A proper definition of recurrent event in probability
I am having hard time understanding the definition of a recurrent event in probability context.
At our lecture it was defined as follows:
Let $X^{[i,n]} = {X_i, ldots, X_n}$ be a random sequence. The random variables $X_j$ can attain countably many values. An event $epsilon$ that appears on the $n$-th and the $m+n$-th place of sequence $X^{[1,m+n]}$ is called recurrent if and only if it occurs on the last place of the sequence $X^{[1,n]}$ and on the last place of the sequence $X^{[n+1,n+m]}$.
Then $Prob(X^{[1,n+m]}) = Prob(X^{[1,n]})cdot Prob(X^{[n+1,n+m]})$.
An event $epsilon$ could be for example the event when $S_n = sum_{j=i}^nX_j = 0$.
The variables $X_i$ may be dependent in some way. It is clear that the multiplication rule holds if the events are independent. However, I do not understand, why the multiplication rule should apply in general.
Is it possible, that the event is recurrent if the multiplication rule applies?
The closest definition I could find online to what we defined in the lecture can be seen on this webpage. It is probably more exact, but I really struggle tu wrap my head around it.
Could anyone please explain, why the multiplication rule holds for recurrent events and potentialy elaborate on the definition from the link? Any help would be very appreciated!
probability probability-theory stochastic-processes recurrence-relations random-walk
asked Jan 4 at 18:39
Jan VainerJan Vainer
206
add a comment |
I am having hard time understanding the definition of a recurrent event in probability context.
At our lecture it was defined as follows:
Let $X^{[i,n]} = {X_i, ldots, X_n}$ be a random sequence. The random variables $X_j$ can attain countably many values. An event $epsilon$ that appears on the $n$-th and the $m+n$-th place of sequence $X^{[1,m+n]}$ is called recurrent if and only if it occurs on the last place of the sequence $X^{[1,n]}$ and on the last place of the sequence $X^{[n+1,n+m]}$.
Then $Prob(X^{[1,n+m]}) = Prob(X^{[1,n]})cdot Prob(X^{[n+1,n+m]})$.
An event $epsilon$ could be for example the event when $S_n = sum_{j=i}^nX_j = 0$.
The variables $X_i$ may be dependent in some way. It is clear that the multiplication rule holds if the events are independent. However, I do not understand, why the multiplication rule should apply in general.
Is it possible, that the event is recurrent if the multiplication rule applies?
The closest definition I could find online to what we defined in the lecture can be seen on this webpage. It is probably more exact, but I really struggle tu wrap my head around it.
Could anyone please explain, why the multiplication rule holds for recurrent events and potentialy elaborate on the definition from the link? Any help would be very appreciated!
probability probability-theory stochastic-processes recurrence-relations random-walk
asked Jan 4 at 18:39
Jan VainerJan Vainer
206
add a comment |
I am having hard time understanding the definition of a recurrent event in probability context.
At our lecture it was defined as follows:
Let $X^{[i,n]} = {X_i, ldots, X_n}$ be a random sequence. The random variables $X_j$ can attain countably many values. An event $epsilon$ that appears on the $n$-th and the $m+n$-th place of sequence $X^{[1,m+n]}$ is called recurrent if and only if it occurs on the last place of the sequence $X^{[1,n]}$ and on the last place of the sequence $X^{[n+1,n+m]}$.
Then $Prob(X^{[1,n+m]}) = Prob(X^{[1,n]})cdot Prob(X^{[n+1,n+m]})$.
An event $epsilon$ could be for example the event when $S_n = sum_{j=i}^nX_j = 0$.
The variables $X_i$ may be dependent in some way. It is clear that the multiplication rule holds if the events are independent. However, I do not understand, why the multiplication rule should apply in general.
Is it possible, that the event is recurrent if the multiplication rule applies?
The closest definition I could find online to what we defined in the lecture can be seen on this webpage. It is probably more exact, but I really struggle tu wrap my head around it.
Could anyone please explain, why the multiplication rule holds for recurrent events and potentialy elaborate on the definition from the link? Any help would be very appreciated!
probability probability-theory stochastic-processes recurrence-relations random-walk
asked Jan 4 at 18:39
Jan VainerJan Vainer
206
I am having hard time understanding the definition of a recurrent event in probability context.
At our lecture it was defined as follows:
Let $X^{[i,n]} = {X_i, ldots, X_n}$ be a random sequence. The random variables $X_j$ can attain countably many values. An event $epsilon$ that appears on the $n$-th and the $m+n$-th place of sequence $X^{[1,m+n]}$ is called recurrent if and only if it occurs on the last place of the sequence $X^{[1,n]}$ and on the last place of the sequence $X^{[n+1,n+m]}$.
Then $Prob(X^{[1,n+m]}) = Prob(X^{[1,n]})cdot Prob(X^{[n+1,n+m]})$.
An event $epsilon$ could be for example the event when $S_n = sum_{j=i}^nX_j = 0$.
The variables $X_i$ may be dependent in some way. It is clear that the multiplication rule holds if the events are independent. However, I do not understand, why the multiplication rule should apply in general.
Is it possible, that the event is recurrent if the multiplication rule applies?
The closest definition I could find online to what we defined in the lecture can be seen on this webpage. It is probably more exact, but I really struggle tu wrap my head around it.
Could anyone please explain, why the multiplication rule holds for recurrent events and potentialy elaborate on the definition from the link? Any help would be very appreciated!
probability probability-theory stochastic-processes recurrence-relations random-walk
probability probability-theory stochastic-processes recurrence-relations random-walk
asked Jan 4 at 18:39
Jan VainerJan Vainer
206
asked Jan 4 at 18:39
Jan VainerJan Vainer
206
edited Jan 4 at 19:20
Jan Vainer
asked Jan 4 at 18:39
Jan VainerJan Vainer
206
asked Jan 4 at 18:39
Jan VainerJan Vainer
206
asked Jan 4 at 18:39
Jan VainerJan Vainer
206
206
add a comment |
add a comment |
0
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',
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%2f3061953%2fa-proper-definition-of-recurrent-event-in-probability%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
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%2f3061953%2fa-proper-definition-of-recurrent-event-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
