A Course on Large Deviations with an Introduction to Gibbs Measures.

More generalities and
Cramér’s theorem
3.1. Weak large deviation principles
Chapter 3
As we have seen in Example 1.1, when proving the lower large deviation
bound (2.4) it is enough to consider G that are local neighborhoods. It is
also enough to focus on local neighborhoods if one only needs to prove the
upper large deviation bound (2.3) for compact sets F . This often simplifies
the analysis considerably.
For the discussion in this section let X be a Hausdorff space.
Definition 3.1. A sequence of probability measures {µn} ⊂ M1(X ) is said
to satisfy a weak large deviation principle with lower semicontinuous rate
function I : X → [0, ∞] and normalization {rn} if the lower large deviation
bound (2.4) holds for all open sets G ⊂ X and the upper large deviation
bound (2.3) holds for all compact sets F ⊂ X .
With enough control on the tails of the measures µn this is in fact sufficient
for the full LDP to hold.
Definition 3.2. We say {µn} ⊂ M1(X ) is exponentially tight with normalization
rn if for each b > 0 there exists a compact set Kb such that
µn(K c b ) ≤ e−rnb for all n ∈ N.
Note that if {µn} is exponentially tight with normalization rn ↗ ∞,
then {µn} is tight; see Appendix A.1 for the definition of tightness of a
family of measures.
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