Download Limit Supremum, Limit Infimum and Limit of a Sequence of Sets

EE603 Class Notes
03/03/09
John Stensby
Appendix 11B: Limit Superior, Limit Inferior and Limit of a Sequence of Sets (Events)
If A1 ⊂ A2 ⊂ A3 ⊂ ... is a nested increasing sequence of events with
A≡
∞
∪An ,
(11B1)
n =1
Theorem 11-1 states and proves the result
N
⎡N
⎤
⎡
⎤
limit P ⎢ ∪ A n ⎥ = P ⎢ limit ∪ A n ⎥ = P [ A ] .
N →∞ ⎢⎣ n =1
⎥⎦
⎢⎣ N →∞ n =1 ⎥⎦
(11B2)
That is, it is permissible to move the limit operation from “outside” to “inside” of the probability
measure P. A similar result was shown for a nested decreasing sequence of events. In this
appendix, these simple continuity results are extended to sequences of arbitrary events.
Let A1, A2, … be a sequence of arbitrary events (not necessarily nested in any manner).
Define the events
Bn ≡
Cn ≡
∞
∪
m=n
∞
∩
m=n
Am ,
Bn is a nested decreasing sequence of events, and
(11B3)
Am ,
Cn is a nested increasing sequence of events.
Note that
Cn ≡
∞
∩
∞
∪
A m ≡ Bn
(11B4)
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11b-1
m=n
Am ⊆ An ⊆
m=n
for all n.
EE603 Class Notes
03/03/09
John Stensby
As defined, Bn and Cn are legitimate events since countable intersections and unions of
events are always events (recall the definition of a σ-algebra of events). Note that Bn is a nested
decreasing sequence of events, and Cn is a nested increasing sequence of events. Because of
this monotone nature, we can write
Bn =
Cn =
n
∩
m =1
Bm =
n
n
∩ ∪
m =1 m′ = m
n
∪ Cm = ∪
m =1
∞
A m′
∞
∩
m =1 m′ = m
(11B5)
A m′
Now, define events B and C as the limits of Bn and Cn, respectively; that is, define
B ≡ limit Bn =
n →∞
C ≡ limit Cn =
n →∞
∞
∩
m =1
∞
Bm =
∞
∞
∩ ∪
m =1 m′ = m
∞
∞
∪ Cm = ∪ ∩
m =1
m =1 m′=m
A m′
(11B6)
A m′ .
(11B7)
Note that
ρ ∈ B ⇒ ρ is in infinitely many of the A n
(11B8)
ρ ∈ C ⇒ ρ is in all but a finite number of the A n
(note: the phrase “infinitely many” is not the same as the phrase “all but a finite”).
That B and C exist as events follows from the fact that countable unions and intersections
of events are events. That is, in the terminology of Chapter 1, σ-Algebra F is closed under
countable intersections and unions (once you are “inside” F it’s hard to get “outside”).
Updates at http://www.ece.uah.edu/courses/ee420-500/
11b-2
EE603 Class Notes
03/03/09
John Stensby
In the literature, B is called the limit supremum (also called lim sup, limit superior or
upper limit) of the sequence An, and it is denoted symbolically as
B = lim sup A n .
(11B9)
n →∞
A little thought will lead to the conclusion that each element of B is in infinitely many of the An.
That is, if event B occurs then infinitely many of the An occur (we say that the An occur
infinitely often, or An i.o.).
In the literature, C is called the limit infimum (also called lim inf, limit inferior or lower
limit) of the sequence An, and it is denoted symbolically as
C = lim inf A n .
(11B10)
n →∞
A little thought will lead to the conclusion that each element of C is in all but a finite number of
the An. That is, if event C occurs then all but a finite of the An occur.
Given a sequence of events An, then B = lim sup An and C = lim inf An always exist;
however, they may not be equal. However, it is easily seen that C ⊂ B always.
The infinite event sequence A1, A2, … is said to have a limit event A, and we write
A = limit A n ,
(11B11)
n →∞
if lim inf An = lim sup An. That is, the infinite event sequence A1, A2, … has limit event A if
events B and C, given by (11B6) and (11B7), respectively, are equal (i.e., B ⊆ C and C ⊆ B). In
this case, we write
A = limit A n = lim sup A n = lim inf A n .
n →∞
n →∞
n →∞
Updates at http://www.ece.uah.edu/courses/ee420-500/
(11B12)
11b-3
EE603 Class Notes
03/03/09
John Stensby
Given convergence of An to A as defined by (11B12), it is not difficult to show that
F
I
H n→∞ K
limit P(A n ) = P limit A n = P(A)
n →∞
(11B13)
(this is a good homework problem!). So, in the sense described by (11B13), we can say that
probability measures are continuous (often, (11B13) is taken as the definition of sequential
continuity, and P is said to be sequentially continuous).
