First Borel-Cantelli-Lemma - Two different proofs
up vote
1
down vote
favorite
The Borel Cantelli Lemma states that for a sequence of events $(A_n)$ with finite sum $sum_{k=1}^{infty}P(A_k)< infty$ the probability of infinite many events happening is $0$, thus $P(limsup_{n to infty} A_n)=0.$
Before getting to my question, here are two different proofs that make sense to me:
First one:
Let $B_m=bigcup_{k geq m}A_k$, then $lim_{m to infty} B_m=limsup_{n toinfty}A_n$ from above. Hence, by the continuity, subadditivity of measures
$$P(limsup_{n to infty} A_n)=P(lim_{m to infty} B_m)=lim_{m to infty}P( bigcup_{k geq m}A_k)leq lim_{m to infty}sum_{k=m}^{infty}P(A_k)=0.$$
Second one:
$P(limsup_{n to infty} A_n)=0$ is equivalent to $sum_{k=1}^{infty}1_{A_k}(omega) < infty$. So we need to show that $P(sum_{k=1}^{infty}1_{A_k}(omega))geq infty)=0$.
First of all, by linearity of the expectation we have $$Ebigl (sum_{k=1}^{infty}1_{A_k}bigr )=sum_{k=1}^{infty}E(1_{A_k})=sum_{k=1}^{infty}P(A_k) < infty$$
Thus, by using Markov's inequality we obtain
$$P(sum_{k=1}^{infty}1_{A_k}(omega) geq epsilon) leq {1 over epsilon}sum_{k=1}^{infty}P(A_k) rightarrow0$$ for $epsilon to infty$ which completes the proof.
Question:
If both proofs are correct (please check), does any of those two proofs require something that is not needed in the other proof?
probability-theory proof-verification borel-cantelli-lemmas
asked Nov 22 at 12:02
Tesla
928426
|
show 1 more comment
up vote
1
down vote
favorite
The Borel Cantelli Lemma states that for a sequence of events $(A_n)$ with finite sum $sum_{k=1}^{infty}P(A_k)< infty$ the probability of infinite many events happening is $0$, thus $P(limsup_{n to infty} A_n)=0.$
Before getting to my question, here are two different proofs that make sense to me:
First one:
Let $B_m=bigcup_{k geq m}A_k$, then $lim_{m to infty} B_m=limsup_{n toinfty}A_n$ from above. Hence, by the continuity, subadditivity of measures
$$P(limsup_{n to infty} A_n)=P(lim_{m to infty} B_m)=lim_{m to infty}P( bigcup_{k geq m}A_k)leq lim_{m to infty}sum_{k=m}^{infty}P(A_k)=0.$$
Second one:
$P(limsup_{n to infty} A_n)=0$ is equivalent to $sum_{k=1}^{infty}1_{A_k}(omega) < infty$. So we need to show that $P(sum_{k=1}^{infty}1_{A_k}(omega))geq infty)=0$.
First of all, by linearity of the expectation we have $$Ebigl (sum_{k=1}^{infty}1_{A_k}bigr )=sum_{k=1}^{infty}E(1_{A_k})=sum_{k=1}^{infty}P(A_k) < infty$$
Thus, by using Markov's inequality we obtain
$$P(sum_{k=1}^{infty}1_{A_k}(omega) geq epsilon) leq {1 over epsilon}sum_{k=1}^{infty}P(A_k) rightarrow0$$ for $epsilon to infty$ which completes the proof.
Question:
If both proofs are correct (please check), does any of those two proofs require something that is not needed in the other proof?
probability-theory proof-verification borel-cantelli-lemmas
asked Nov 22 at 12:02
Tesla
928426
1
Both proofs are valid.
– Kavi Rama Murthy
Nov 22 at 12:13
@KaviRamaMurthy IMO the second proof should be rewritten before it can be judged. The idea might be okay, but very weird things are said in that proof.
– drhab
Nov 22 at 12:42
@drhab thanks for checking. Whats parts should be rewritten?
– Tesla
Nov 22 at 12:44
Start with looking at spots where your write an $omega$ (so just one arbitrary outcome) and wonder whether it is on its place. E.g. we can talk about $mathbb E1_A$ but not about $mathbb E1_A(omega)$.
– drhab
Nov 22 at 12:47
Also note that $P(limsup_{ntoinfty}A_n=0)iffsum_{k=1}^{infty}1_{A_k}(omega)<infty$ must become something like $omeganotinlimsup_{ntoinfty}A_niffsum_{k=1}^{infty}1_{A_k}(omega)<infty$.
