1.1. GARTNER-ELLIS ¨ THEOREM 5 According to the assumption there is θ θ such that Λ(θ ) ∞. Using Chebyshev inequality and (1.1.4) one can show that lim M→∞ lim sup n→∞ 1 bn log E exp θbnYn 1{Y n ≥M} = −∞. Letting b on the right hand side of the previous display leads to (1.1.6). We now come to the proof of (1.1.5). It suffices to show that for any λ0 G, lim inf n→∞ 1 bn log P{Yn G} −Λ∗(λ0). (1.1.7) We may assume that Λ∗(λ0) ∞. By essential smoothness of Λ(·) and by the definition of Λ∗(·) there is a θ0 Do Λ such that Λ∗(θ) = λ0θ0 Λ(θ0) and λ0 = Λ (θ0). Let δ 0 be so small that (x0 δ, x0 + δ) G. P{Yn G} P{Yn (x0 δ, x0 + δ)} e−(θ0λ0+|θ0|δ)bnE exp θ0bnYn 1{Y n ∈(x0−δ, x0+δ)} . Given λ = λ0, Λ∗(λ) λθ0 Λ(θ0), for otherwise λ = Λ (θ0) = λ0. The goodness and the lower semi-continuity of Λ∗(·) imply that Λ(θ0) sup λ∈(x0−δ, x0+δ) λθ0 Λ∗(λ) . Write E exp θ0bnYn = E exp θ0bnYn 1{Y n ∈(x0−δ, x0+δ)} + E exp θ0bnYn 1{Y n ∈(x0−δ, x0+δ)} . By Assumption 1.1.1 and by (1.1.6), we get Λ(θ0) max lim inf n→∞ 1 bn log E exp θ0bnYn 1{Y n ∈(x0−δ, x0+δ)} , sup λ∈(x0−δ,x0+δ) λθ0 Λ∗(λ) . Consequently, lim inf n→∞ 1 bn log E exp θ0bnYn 1{Y n ∈(x0−δ, x0+δ)} Λ(θ0). Thus, lim inf n→∞ 1 bn log P{Yn G} −θ0λ0 |θ0|δ + Λ(θ0) = −Λ∗(θ0) |θ0|δ. Finally, letting δ 0+ on the right hand side proves (1.1.7). The limit form described in (1.1.4) and (1.1.5) is called large deviation principle (LDP) in the literature. Theorem 1.1.4 is known as the artner-Ellis theorem on large deviations. In application, the main focus is on the event that the random
Previous Page Next Page