real sequence and convergence in probability
$X_n$ is a sequence of random variables.$X_n equiv a_n$, $a_n $ is a real sequence.
Then prove that $X_n $ converges in probability iff $a_n$ converges and then $X_n to lim_{ntoinfty} a_n$ in probability.
I have a feeling that the above statement is trivially true but I am not so sure. If anyone can prove otherwise,please do so.
convergence
asked May 30 '14 at 11:43
kris91kris91
18612
add a comment |
$X_n$ is a sequence of random variables.$X_n equiv a_n$, $a_n $ is a real sequence.
Then prove that $X_n $ converges in probability iff $a_n$ converges and then $X_n to lim_{ntoinfty} a_n$ in probability.
I have a feeling that the above statement is trivially true but I am not so sure. If anyone can prove otherwise,please do so.
convergence
asked May 30 '14 at 11:43
kris91kris91
18612
1
write down the definition of convergence in probability and see what you can do
– mm-aops
May 30 '14 at 11:45
Hrmm if you're working with a general measure (which is a generalization of a probability measure) then convergence in measure is not equivalent to convergence almost everywhere. Is there something I'm missing about this specific question that allows for those two things to be equivalent?
– DanZimm
May 30 '14 at 11:55
no.This was all there was to the question.I did write down the definition of convergence in probability and I think that this statement is trivially true but the thing is nothing is said about the nature of convergence of real sequence.
– kris91
May 30 '14 at 14:07
add a comment |
$X_n$ is a sequence of random variables.$X_n equiv a_n$, $a_n $ is a real sequence.
Then prove that $X_n $ converges in probability iff $a_n$ converges and then $X_n to lim_{ntoinfty} a_n$ in probability.
I have a feeling that the above statement is trivially true but I am not so sure. If anyone can prove otherwise,please do so.
convergence
asked May 30 '14 at 11:43
kris91kris91
18612
$X_n$ is a sequence of random variables.$X_n equiv a_n$, $a_n $ is a real sequence.
Then prove that $X_n $ converges in probability iff $a_n$ converges and then $X_n to lim_{ntoinfty} a_n$ in probability.
I have a feeling that the above statement is trivially true but I am not so sure. If anyone can prove otherwise,please do so.
convergence
convergence
asked May 30 '14 at 11:43
kris91kris91
18612
asked May 30 '14 at 11:43
kris91kris91
18612
asked May 30 '14 at 11:43
kris91kris91
18612
asked May 30 '14 at 11:43
kris91kris91
18612
asked May 30 '14 at 11:43
kris91kris91
18612
18612
1
write down the definition of convergence in probability and see what you can do
– mm-aops
May 30 '14 at 11:45
Hrmm if you're working with a general measure (which is a generalization of a probability measure) then convergence in measure is not equivalent to convergence almost everywhere. Is there something I'm missing about this specific question that allows for those two things to be equivalent?
– DanZimm
May 30 '14 at 11:55
no.This was all there was to the question.I did write down the definition of convergence in probability and I think that this statement is trivially true but the thing is nothing is said about the nature of convergence of real sequence.
– kris91
May 30 '14 at 14:07
add a comment |
1
write down the definition of convergence in probability and see what you can do
– mm-aops
May 30 '14 at 11:45
Hrmm if you're working with a general measure (which is a generalization of a probability measure) then convergence in measure is not equivalent to convergence almost everywhere. Is there something I'm missing about this specific question that allows for those two things to be equivalent?
– DanZimm
May 30 '14 at 11:55
no.This was all there was to the question.I did write down the definition of convergence in probability and I think that this statement is trivially true but the thing is nothing is said about the nature of convergence of real sequence.
– kris91
May 30 '14 at 14:07
1
1
write down the definition of convergence in probability and see what you can do
– mm-aops
May 30 '14 at 11:45
write down the definition of convergence in probability and see what you can do
– mm-aops
May 30 '14 at 11:45
Hrmm if you're working with a general measure (which is a generalization of a probability measure) then convergence in measure is not equivalent to convergence almost everywhere. Is there something I'm missing about this specific question that allows for those two things to be equivalent?