Continuity property (11B13) held by P is analogous to a well-known continuity property
enjoyed by functions. Function f(x) is continuous at x = x0 if, and only if, f(xn) → f(x0) for any
sequence (and all sequences) xn → x0.
Borel-Cantelli Lemma
Let A1, A2, … be a sequence of events. Let
B = lim sup A n ,
n →∞
the event that infinitely many of the An occur. Then it follows that
∞
∑ P(A n ) < ∞
⇒
P(B) = 0 .
(11B14)
n =1
Furthermore, if all of the Ai are independent (i.e., we have an independent sequence) then
∞
∑ P(An ) = ∞
⇒
P(B) = 1 .
(11B15)
n =1
Proof: We show (11B14) first. Since B =
∞
∞
∩ ∪
n =1 m = n
Updates at http://www.ece.uah.edu/courses/ee420-500/
A m ≡ lim sup A n , we see that
n →∞
11b-4
EE603 Class Notes
B⊂
∞
∪
m=n
03/03/09
John Stensby
Am ,
(11B16)
for all n. This last equation implies that
P(B) ≤
∞
∑ P(Am )
(11B17)
m=n
for all n. Now, the hypothesis
∞
∑ P(A n ) < ∞
implies that the right-hand side of (11B17)
n =1
approaches zero as n approaches infinity. Hence, take the limit in (11B17) to see that P(B) = 0
as claimed by (11B14).
Now we show (11B15). From DeMorgans Law’s, we have
B=
∞
∞
∪ ∩
n =1 m = n
Am ,
where over-bar denotes the complementation. However, as indexed on r, r ≥ n,
(11B18)
r
∩
m=n
A m is a
nested, decreasing sequence of sets. By Theorem 11-1 we have
⎛ ∞
⎞
⎛ r
⎞
P ⎜ ∩ A m ⎟ = limit P ⎜ ∩ A m ⎟ .
⎜
⎟ r →∞ ⎜
⎟
⎝ m=n
⎠
⎝ m=n
⎠
(11B19)
Due to the independence of the sequence, we have
r
r
⎛ ∞
⎞
⎛ r
⎞
P ⎜ ∩ A m ⎟ = limit P ⎜ ∩ A m ⎟ = limit ∏ P ( A m ) = limit ∏ ⎡⎣1- P ( A m ) ⎤⎦ .
⎜
⎟ r →∞ ⎜
⎟ r →∞
r →∞ m = n
m=n
⎝ m=n
⎠
⎝ m=n
⎠
Updates at http://www.ece.uah.edu/courses/ee420-500/
(11B20)
11b-5
EE603 Class Notes
03/03/09
John Stensby
For all x ≥ 0, note that 1 - x ≤ e-x. Apply this inequality to (11B20) and obtain
∞
⎛ ∞
⎞
⎛ ∞
⎞
P ⎜ ∩ A m ⎟ = ∏ exp ⎡⎣- P ( A m ) ⎤⎦ = exp ⎜ − ∑ P ( A m ) ⎟
⎜
⎟
⎜
⎟
⎝ m=n
⎠
⎝ m=n
⎠ m=n
Use the hypothesis
(11B21)
∞
∑ P(A n ) = ∞ stated in (11B15) to see that (11B21) implies that
n =1
⎛ ∞
⎞
P ⎜ ∩ Am ⎟ = 0
⎜
⎟
⎝ m=n
⎠
(11B22)
for all n. Finally, from (11B18) we have
⎛ ∞
P(B) = limit P ⎜ ∩ A m
n →∞ ⎜⎝ m = n
⎞
⎟ = 0,
⎟
⎠
(11B23)
so that P(B) = 1 as claimed by (11B15).♥
In general, (11B15) is false if the sequence of Ai is not independent. To show this,
consider any event E, 0 < P(E) < 1, and define An = E for all n. Then, B = E and P(B) = P(E) ≠
0.
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11b-6