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 θc−1Yn n ≤ E exp θc−1Y[n/m] n 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} ∞

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