– DanZimm
May 30 '14 at 11:55
Hrmm if you're working with a general measure (which is a generalization of a probability measure) then convergence in measure is not equivalent to convergence almost everywhere. Is there something I'm missing about this specific question that allows for those two things to be equivalent?
– DanZimm
May 30 '14 at 11:55
no.This was all there was to the question.I did write down the definition of convergence in probability and I think that this statement is trivially true but the thing is nothing is said about the nature of convergence of real sequence.
– kris91
May 30 '14 at 14:07
no.This was all there was to the question.I did write down the definition of convergence in probability and I think that this statement is trivially true but the thing is nothing is said about the nature of convergence of real sequence.
– kris91
May 30 '14 at 14:07
add a comment |
1 Answer
1
active
oldest
votes
For any real number $a$ and $varepsilon > 0$,
$$
Pr{leftlvert X_n-arightrvertgt varepsilon }=
begin{cases}
1&mbox{ if }leftlvert a_n-arightrvertgt varepsilon;\
0&mbox{ if }leftlvert a_n-arightrvertleq varepsilon.
end{cases}
$$
answered 2 days ago
Davide GiraudoDavide Giraudo
125k16150260
add a comment |
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',
autoActivateHeartbeat: false,
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%2f814770%2freal-sequence-and-convergence-in-probability%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
For any real number $a$ and $varepsilon > 0$,
$$
Pr{leftlvert X_n-arightrvertgt varepsilon }=
begin{cases}
1&mbox{ if }leftlvert a_n-arightrvertgt varepsilon;\
0&mbox{ if }leftlvert a_n-arightrvertleq varepsilon.
end{cases}
$$
answered 2 days ago
Davide GiraudoDavide Giraudo
125k16150260
add a comment |
For any real number $a$ and $varepsilon > 0$,
$$
Pr{leftlvert X_n-arightrvertgt varepsilon }=
begin{cases}
1&mbox{ if }leftlvert a_n-arightrvertgt varepsilon;\
0&mbox{ if }leftlvert a_n-arightrvertleq varepsilon.
end{cases}
$$
answered 2 days ago
Davide GiraudoDavide Giraudo
125k16150260
add a comment |
For any real number $a$ and $varepsilon > 0$,
$$
Pr{leftlvert X_n-arightrvertgt varepsilon }=
begin{cases}
1&mbox{ if }leftlvert a_n-arightrvertgt varepsilon;\
0&mbox{ if }leftlvert a_n-arightrvertleq varepsilon.
end{cases}
$$
answered 2 days ago
Davide GiraudoDavide Giraudo
125k16150260
For any real number $a$ and $varepsilon > 0$,
$$
Pr{leftlvert X_n-arightrvertgt varepsilon }=
begin{cases}
1&mbox{ if }leftlvert a_n-arightrvertgt varepsilon;\
0&mbox{ if }leftlvert a_n-arightrvertleq varepsilon.
end{cases}
$$
answered 2 days ago
Davide GiraudoDavide Giraudo
125k16150260
answered 2 days ago
Davide GiraudoDavide Giraudo
125k16150260
answered 2 days ago
Davide GiraudoDavide Giraudo
125k16150260
answered 2 days ago
Davide GiraudoDavide Giraudo
125k16150260
125k16150260
add a comment |
add a comment |
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%2f814770%2freal-sequence-and-convergence-in-probability%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
write down the definition of convergence in probability and see what you can do
– mm-aops
May 30 '14 at 11:45
Hrmm if you're working with a general measure (which is a generalization of a probability measure) then convergence in measure is not equivalent to convergence almost everywhere. Is there something I'm missing about this specific question that allows for those two things to be equivalent?
– DanZimm
May 30 '14 at 11:55
no.This was all there was to the question.I did write down the definition of convergence in probability and I think that this statement is trivially true but the thing is nothing is said about the nature of convergence of real sequence.
– kris91
May 30 '14 at 14:07