Mixing
Variance of first return time in Markov chain
$begingroup$
Good morning to everyone. I have a problem on proving a fact about an irreducible,finite state space Markov chain.
Specifically,suppose we have an irreducible,finite state space Markov chain. We know that it is positively recurrent, i.e. for every state i $ E_i[T_i] $ is finite,where $T_i$ is defined as the $ min { n>0 : X_n= i } $. I need to prove that this random variable has also finite variance ( and in fact finite moments of every order.
Has anybody any ideas about that??
Thank you a lot.
markov-chains
asked Jan 23 at 12:25
Petros KarajanPetros Karajan
113
$endgroup$
add a comment |
$begingroup$
Good morning to everyone. I have a problem on proving a fact about an irreducible,finite state space Markov chain.
Specifically,suppose we have an irreducible,finite state space Markov chain. We know that it is positively recurrent, i.e. for every state i $ E_i[T_i] $ is finite,where $T_i$ is defined as the $ min { n>0 : X_n= i } $. I need to prove that this random variable has also finite variance ( and in fact finite moments of every order.
Has anybody any ideas about that??
Thank you a lot.
markov-chains
asked Jan 23 at 12:25
Petros KarajanPetros Karajan
113
$endgroup$
add a comment |
$begingroup$
Good morning to everyone. I have a problem on proving a fact about an irreducible,finite state space Markov chain.
Specifically,suppose we have an irreducible,finite state space Markov chain. We know that it is positively recurrent, i.e. for every state i $ E_i[T_i] $ is finite,where $T_i$ is defined as the $ min { n>0 : X_n= i } $. I need to prove that this random variable has also finite variance ( and in fact finite moments of every order.
Has anybody any ideas about that??
Thank you a lot.
markov-chains
asked Jan 23 at 12:25
Petros KarajanPetros Karajan
113
$endgroup$
Good morning to everyone. I have a problem on proving a fact about an irreducible,finite state space Markov chain.
Specifically,suppose we have an irreducible,finite state space Markov chain. We know that it is positively recurrent, i.e. for every state i $ E_i[T_i] $ is finite,where $T_i$ is defined as the $ min { n>0 : X_n= i } $. I need to prove that this random variable has also finite variance ( and in fact finite moments of every order.
Has anybody any ideas about that??
Thank you a lot.
markov-chains
markov-chains
asked Jan 23 at 12:25
Petros KarajanPetros Karajan
113
asked Jan 23 at 12:25
Petros KarajanPetros Karajan
113
asked Jan 23 at 12:25
Petros KarajanPetros Karajan
113
asked Jan 23 at 12:25
Petros KarajanPetros Karajan
113
asked Jan 23 at 12:25
Petros KarajanPetros Karajan
113
113
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "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%2f3084409%2fvariance-of-first-return-time-in-markov-chain%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.
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%2f3084409%2fvariance-of-first-return-time-in-markov-chain%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