– drhab
Nov 22 at 12:52
|
show 1 more comment
up vote
1
down vote
favorite
up vote
1
down vote
favorite
The Borel Cantelli Lemma states that for a sequence of events $(A_n)$ with finite sum $sum_{k=1}^{infty}P(A_k)< infty$ the probability of infinite many events happening is $0$, thus $P(limsup_{n to infty} A_n)=0.$
Before getting to my question, here are two different proofs that make sense to me:
First one:
Let $B_m=bigcup_{k geq m}A_k$, then $lim_{m to infty} B_m=limsup_{n toinfty}A_n$ from above. Hence, by the continuity, subadditivity of measures
$$P(limsup_{n to infty} A_n)=P(lim_{m to infty} B_m)=lim_{m to infty}P( bigcup_{k geq m}A_k)leq lim_{m to infty}sum_{k=m}^{infty}P(A_k)=0.$$
Second one:
$P(limsup_{n to infty} A_n)=0$ is equivalent to $sum_{k=1}^{infty}1_{A_k}(omega) < infty$. So we need to show that $P(sum_{k=1}^{infty}1_{A_k}(omega))geq infty)=0$.
First of all, by linearity of the expectation we have $$Ebigl (sum_{k=1}^{infty}1_{A_k}bigr )=sum_{k=1}^{infty}E(1_{A_k})=sum_{k=1}^{infty}P(A_k) < infty$$
Thus, by using Markov's inequality we obtain
$$P(sum_{k=1}^{infty}1_{A_k}(omega) geq epsilon) leq {1 over epsilon}sum_{k=1}^{infty}P(A_k) rightarrow0$$ for $epsilon to infty$ which completes the proof.
Question:
If both proofs are correct (please check), does any of those two proofs require something that is not needed in the other proof?
probability-theory proof-verification borel-cantelli-lemmas
asked Nov 22 at 12:02
Tesla
928426
The Borel Cantelli Lemma states that for a sequence of events $(A_n)$ with finite sum $sum_{k=1}^{infty}P(A_k)< infty$ the probability of infinite many events happening is $0$, thus $P(limsup_{n to infty} A_n)=0.$
Before getting to my question, here are two different proofs that make sense to me:
First one:
Let $B_m=bigcup_{k geq m}A_k$, then $lim_{m to infty} B_m=limsup_{n toinfty}A_n$ from above. Hence, by the continuity, subadditivity of measures
$$P(limsup_{n to infty} A_n)=P(lim_{m to infty} B_m)=lim_{m to infty}P( bigcup_{k geq m}A_k)leq lim_{m to infty}sum_{k=m}^{infty}P(A_k)=0.$$
Second one:
$P(limsup_{n to infty} A_n)=0$ is equivalent to $sum_{k=1}^{infty}1_{A_k}(omega) < infty$. So we need to show that $P(sum_{k=1}^{infty}1_{A_k}(omega))geq infty)=0$.
First of all, by linearity of the expectation we have $$Ebigl (sum_{k=1}^{infty}1_{A_k}bigr )=sum_{k=1}^{infty}E(1_{A_k})=sum_{k=1}^{infty}P(A_k) < infty$$
Thus, by using Markov's inequality we obtain
$$P(sum_{k=1}^{infty}1_{A_k}(omega) geq epsilon) leq {1 over epsilon}sum_{k=1}^{infty}P(A_k) rightarrow0$$ for $epsilon to infty$ which completes the proof.
Question:
If both proofs are correct (please check), does any of those two proofs require something that is not needed in the other proof?
probability-theory proof-verification borel-cantelli-lemmas
probability-theory proof-verification borel-cantelli-lemmas
asked Nov 22 at 12:02
Tesla
928426
asked Nov 22 at 12:02
Tesla
928426
edited Nov 26 at 11:00
asked Nov 22 at 12:02
Tesla
928426
asked Nov 22 at 12:02
Tesla
928426
asked Nov 22 at 12:02
Tesla
928426
928426
1
Both proofs are valid.
– Kavi Rama Murthy
Nov 22 at 12:13
@KaviRamaMurthy IMO the second proof should be rewritten before it can be judged. The idea might be okay, but very weird things are said in that proof.
– drhab
Nov 22 at 12:42
@drhab thanks for checking. Whats parts should be rewritten?
– Tesla
Nov 22 at 12:44
Start with looking at spots where your write an $omega$ (so just one arbitrary outcome) and wonder whether it is on its place. E.g. we can talk about $mathbb E1_A$ but not about $mathbb E1_A(omega)$.
