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.
