1.3. LDP BY SUB-ADDITIVITY 21

Consequently, (1.3.2) holds for some θ 0.

Assuming (1.3.3), we now show that (1.3.2) holds for every θ 0. Let θ0 0

satisfy (1.3.2). By (1.3.3) there is m 1 such that

c[n/m]/cn ≤ θ0/θ

for all n ≥ 1. By sub-additivity we have that

E exp θcn

−1Yn

≤ E exp θcn

−1Y[n/m]

m

.

By the choice of m and by H¨ older inequality, the right hand side is bounded uni-

formly over n ≥ 1.

The notion of sub-additivity can be extended to the stochastic processes with

continuous time. A stochastic process Zt (t ≥ 0) is said to be sub-additive, if for any

s, t ≥ 0, Zs+t ≤ Zs + Zt for a random variable Zt independent of {Zu; 0 ≤ u ≤ s}

with Zt

d

= Zt. With a completely parallel argument we have

Lemma 1.3.4. For any deterministic sub-additive function {a(t)} (t ∈ R+), the

equality

lim

t→∞

t−1a(t)

= inf

s0

s−1a(s)

holds in the extended real line [−∞, ∞).

We may restate Theorem 1.3.3 in the setting of continuous time. Instead, we

give the following slightly different version.

Theorem 1.3.5. For any non-decreasing sub-additive process Zt with continuous

path and with Z0 = 0,

E exp{θZt} ∞ ∀θ, t 0. (1.3.5)

Consequently,

lim

t→∞

1

t

log E exp θZt = Ψ(θ) (1.3.6)

exists with 0 ≤ Ψ(θ) ∞ for every θ 0.

Proof. Clearly, we need only to establish (1.3.5). By sample path continuity and

by monotonicity, an argument used in achieving (1.3.4) leads to

P Zt ≥ a + b ≤ P Zt ≥ a P Zt ≥ b (1.3.7)

holds for any a, b 0.

Consequently,

P Zt ≥ ma ≤ P Zt ≥ a

m

m = 1, 2, · · · .

For any a 0, by the fact that Z0 = 0 and by sample path continuity, one can

have P Zt ≥ a ≤ e−2 by making t 0 suﬃciently small. Consequently, for any

θ 0,

E exp{θZt} ∞