CHAPTER 1

Basics on large deviations

In this chapter we introduce some general theorems on large deviations which

will be frequently used in this book. In most of the cases, the state space we deal

with is the real line. Indeed, a substantial portion of the discussion is limited to the

random variables taking non-negative values. Sometimes, the underlying stochastic

processes are sub-additive (see Section 1.3). Unlike most textbooks on this subject,

we put more attention on the tail probability

P{Yn ≥ λ}

than the probability of the form

P{Yn ∈ A}.

The unique structure of the models we deal with in this book requires some non-

conventional treatments. The topics we chose in this chapter reflect this demand.

As a consequence, most theorems introduced in Section 1.2 and in Section 1.3 are

non-standard and are not usually seen in the textbooks on large deviations.

1.1. G¨ artner-Ellis theorem

In the area of large deviations, we are concerned about asymptotic computation

of small probabilities on an exponential scale. The general form of large deviation

can be roughly described as

P{Yn ∈ A} ≈ exp{−bnI(A)} (n → ∞)

for a random sequence {Yn}, a positive sequence {bn} with bn → ∞, and a coeﬃ-

cient I(A) ≥ 0. In the application, we are often concerned with the probability that

the random variable(s) takes large values. Since the remarkable works by Donsker

and Varadhan (and others) in the 1970s and 1980s, this area has developed into a

relatively complete system. There have been several standard approaches in dealing

with large deviation problems. Perhaps the most useful tool is the G¨artner-Ellis

theorem.

We have no intention to state the large deviation theory in its full generality.

Let {Yn} be a sequence of real random variables and let {bn} be a positive sequence

such that bn −→ ∞.

Assumption 1.1.1. For each θ ∈ R, the logarithmic moment generating function

Λ(θ), defined as the limit

(1.1.1) Λ(θ) = lim

n→∞

1

bn

log E exp θbnYn θ ∈ R

1

http://dx.doi.org/10.1090/surv/157/01