8 1. BASICS ON LARGE DEVIATIONS
Recall that a set K B is said to be positively balanced, if λx K whenever
x K and 0 λ 1. The Minkowski functional of a convex and positively
balanced set K is defined by
qK (x) = inf{λ 0; x λK}
with the customary convention that inf ϕ = ∞. The Minkowski functional qK (·) is
sub-additive and positively homogeneous:
(1.1.10) qK (x + y) qK (x) + qK (y) and qK (λx) = λqK (x)
where x, y B and λ 0.
A family {µα; α Θ} of probability measures on B is said to be uniformly
tight, if for any 0, there is a compact set K B such that
µα(K) 1 α Θ.
The following result is given in Theorem 3.1, [1]. We state it without proof.
Theorem 1.1.7. Let {µα, α Θ} be a family of probability measures on the
separable Banach space B and assume that {µα; α Θ} is uniformly tight and
that
sup
α∈Θ
B
exp λ||x|| µα(dx) ∀λ 0.
There is a convex, positively balanced and compact set K B such that
sup
α∈Θ
B
exp qK (x) µα(dx) ∞.
1.2. LDP for non-negative random variables
In this section we assume that {Yn} take non-negative values. Recall that the
full large deviation principle is stated as: For every closed set F R+,
(1.2.1) lim sup
n→∞
1
bn
log P Yn F inf
λ∈F
I(λ)
and, for every open set G
R+,
(1.2.2) lim inf
n→∞
1
bn
log P Yn G inf
λ∈G
I(λ).
In application, (1.2.1) and (1.2.2) are often replaced by our concern of the tail
probability of the form
P{Yn λ} λ 0.
Under some mild conditions on the rate function I(λ), the following theorem
shows that large deviation principle is determined by the asymptotic behavior of
tail probabilities.
Theorem 1.2.1. Assume that the rate function I(λ) is strictly increasing and
continuous on
R+.
The following two statements are equivalent:
(1) The large deviation principle stated by (1.2.1) and (1.2.2) holds, respec-
tively, for every closed set F
R+
and for every open set G
R+.
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