1.2. LDP FOR NON-NEGATIVE RANDOM VARIABLES 9

(2) For every λ 0,

(1.2.3) lim

n→∞

1

bn

log P Yn ≥ λ = −I(λ).

Proof. Assume that (1) holds. Taking F = [λ, ∞) in (1.2.1) gives that

lim sup

n→∞

1

bn

log P Yn ≥ λ ≤ − inf

x≥λ

I(x) = −I(λ).

Similarly, by (1.2.2) with G = (λ, ∞) we get

lim inf

n→∞

1

bn

log P Yn ≥ λ ≥ − inf

xλ

I(x) = −I(λ)

where the last step follows from the continuity.

Assume that (2) holds. For a closed set F ⊂ R+, let λ0 = inf F . We have that

P{Yn ∈ F } ≤ P{Yn ≥ λ0}.

By (1.2.3) we have

lim sup

n→∞

1

bn

log P Yn ∈ F ≤ −I(λ0) = − inf

λ∈F

I(λ)

where the last step follows from the monotonicity of I(·).

To establish (1.2.2), let λ0 ∈ G. There is a δ 0 such that [λ0,λ0 + δ) ⊂ G. By

the fact that

P{Yn ≥ λ0} ≤ P{Yn ≥ λ0 + δ} + P{Yn ∈ [λ0,λ0 + δ)}

and by (1.2.3) we have

−I(λ0) ≤ max − I(λ0 + δ), lim inf

n→∞

1

bn

log P{Yn ∈ [λ0,λ0 + δ)} .

By assumption we have that I(λ0 + δ) I(λ0). Consequently,

lim inf

n→∞

1

bn

log P Yn ∈ G ≥ −I(λ0).

Taking supremum over λ0 ∈ G on the right hand side gives (1.2.2).

The following theorem shows that under certain conditions, the tail probability

of the fixed sum of independent non-negative random variables is dominated by the

tail probabilities of individual terms.

Theorem 1.2.2. Let Z1(n), · · · , Zl(n) be independent non-negative random vari-

ables with l ≥ 2 being fixed.

(a) If there are constant C1 0 and 0 a ≤ 1 such that

lim sup

n→∞

1

bn

log P Zj (n) ≥ λ ≤

−C1λa

∀λ 0

for j = 1, · · · , l, then

lim sup

n→∞

1

bn

log P Z1(n) + · · · + Zl(n) ≥ λ ≤

−C1λa

∀λ 0.