Theorems like the Lovász Local Lemma?analog of principle of inclusion-exclusionprobabilities of increasing events under different product measures.Can you explain the description of the Lovasz Local Lemma by Moser+Tardos?To what extent can the following zero-one laws be relaxed?A question on independenceHelp with derivation of probability density of event generation & event detectionWhat is the early history of the concepts of probabilistic independence and conditional probability/expectation?Fixing (non)-independency of a the subfamilies of finitely many events.Negative association in a “k out of n” processWhat's the probability of two independent events in time domain?
Theorems like the Lovász Local Lemma?
analog of principle of inclusion-exclusionprobabilities of increasing events under different product measures.Can you explain the description of the Lovasz Local Lemma by Moser+Tardos?To what extent can the following zero-one laws be relaxed?A question on independenceHelp with derivation of probability density of event generation & event detectionWhat is the early history of the concepts of probabilistic independence and conditional probability/expectation?Fixing (non)-independency of a the subfamilies of finitely many events.Negative association in a “k out of n” processWhat's the probability of two independent events in time domain?
$begingroup$
The Lovász Local Lemma gives a probability bound in a context where there are many events that are "not quite" independent.
What other theorems exist in this genre? That is, what other theorems have a hypothesis of the form "Let events E_1, E_2, ... satisfy [relaxed form of independence]" and a conclusion of the form "Then the probability of [compound event] satisfies [inequality]"?
(I hope this question isn't too broad. I frequently encounter problems with events that are "almost independent", either in the sense that most subsets are independent or in the sense that the probabilities of compound events are well-approximated by assuming independence, and I am looking for general tools that may be helpful when these situations come up.)
pr.probability probabilistic-method
asked 4 hours ago
AustinAustin
1813
$endgroup$
add a comment |
$begingroup$
The Lovász Local Lemma gives a probability bound in a context where there are many events that are "not quite" independent.
What other theorems exist in this genre? That is, what other theorems have a hypothesis of the form "Let events E_1, E_2, ... satisfy [relaxed form of independence]" and a conclusion of the form "Then the probability of [compound event] satisfies [inequality]"?
(I hope this question isn't too broad. I frequently encounter problems with events that are "almost independent", either in the sense that most subsets are independent or in the sense that the probabilities of compound events are well-approximated by assuming independence, and I am looking for general tools that may be helpful when these situations come up.)
pr.probability probabilistic-method
asked 4 hours ago
AustinAustin
1813
$endgroup$
1
$begingroup$
See pdfs.semanticscholar.org/6631/…
$endgroup$
– Sam Hopkins
4 hours ago
2
$begingroup$
Talagrand’s concentration inequality in particular is very powerful for this kind of thing.
$endgroup$
– Sam Hopkins
3 hours ago
add a comment |
$begingroup$
The Lovász Local Lemma gives a probability bound in a context where there are many events that are "not quite" independent.
What other theorems exist in this genre? That is, what other theorems have a hypothesis of the form "Let events E_1, E_2, ... satisfy [relaxed form of independence]" and a conclusion of the form "Then the probability of [compound event] satisfies [inequality]"?
(I hope this question isn't too broad. I frequently encounter problems with events that are "almost independent", either in the sense that most subsets are independent or in the sense that the probabilities of compound events are well-approximated by assuming independence, and I am looking for general tools that may be helpful when these situations come up.)
pr.probability probabilistic-method
asked 4 hours ago
AustinAustin
1813
$endgroup$
The Lovász Local Lemma gives a probability bound in a context where there are many events that are "not quite" independent.
What other theorems exist in this genre? That is, what other theorems have a hypothesis of the form "Let events E_1, E_2, ... satisfy [relaxed form of independence]" and a conclusion of the form "Then the probability of [compound event] satisfies [inequality]"?
(I hope this question isn't too broad. I frequently encounter problems with events that are "almost independent", either in the sense that most subsets are independent or in the sense that the probabilities of compound events are well-approximated by assuming independence, and I am looking for general tools that may be helpful when these situations come up.)
pr.probability probabilistic-method
pr.probability probabilistic-method
asked 4 hours ago
AustinAustin
1813
asked 4 hours ago
AustinAustin
1813
asked 4 hours ago
AustinAustin
1813
asked 4 hours ago
AustinAustin
1813
asked 4 hours ago
AustinAustin
1813
1813
1
$begingroup$
See pdfs.semanticscholar.org/6631/…
$endgroup$
– Sam Hopkins
4 hours ago
2
$begingroup$
Talagrand’s concentration inequality in particular is very powerful for this kind of thing.
