is the topological dual of B. In addition to some

smoothness assumptions on Λ(f), it is required that {Yn} be exponentially tight .

That is, for any l 0 there is a compact set K ⊂ B such that

lim sup

n→∞

1

bn

log P{Yn ∈ K} ≤ −l.

In the finite dimensional setting, the exponential tightness automatically holds

under the existence of Λ(f).

In dealing with the large deviations in infinite dimensional spaces, the main

challenge often lies in the issue of exponential tightness. In the rest of this section

we introduce a result of de Acosta [1].