1.3. LDP BY SUB-ADDITIVITY 19
1.3. LDP by sub-additivity
A random sequence {Yn}n≥1 is said to be sub-additive, if for any n, m 1, there
is a random variable Ym such that
Yn+m Yn + Ym,
Ym
d
= Ym and Ym is independent of {Y1, · · · , Yn}.
In particular, a deterministic sub-additive sequence {a(n)} is defined by the
inequality a(n + m) a(n) + a(m).
Lemma 1.3.1. For any deterministic sub-additive sequence {a(n)}, the equality
lim
n→∞
n−1a(n)
= inf
m≥1
m−1a(m)
holds in the extended real line [−∞, ∞).
Proof. All we need is to show that
lim sup
n→∞
n−1a(n)
inf
m≥1
m−1a(m).
(1.3.1)
Let m 1 be fixed but arbitrary. For any big n, write n = km + r, where k 1
and 0 r m are integers. By sub-additivity,
a(n) ka(m) + a(r).
Consequently,
lim sup
n→∞
n−1a(n)

m−1a(m).
Taking infimum over m on the right hand side leads to (1.3.1).
Given a sub-additive random sequence {Yn}n≥1 and θ 0, we have that
E exp θYn+m E exp θYn E exp θYm .
Consequently,
a(n) log E exp θYn n = 1, 2, · · ·
is a deterministic sub-additive sequence. Consequently, we have the following corol-
lary.
Corollary 1.3.2. Let {Yn}n≥1 be sub-additive. For any θ 0, the equality
lim
n→∞
1
n
log E exp θYn = inf
m≥1
1
m
log E exp θYm
holds in the extended real line [−∞, ∞].
In the following theorem, we establish exponential integrability for Yn.
Theorem 1.3.3. Let {Yn} be a sub-additive random sequence such that Y1 C
a.s. for some constant C 0. Let cn 1 be a deterministic sequence such that the
normalized sequence
max
k≤n
Yk/cn n = 1, 2, · · ·
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