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

3.2. Cramér’s theorem 23
though we will not prove it. A proof can be found in Dembo and Zeitouni
[8].
Cramér’s theorem on R. Let {Xn} be a sequence of i.i.d. real-valued
random variables. Let µn be the law of the sample mean Sn/n. Then, the
large deviation principle LDP (µn, n, I) is satisfied with I defined in (3.5).
While Cramér’s theorem is valid in general, it does not give much information
unless the variables have exponentially decaying tails. This point is
explored in the next exercise.
Exercise 3.11. Let {Xi} be an i.i.d. real-valued sequence. Assume E[X 2 1 ] <
∞ but, for all ε > 0, P {X1 > b} > e−εb for all large enough b. Show that
(a) limn→∞ 1
n log P {Sn/n > E[X1] + δ} = 0 for any δ > 0.
(b) The rate function is identically 0 on [E(X1), ∞).
Hint: For (a), deduce
P {Sn/n ≥ E[X1] + δ} ≥ P {Sn−1 ≥ (n − 1)E[X1]}P {X1 ≥ nδ + E[X1]}
and apply the central limit theorem (see page 112 of [15] or page 100 of
[26]). For (b), first find M(θ) for θ > 0. Then observe that for θ ≤ 0 and
x ≥ E[X1],
θx − log M(θ) ≤ θ(x − E[X1]) ≤ 0.
Exercise 3.12. Let {Xi} be an i.i.d. real-valued sequence. Prove that the
closure of the set {I < ∞} is the same as the closure of the convex hull of
the support of the law of X.
Hint: Let K be the latter set and y /∈ K. To show that I(y) = ∞, find
θ ∈ R such that θy − ε > sup x∈K xθ. For the other direction, take y in the
interior of {I = ∞}. To get y ∈ K, show first that φy(θ) = θy − log M(θ)
converges to infinity as θ → ∞ or −∞. Assume the former. Show that
for some ε, |x − y| ≤ ε implies φx(θ) → ∞ as θ → ∞. Then, for θ > 0,
θ(y − ε) − log M(θ) ≤ − log µ{x : |x − y| ≤ ε}. Let θ → ∞.
The information contained in Cramér’s theorem is quite crude because
only the exponentially decaying terms of a full expansion affect the result.
In some cases one can easily derive much more precise asymptotics.
Exercise 3.13. Prove that if {Xk} are i.i.d. standard normal, then for any
k ∈ N and a > 0
log P {Sn ≥ an} ∼ − a2n 1
−
2 2 log(2πna2 )
+ log 1 − 1
a2 1 × 3
+
n a4 1 × 3 × · · · × (2k − 1)
− · · · + (−1)k
n2 a2knk
.