Mixing
Negatively correlated random variables Chebyshev bound?
$begingroup$
It is quite well known that the Chernoff bound applies to negatively correlated binary random variables (see e.g. Theorem 1.16 here). Does there exist a reference for Chebyshev-type bound for negatively correlated binary variables?
probability inequality distribution-tails
edited Jan 9 at 3:45
Clement C.
50.2k33889
asked Jan 9 at 3:20
user1246462user1246462
1503
$endgroup$
add a comment |
$begingroup$
It is quite well known that the Chernoff bound applies to negatively correlated binary random variables (see e.g. Theorem 1.16 here). Does there exist a reference for Chebyshev-type bound for negatively correlated binary variables?
probability inequality distribution-tails
edited Jan 9 at 3:45
Clement C.
50.2k33889
asked Jan 9 at 3:20
user1246462user1246462
1503
$endgroup$
$begingroup$
As mentioned in the comment on my answer, the main question is why you need a reference, if you agree it's "fairly straightforward." Is it for a research paper? In which case you can just state it as a fact (or, if you feel you owe it to the reader, include a short proof "for completeness"). There is no need to provide a reference for all the statements -- for instance, you wouldn't provide one for Markov's inequality or Chebyshev, and this is of the same level.
$endgroup$
– Clement C.
Jan 11 at 3:42
add a comment |
$begingroup$
It is quite well known that the Chernoff bound applies to negatively correlated binary random variables (see e.g. Theorem 1.16 here). Does there exist a reference for Chebyshev-type bound for negatively correlated binary variables?
probability inequality distribution-tails
edited Jan 9 at 3:45
Clement C.
50.2k33889
asked Jan 9 at 3:20
user1246462user1246462
1503
$endgroup$
It is quite well known that the Chernoff bound applies to negatively correlated binary random variables (see e.g. Theorem 1.16 here). Does there exist a reference for Chebyshev-type bound for negatively correlated binary variables?
probability inequality distribution-tails
probability inequality distribution-tails
edited Jan 9 at 3:45
Clement C.
50.2k33889
asked Jan 9 at 3:20
user1246462user1246462
1503
edited Jan 9 at 3:45
Clement C.
50.2k33889
asked Jan 9 at 3:20
user1246462user1246462
1503
edited Jan 9 at 3:45
Clement C.
50.2k33889
edited Jan 9 at 3:45
Clement C.
50.2k33889
edited Jan 9 at 3:45
Clement C.
50.2k33889
50.2k33889
asked Jan 9 at 3:20
user1246462user1246462
1503
asked Jan 9 at 3:20
user1246462user1246462
1503
asked Jan 9 at 3:20
user1246462user1246462
1503
1503
$begingroup$
As mentioned in the comment on my answer, the main question is why you need a reference, if you agree it's "fairly straightforward." Is it for a research paper? In which case you can just state it as a fact (or, if you feel you owe it to the reader, include a short proof "for completeness"). There is no need to provide a reference for all the statements -- for instance, you wouldn't provide one for Markov's inequality or Chebyshev, and this is of the same level.
$endgroup$
– Clement C.
Jan 11 at 3:42
add a comment |
$begingroup$
As mentioned in the comment on my answer, the main question is why you need a reference, if you agree it's "fairly straightforward." Is it for a research paper? In which case you can just state it as a fact (or, if you feel you owe it to the reader, include a short proof "for completeness"). There is no need to provide a reference for all the statements -- for instance, you wouldn't provide one for Markov's inequality or Chebyshev, and this is of the same level.
$endgroup$
– Clement C.
Jan 11 at 3:42
$begingroup$
As mentioned in the comment on my answer, the main question is why you need a reference, if you agree it's "fairly straightforward." Is it for a research paper? In which case you can just state it as a fact (or, if you feel you owe it to the reader, include a short proof "for completeness"). There is no need to provide a reference for all the statements -- for instance, you wouldn't provide one for Markov's inequality or Chebyshev, and this is of the same level.
$endgroup$
– Clement C.
Jan 11 at 3:42
$begingroup$
As mentioned in the comment on my answer, the main question is why you need a reference, if you agree it's "fairly straightforward." Is it for a research paper? In which case you can just state it as a fact (or, if you feel you owe it to the reader, include a short proof "for completeness"). There is no need to provide a reference for all the statements -- for instance, you wouldn't provide one for Markov's inequality or Chebyshev, and this is of the same level.
