Università Degli Studi Di Milano-bicocca

Theory The abstract theory of large deviations was first formulated in 1966 by the Indian American mathematician Srinivasa Varadhan (see [Var]), who was awarded the prestigious Abel prize for 2007 “for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviations”. The American mathematician Monroe David Donsker also made a significant contribution, publishing several papers together with Varadhan. The abstract theory, which we present in this chapter (mainly based on [Swa, ch.1] and [Hol, ch.III]), allows to unify and extend Cramér’s theorem and other important results under the same theoretical framework, as we will see in the next chapter. 2.1 Semi-continuous functions Semi-continuous functions play a fundamental role in large deviations theory, so in this section we intend to define them and study their main properties. Proposition 2.1 (Topologies of semi-continuity). The following collections of sets are non-Hausdorff topologies on R: (i) Oup = { [−∞, a) : −∞ < a ≤ +∞ } ∪ { ∅,R } . (ii) Olow = { (a,+∞] : −∞ ≤ a < +∞ } ∪ { ∅,R } . Proof. (i) Let us prove the three axioms of topology: • ∅,R ∈ Oup by definition. • If −∞ < a1, a2 ≤ +∞: [−∞, a1)∩ [−∞, a2) = [−∞, a1 ∧ a2) ∈ Oup. 26 2.1. Semi-continuous functions • If −∞ < ai ≤ +∞ for all i in an index set S: ⋃ i∈S [−∞, ai) = [ −∞,sup i∈S ai ) ∈ Oup. The topology Oup is not Hausdorff, since open neighborhoods of two different points always intersect: if b,c ∈ R with b < c, the open neighborhoods of c are [−∞, a) with c < a ≤ +∞ and R, all of which intersect {b}. (ii) The proof is analogous. We call O the Euclidean topology on R, i.e. the topology whose base is composed of the open intervals (a,b), −∞ < a < b < +∞, and the neighborhoods of −∞ and +∞ (namely, the sets [−∞, a) and (a,+∞], a ∈ R). Recall that this topology is not only Hausdorff, but it is also metrizable: to be more precise, it is induced by a metric whose balls form the base we have just defined, and whose restriction to R generates the Euclidean topology of R (see [Soa, §3.9 and §3.10.3]). Definition 2.2. Let (X, τ) be a topological space and let f : X→ R be a function. • f is said to be upper semi-continuous if it is continuous as function (X, τ)→ (R,Oup). • f is said to be lower semi-continuous if it is continuous as function (X, τ)→ (R,Olow). • f is said to be continuous if it is continuous as function (X, τ)→ (R,O). Remark 2.3. • Assume that f −1([−∞, a)) is open for all a ∈ R; then f −1 ( [−∞,+∞) ) = f −1 ( ∞ ⋃