22 1. BASICS ON LARGE DEVIATIONS

for small t. This can be extended to arbitary t 0 by sub-additivity.

1.4. Notes and comments

Section 1.1.

The earliest recorded work in large deviation theory is due to Cram´ er ([41])

and was published in 1938. The literature on large deviations is massive and it is

impossible to list even a small portion of it. We point to the fundamental roles

played by Donsker and Varadhan, and Freidlin and Wentzell in the birth of the

modern theory of large deviations. The idea that the limit of the logarithmic

moment generating function decides the large deviation goes back to Cram´ er ([41]).

It has been formulated by G¨ artner ([84]) and Ellis ([73]) into a general theorem later

known as the G¨ artner-Ellis theorem (Theorem 1.1.4). There are many excellent

book accounts available in the theory of large deviations. We mention here the

books by Varadhan [159], Freidlin and Wentzell [80], Ellis [74], Stroock [156],

Deuschel and Stroock [53], Bucklew [20], Dembo and Zeitouni [47], den Hollander

[97], Feng and Kurtz [77]. Finally, we refer an interested reader to the recent

survey by Varadhan [161] for an overview on the latest development in the area of

large deviations.

Most of the material in this section comes from the book by Dembo and Zeitouni

([47]).

For the large deviations in infinite dimensional space, a challenging part is to

establish the exponential tightness. Theorem 1.1.7 provides a practical way of

examining the exponential tightness. This useful result is due to de Acosta ([1]).

Exercise 1.4.1. Let {Yn} and {Zn} are two sequences of real random variables

such that

lim

n→∞

1

bn

log P |Yn − Zn| ≥ = −∞.

Show that if Yn obeys the large deviation principle with the scale bn and the good

rate function I(·), then the same large deviation principle holds for Zn.

Exercise 1.4.2. Recall that a Poisson process Nt is a stochastic process taking

non-integer values such that

(1) N0 = 0,

(2) For any t 0,

P Nt = m =

e−t

tm

m!

m = 0, 1, · · · ,

(3) For any s, t ∈

R+

with s t, Nt −Ns is independent of {Nu; 0 ≤ u ≤ s}

and has same distribution as Nt−s.

Prove the following LDP: For any closed set F ⊂ R and open set G ∈ R,

lim sup

t→∞

1

t

log P Nt/t ∈ F ≤ − inf

λ∈F

I(λ),

lim inf

t→∞

1

t

log P Nt/t ∈ G ≥ − inf

λ∈G

I(λ)