MATH 529
Liminf and Limsup
Let (Ω, F ) be given, where F is a σ -algebra of events.
limsup of a sequence of events
∞
{ Ai }i=1
We define the liminf and
by
∞ ⎛ ∞
⎞
lim inf { Ai } = ∪ ⎜⎜ ∩ Ai ⎟⎟
n =1 ⎝ i=n ⎠
and
∞ ⎛ ∞
⎞
lim sup { Ai } = ∩ ⎜⎜ ∪ Ai ⎟⎟ .
n=1 ⎝ i=n ⎠
To see what these sets are, first consider the events created by the interior operations:
∞
B1 = ∩ Ai = A1 ∩ A2 ∩ A3 ∩ A4 ∩. . .
i=1
∞
B2 = ∩ Ai =
i=2
∞
B3 = ∩ Ai =
i=3
A2 ∩ A3 ∩ A4 ∩. . .
A3 ∩ A4 ∩. . .
etc.
∞
Bn = ∩ Ai
i=n
We see that B1 ⊆ B2 ⊆ B3 ⊆ . . . ⊆ Bn ⊆ . . .
∞ ⎛ ∞
∞
⎞
We then take the union of these events: B = ∪ Bn = ∪ ⎜⎜ ∩ Ai ⎟⎟ = lim inf {Ai}.
n =1 ⎝ i=n ⎠
n=1
For the limsup, consider first
∞
C1 = ∪ Ai = A1 ∪ A2 ∪ A3 ∪ A4 ∪. . .
i=1
∞
C2 = ∪ Ai =
i=2
∞
C3 = ∪ Ai =
i=3
A2 ∪ A3 ∪ A4 ∪. . .
A3 ∪ A4 ∪. . .
etc.
∞
Cn = ∪ Ai
i=n
We see that C1 ⊇ C2 ⊇ C3 ⊇ . . . ⊇ Cn ⊇ . . .
∞ ⎛ ∞
∞
⎞
We then take the intersection of these events: C = ∩ Ci = ∩ ⎜⎜ ∪ Ai ⎟⎟ = lim sup {Ai } .
n=1 ⎝ i=n ⎠
n=1
Because the { Bn } are nested increasing and {Cn } are nested decreasing, we have
⎛∞
⎞
P(liminf {Ai }) = lim P(Bn ) = lim P ⎜⎜ ∩ Ai ⎟⎟
n→∞
n→∞ ⎝ i=n ⎠
and
⎛∞
⎞
P(limsup {Ai }) = lim P(Cn ) = lim P ⎜⎜ ∪ Ai ⎟⎟ .
n→∞
n→∞ ⎝ i=n ⎠
The following result characterizes what is means to for elements to be in lim inf {Ai }
and lim sup {Ai } .
∞
Theorem. Let {Ai }i=1 be a sequence of events.
(a) A point ω is in lim inf {Ai} if and only if ω belongs to all but a finite number of the Ai .
(b) A point ω is in lim sup {Ai} if and only if ω belongs to an infinite number of the Ai .
∞
Proof. (a) We have ω ∈ lim inf {Ai } iff ω ∈ ∩ Ai for some n ≥ 1 iff ω is in all Ai except
i=n
for possibly some of the sets A1 , . . . , An−1 .
(b)
∞
If ω ∈ lim sup {Ai } then ω ∈ ∪ Ai for all n ≥ 1 .
i=n
For n = 1 , choose i1 ≥ 1 such that
ω ∈ Ai1 . For n = i1 +1 , choose i2 ≥ i1 +1 such that ω ∈ Ai2 , etc. We see that ω is in Ai1 ,
ω ∈ Ai2 , . . .
Conversely, if ω belongs to an infinite number of the Ai , we can choose a
subsequence {ik } such that ω ∈ Aik for all k . Then for any n ≥ 1 , we can choose an
∞ ⎛ ∞
∞
⎞
index ik ≥ n . Then ω ∈ Aik , therefore ω ∈ ∪ Ai . Hence, ω ∈ ∩ ⎜⎜ ∪ Ai ⎟⎟ .
n=1 ⎝ i=n ⎠
i=n
Note: If ω ∈ lim sup {Ai } , we also say that ω ∈ {Ai i.o.} , because ω is in a set Ai
infinitely often.
Exercises
1. Prove that lim inf {Ai } ⊆ lim sup { Ai} .
2. Prove that (lim inf {Ai})c = lim sup {Aic } and (lim sup {Ai})c = lim inf{Aic } .
∞
∞
∞
∞
3. Assume {Ai }i=1 is nested increasing. Prove that lim inf {Ai} = lim sup {Ai} = ∪ An .
n =1
4. Assume {Ai }i=1 is nested decreasing. Prove that lim inf {Ai} = lim sup {Ai} = ∩ An .
n =1