Skip to main content
Theorem
Path
Curriculum
Paths
Labs
Diagnostic
Blog
Search
Sign in
Quiz Hub
/
Borel-Cantelli Lemmas
Borel-Cantelli Lemmas
5 selected
Difficulty 5-7
5 unseen
View topic
Intermediate
New
0 answered
4 intermediate
1 advanced
Adapts to your performance
Question 1 of 5
120s
intermediate (5/10)
state theorem
The first Borel-Cantelli lemma says that if
∑
n
=
1
∞
P
(
A
n
)
<
∞
, then
P
(
A
n
infinitely often
)
=
0
. What does the conclusion mean operationally?
Hide and think first
A.
With probability one, all events
A
n
stop occurring after some random index
N
that depends only on the convergence rate of the series.
B.
With probability one, only finitely many of the events
A
n
occur in any single realization of the underlying sample path.
C.
With probability one, no
A
n
for
n
greater than some fixed constant
C
ever occurs, regardless of the realization.
D.
The expected number of events that occur is finite,
E
[
∑
n
1
A
n
]
<
∞
, but individual realizations can still see infinitely many events.
Submit Answer
I don't know