$endgroup$
– Clement C.
Jan 11 at 3:42
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Why do you need a reference, instead of just deriving it or stating it as a fact? If the r.v.'s $(X_k)_{1leq k leq n}$ are negatively correlated (and have finite variance), then you have
$$
operatorname{Var}sum_{k=1}^n X_k leq sum_{k=1}^n operatorname{Var} X_k
$$
(since all covariances are non-positive: expand the variance on the LHS); and therefore you can use Chebyshev's inequality directly:
$$
forall a>0,qquad mathbb{P}left{leftlvertsum_{k=1}^n X_k - sum_{k=1}^n mathbb{E}X_krightrvert > aright}leq frac{operatorname{Var}sum_{k=1}^n X_k}{a^2} leq frac{sum_{k=1}^n operatorname{Var}X_k}{a^2} tag{$dagger$}
$$
answered Jan 9 at 3:31
Clement C.Clement C.
50.2k33889
$endgroup$
$begingroup$
The derivation is fairly straightforward and I was more concerned about providing the appropriate citation.
$endgroup$
– user1246462
Jan 9 at 18:15
$begingroup$
As you say, it is pretty straightforward. Either state it as a fact (folklore? Immediate fact?) or include a short proof "for completeness". (Not everything needs a reference...) @user1246462
$endgroup$
– Clement C.
Jan 9 at 18:18
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%2f3067033%2fnegatively-correlated-random-variables-chebyshev-bound%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
$begingroup$
Why do you need a reference, instead of just deriving it or stating it as a fact? If the r.v.'s $(X_k)_{1leq k leq n}$ are negatively correlated (and have finite variance), then you have
$$
operatorname{Var}sum_{k=1}^n X_k leq sum_{k=1}^n operatorname{Var} X_k
$$
(since all covariances are non-positive: expand the variance on the LHS); and therefore you can use Chebyshev's inequality directly:
$$
forall a>0,qquad mathbb{P}left{leftlvertsum_{k=1}^n X_k - sum_{k=1}^n mathbb{E}X_krightrvert > aright}leq frac{operatorname{Var}sum_{k=1}^n X_k}{a^2} leq frac{sum_{k=1}^n operatorname{Var}X_k}{a^2} tag{$dagger$}
$$
answered Jan 9 at 3:31
Clement C.Clement C.
50.2k33889
$endgroup$
$begingroup$
The derivation is fairly straightforward and I was more concerned about providing the appropriate citation.
$endgroup$
– user1246462
Jan 9 at 18:15
$begingroup$
As you say, it is pretty straightforward. Either state it as a fact (folklore? Immediate fact?) or include a short proof "for completeness". (Not everything needs a reference...) @user1246462
$endgroup$
– Clement C.
Jan 9 at 18:18
add a comment |
$begingroup$
Why do you need a reference, instead of just deriving it or stating it as a fact? If the r.v.'s $(X_k)_{1leq k leq n}$ are negatively correlated (and have finite variance), then you have
$$
operatorname{Var}sum_{k=1}^n X_k leq sum_{k=1}^n operatorname{Var} X_k
$$
(since all covariances are non-positive: expand the variance on the LHS); and therefore you can use Chebyshev's inequality directly:
$$
forall a>0,qquad mathbb{P}left{leftlvertsum_{k=1}^n X_k - sum_{k=1}^n mathbb{E}X_krightrvert > aright}leq frac{operatorname{Var}sum_{k=1}^n X_k}{a^2} leq frac{sum_{k=1}^n operatorname{Var}X_k}{a^2} tag{$dagger$}
$$
answered Jan 9 at 3:31
Clement C.Clement C.
50.2k33889
$endgroup$
$begingroup$
The derivation is fairly straightforward and I was more concerned about providing the appropriate citation.
$endgroup$
– user1246462
Jan 9 at 18:15
$begingroup$
As you say, it is pretty straightforward. Either state it as a fact (folklore? Immediate fact?) or include a short proof "for completeness". (Not everything needs a reference...) @user1246462
$endgroup$
– Clement C.