$endgroup$
– Sam Hopkins
3 hours ago
add a comment |
1
$begingroup$
See pdfs.semanticscholar.org/6631/…
$endgroup$
– Sam Hopkins
4 hours ago
2
$begingroup$
Talagrand’s concentration inequality in particular is very powerful for this kind of thing.
$endgroup$
– Sam Hopkins
3 hours ago
1
1
$begingroup$
See pdfs.semanticscholar.org/6631/…
$endgroup$
– Sam Hopkins
4 hours ago
$begingroup$
See pdfs.semanticscholar.org/6631/…
$endgroup$
– Sam Hopkins
4 hours ago
2
2
$begingroup$
Talagrand’s concentration inequality in particular is very powerful for this kind of thing.
$endgroup$
– Sam Hopkins
3 hours ago
$begingroup$
Talagrand’s concentration inequality in particular is very powerful for this kind of thing.
$endgroup$
– Sam Hopkins
3 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
A large number of results for sums $W$ of possibly dependent indicators of events (that is, for sums of possibly dependent Bernoulli random variables) $X_i$ have been obtained by the Chen--Stein method. See e.g. Theorem 1, which gives an upper bound on the total variation distance between the distribution of such a sum $W$ and a corresponding Poisson distribution in terms of certain characteristics $b_1,b_2,b_3$ of the strength of the dependence between the $X_i$'s (defined in formulas (4)--(6) of that paper).
answered 3 hours ago
Iosif PinelisIosif Pinelis
19.8k22259
$endgroup$
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: "504"
;
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%2fmathoverflow.net%2fquestions%2f325465%2ftheorems-like-the-lov%25c3%25a1sz-local-lemma%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
$begingroup$
A large number of results for sums $W$ of possibly dependent indicators of events (that is, for sums of possibly dependent Bernoulli random variables) $X_i$ have been obtained by the Chen--Stein method. See e.g. Theorem 1, which gives an upper bound on the total variation distance between the distribution of such a sum $W$ and a corresponding Poisson distribution in terms of certain characteristics $b_1,b_2,b_3$ of the strength of the dependence between the $X_i$'s (defined in formulas (4)--(6) of that paper).
answered 3 hours ago
Iosif PinelisIosif Pinelis
19.8k22259
$endgroup$
add a comment |
$begingroup$
A large number of results for sums $W$ of possibly dependent indicators of events (that is, for sums of possibly dependent Bernoulli random variables) $X_i$ have been obtained by the Chen--Stein method. See e.g. Theorem 1, which gives an upper bound on the total variation distance between the distribution of such a sum $W$ and a corresponding Poisson distribution in terms of certain characteristics $b_1,b_2,b_3$ of the strength of the dependence between the $X_i$'s (defined in formulas (4)--(6) of that paper).
answered 3 hours ago
Iosif PinelisIosif Pinelis
19.8k22259
$endgroup$
add a comment |
$begingroup$
A large number of results for sums $W$ of possibly dependent indicators of events (that is, for sums of possibly dependent Bernoulli random variables) $X_i$ have been obtained by the Chen--Stein method. See e.g. Theorem 1, which gives an upper bound on the total variation distance between the distribution of such a sum $W$ and a corresponding Poisson distribution in terms of certain characteristics $b_1,b_2,b_3$ of the strength of the dependence between the $X_i$'s (defined in formulas (4)--(6) of that paper).
answered 3 hours ago
Iosif PinelisIosif Pinelis
19.8k22259
$endgroup$
A large number of results for sums $W$ of possibly dependent indicators of events (that is, for sums of possibly dependent Bernoulli random variables) $X_i$ have been obtained by the Chen--Stein method. See e.g. Theorem 1, which gives an upper bound on the total variation distance between the distribution of such a sum $W$ and a corresponding Poisson distribution in terms of certain characteristics $b_1,b_2,b_3$ of the strength of the dependence between the $X_i$'s (defined in formulas (4)--(6) of that paper).
answered 3 hours ago
Iosif PinelisIosif Pinelis
19.8k22259
answered 3 hours ago
Iosif PinelisIosif Pinelis
19.8k22259
answered 3 hours ago
Iosif PinelisIosif Pinelis
19.8k22259
answered 3 hours ago
Iosif PinelisIosif Pinelis
19.8k22259
19.8k22259
add a comment |
add a comment |
Thanks for contributing an answer to MathOverflow!
- 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%2fmathoverflow.net%2fquestions%2f325465%2ftheorems-like-the-lov%25c3%25a1sz-local-lemma%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
$begingroup$
See pdfs.semanticscholar.org/6631/…
$endgroup$
– Sam Hopkins
4 hours ago
2
$begingroup$
Talagrand’s concentration inequality in particular is very powerful for this kind of thing.
$endgroup$
– Sam Hopkins
3 hours ago