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 Y m such that Yn+m Yn + Ym, Y m d = Ym and Y m 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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