Jan 9 at 18:18
add a comment |
$begingroup$
Why do you need a reference, instead of just deriving it or stating it as a fact? If the r.v.'s $(X_k)_{1leq k leq n}$ are negatively correlated (and have finite variance), then you have
$$
operatorname{Var}sum_{k=1}^n X_k leq sum_{k=1}^n operatorname{Var} X_k
$$
(since all covariances are non-positive: expand the variance on the LHS); and therefore you can use Chebyshev's inequality directly:
$$
forall a>0,qquad mathbb{P}left{leftlvertsum_{k=1}^n X_k - sum_{k=1}^n mathbb{E}X_krightrvert > aright}leq frac{operatorname{Var}sum_{k=1}^n X_k}{a^2} leq frac{sum_{k=1}^n operatorname{Var}X_k}{a^2} tag{$dagger$}
$$
answered Jan 9 at 3:31
Clement C.Clement C.
50.2k33889
$endgroup$
Why do you need a reference, instead of just deriving it or stating it as a fact? If the r.v.'s $(X_k)_{1leq k leq n}$ are negatively correlated (and have finite variance), then you have
$$
operatorname{Var}sum_{k=1}^n X_k leq sum_{k=1}^n operatorname{Var} X_k
$$
(since all covariances are non-positive: expand the variance on the LHS); and therefore you can use Chebyshev's inequality directly:
$$
forall a>0,qquad mathbb{P}left{leftlvertsum_{k=1}^n X_k - sum_{k=1}^n mathbb{E}X_krightrvert > aright}leq frac{operatorname{Var}sum_{k=1}^n X_k}{a^2} leq frac{sum_{k=1}^n operatorname{Var}X_k}{a^2} tag{$dagger$}
$$
answered Jan 9 at 3:31
Clement C.Clement C.
50.2k33889
answered Jan 9 at 3:31
Clement C.Clement C.
50.2k33889
answered Jan 9 at 3:31
Clement C.Clement C.
50.2k33889
answered Jan 9 at 3:31
Clement C.Clement C.
50.2k33889
50.2k33889
$begingroup$
The derivation is fairly straightforward and I was more concerned about providing the appropriate citation.
$endgroup$
– user1246462
Jan 9 at 18:15
$begingroup$
As you say, it is pretty straightforward. Either state it as a fact (folklore? Immediate fact?) or include a short proof "for completeness". (Not everything needs a reference...) @user1246462
$endgroup$
– Clement C.
Jan 9 at 18:18
add a comment |
$begingroup$
The derivation is fairly straightforward and I was more concerned about providing the appropriate citation.
$endgroup$
– user1246462
Jan 9 at 18:15
$begingroup$
As you say, it is pretty straightforward. Either state it as a fact (folklore? Immediate fact?) or include a short proof "for completeness". (Not everything needs a reference...) @user1246462
$endgroup$
– Clement C.
Jan 9 at 18:18
$begingroup$
The derivation is fairly straightforward and I was more concerned about providing the appropriate citation.
$endgroup$
– user1246462
Jan 9 at 18:15
$begingroup$
The derivation is fairly straightforward and I was more concerned about providing the appropriate citation.
$endgroup$
– user1246462
Jan 9 at 18:15
$begingroup$
As you say, it is pretty straightforward. Either state it as a fact (folklore? Immediate fact?) or include a short proof "for completeness". (Not everything needs a reference...) @user1246462
$endgroup$
– Clement C.
Jan 9 at 18:18
$begingroup$
As you say, it is pretty straightforward. Either state it as a fact (folklore? Immediate fact?) or include a short proof "for completeness". (Not everything needs a reference...) @user1246462
$endgroup$
– Clement C.
Jan 9 at 18:18
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.
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%2f3067033%2fnegatively-correlated-random-variables-chebyshev-bound%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
$begingroup$
As mentioned in the comment on my answer, the main question is why you need a reference, if you agree it's "fairly straightforward." Is it for a research paper? In which case you can just state it as a fact (or, if you feel you owe it to the reader, include a short proof "for completeness"). There is no need to provide a reference for all the statements -- for instance, you wouldn't provide one for Markov's inequality or Chebyshev, and this is of the same level.
$endgroup$
– Clement C.
Jan 11 at 3:42