Theory of Statistics - George Mason University

1.3 Sequences of Events and of Random Variables 71
Theorem 1.26
Let {An} be a sequence of events in a probability space. Then
and
Proof.
Consider
and
We see
and likewise
P(lim sup
n
An) ≤ lim supP(An)
(1.146)
n
P(lim inf
n An) ≥ lim inf P(An). (1.147)
n
Bn = ∪ ∞ i=n Ai
Cn = ∩ ∞ i=nAi.
Bn ↘ lim supAn,
n
Cn ↗ lim inf
n An.
Now we use the continuity of the measure to get
and
P(An) ≤ P(Bn) → P(lim supAn)
n
P(An) ≥ P(Cn) → P(lim inf
n An).
For a sequence of sets {An}, we recall the intuitive interpretations of
lim sup n An and lim infn An:
• An element ω is in lim sup n An iff for each n, there is some i ≥ n for which
ω ∈ Ai. This means that ω must lie in infinitely many of the An.
• An element ω is in lim infn An iff there is some n such that for all i ≥ n,
ω ∈ Ai. This means that ω must lie in all but finitely many of the An.
In applications of probability theory, the sets correspond to events, and
generally we are more interested in those events that occur infinitely often;
that is, we are more interested in lim sup n An. We often denote this as “i.o.”,
and we define
Theory of Statistics c○2000–2013 James E. Gentle
{An i.o.} = lim supAn.
(1.148)
n