– drhab
Nov 22 at 12:47
Also note that $P(limsup_{ntoinfty}A_n=0)iffsum_{k=1}^{infty}1_{A_k}(omega)<infty$ must become something like $omeganotinlimsup_{ntoinfty}A_niffsum_{k=1}^{infty}1_{A_k}(omega)<infty$.
– drhab
Nov 22 at 12:52
|
show 1 more comment
1
Both proofs are valid.
– Kavi Rama Murthy
Nov 22 at 12:13
@KaviRamaMurthy IMO the second proof should be rewritten before it can be judged. The idea might be okay, but very weird things are said in that proof.
– drhab
Nov 22 at 12:42
@drhab thanks for checking. Whats parts should be rewritten?
– Tesla
Nov 22 at 12:44
Start with looking at spots where your write an $omega$ (so just one arbitrary outcome) and wonder whether it is on its place. E.g. we can talk about $mathbb E1_A$ but not about $mathbb E1_A(omega)$.
– drhab
Nov 22 at 12:47
Also note that $P(limsup_{ntoinfty}A_n=0)iffsum_{k=1}^{infty}1_{A_k}(omega)<infty$ must become something like $omeganotinlimsup_{ntoinfty}A_niffsum_{k=1}^{infty}1_{A_k}(omega)<infty$.
– drhab
Nov 22 at 12:52
1
1
Both proofs are valid.
– Kavi Rama Murthy
Nov 22 at 12:13
Both proofs are valid.
– Kavi Rama Murthy
Nov 22 at 12:13
@KaviRamaMurthy IMO the second proof should be rewritten before it can be judged. The idea might be okay, but very weird things are said in that proof.
– drhab
Nov 22 at 12:42
@KaviRamaMurthy IMO the second proof should be rewritten before it can be judged. The idea might be okay, but very weird things are said in that proof.
– drhab
Nov 22 at 12:42
@drhab thanks for checking. Whats parts should be rewritten?
– Tesla
Nov 22 at 12:44
@drhab thanks for checking. Whats parts should be rewritten?
– Tesla
Nov 22 at 12:44
Start with looking at spots where your write an $omega$ (so just one arbitrary outcome) and wonder whether it is on its place. E.g. we can talk about $mathbb E1_A$ but not about $mathbb E1_A(omega)$.
– drhab
Nov 22 at 12:47
Start with looking at spots where your write an $omega$ (so just one arbitrary outcome) and wonder whether it is on its place. E.g. we can talk about $mathbb E1_A$ but not about $mathbb E1_A(omega)$.
– drhab
Nov 22 at 12:47
Also note that $P(limsup_{ntoinfty}A_n=0)iffsum_{k=1}^{infty}1_{A_k}(omega)<infty$ must become something like $omeganotinlimsup_{ntoinfty}A_niffsum_{k=1}^{infty}1_{A_k}(omega)<infty$.
– drhab
Nov 22 at 12:52
Also note that $P(limsup_{ntoinfty}A_n=0)iffsum_{k=1}^{infty}1_{A_k}(omega)<infty$ must become something like $omeganotinlimsup_{ntoinfty}A_niffsum_{k=1}^{infty}1_{A_k}(omega)<infty$.
– drhab
Nov 22 at 12:52
|
show 1 more comment
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3009048%2ffirst-borel-cantelli-lemma-two-different-proofs%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3009048%2ffirst-borel-cantelli-lemma-two-different-proofs%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
Both proofs are valid.
– Kavi Rama Murthy
Nov 22 at 12:13
@KaviRamaMurthy IMO the second proof should be rewritten before it can be judged. The idea might be okay, but very weird things are said in that proof.
– drhab
Nov 22 at 12:42
@drhab thanks for checking. Whats parts should be rewritten?
– Tesla
Nov 22 at 12:44
Start with looking at spots where your write an $omega$ (so just one arbitrary outcome) and wonder whether it is on its place. E.g. we can talk about $mathbb E1_A$ but not about $mathbb E1_A(omega)$.
– drhab
Nov 22 at 12:47
Also note that $P(limsup_{ntoinfty}A_n=0)iffsum_{k=1}^{infty}1_{A_k}(omega)<infty$ must become something like $omeganotinlimsup_{ntoinfty}A_niffsum_{k=1}^{infty}1_{A_k}(omega)<infty$.
– drhab
Nov 22 at 12:52