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).