24 1. BASICS ON LARGE DEVIATIONS
Exercise 1.4.5 (hard). Let Y ≥ 0 be a random variable such that
lim
t→∞
1
t
log E exp −
tγY
= −κ
for some γ 1 and κ 0. Prove that
lim
→0+
1
γ−1
log P{Y ≤ } = −(γ −
1)γ−
γ
γ−1
κ
γ
γ−1
. (1.4.1)
This result is known as exponential Tauberian theorem (see Theorem 3.5, [133] for
a more general form).
Hint: Use Chebyshev inequality to get the upper bound.
The lower bound can be proved in a way similar to some argument used in the
proof of Theorem 1.1.4. In particular, you may need to prove that for any interval
I ⊂
R+
and θ 0,
lim sup
→0+
1
γ−1
log E exp −
θ−
γ
γ−1
Y 1{Y
∈I}
≤ − inf
λ∈I
θλ + (γ −
1)γ−
γ
γ−1
κ
γ
γ−1
λ− 1
γ−1
.
Section 1.3.
Argument by sub-additivity has become a sophisticated tool in the general
frame work of large deviations. Very often in literature, it is the deterministic
sub-additivity formulated in Lemma 1.3.1 that is used to prove the existence of the
logarithmic moment generating function.
Exercise 1.4.6. Prove (1.3.7).
