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 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}
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