Question about proof with doubly stochastic matrix. Why do we need the uniqueness of $pi_j?$
$begingroup$
A stochastic matrix is called doubly stochastic if its rows and
columns sum to 1. Show that a Markov chain whose transition matrix is
doubly stochastic has a stationary distribution, which is uniform on
the statespace.
I'm trying to under stand the related problem but I don't understand how the uniqueness theorem comes in in the first row of the accepted answer, shouldm't it be $pi_i$? What do those sums even mean? They don't have any upper bound but i suppose the upper bound should be the upper bound of the statespace?
markov-chains
asked Dec 27 '18 at 16:53
ParsevalParseval
3,0741719
$endgroup$
add a comment |
$begingroup$
A stochastic matrix is called doubly stochastic if its rows and
columns sum to 1. Show that a Markov chain whose transition matrix is
doubly stochastic has a stationary distribution, which is uniform on
the statespace.
I'm trying to under stand the related problem but I don't understand how the uniqueness theorem comes in in the first row of the accepted answer, shouldm't it be $pi_i$? What do those sums even mean? They don't have any upper bound but i suppose the upper bound should be the upper bound of the statespace?
markov-chains
asked Dec 27 '18 at 16:53
ParsevalParseval
3,0741719
$endgroup$
add a comment |
$begingroup$
A stochastic matrix is called doubly stochastic if its rows and
columns sum to 1. Show that a Markov chain whose transition matrix is
doubly stochastic has a stationary distribution, which is uniform on
the statespace.
I'm trying to under stand the related problem but I don't understand how the uniqueness theorem comes in in the first row of the accepted answer, shouldm't it be $pi_i$? What do those sums even mean? They don't have any upper bound but i suppose the upper bound should be the upper bound of the statespace?
markov-chains
asked Dec 27 '18 at 16:53
ParsevalParseval
3,0741719
$endgroup$
A stochastic matrix is called doubly stochastic if its rows and
columns sum to 1. Show that a Markov chain whose transition matrix is
doubly stochastic has a stationary distribution, which is uniform on
the statespace.
I'm trying to under stand the related problem but I don't understand how the uniqueness theorem comes in in the first row of the accepted answer, shouldm't it be $pi_i$? What do those sums even mean? They don't have any upper bound but i suppose the upper bound should be the upper bound of the statespace?
markov-chains
markov-chains
asked Dec 27 '18 at 16:53
ParsevalParseval
3,0741719
asked Dec 27 '18 at 16:53
ParsevalParseval
3,0741719
asked Dec 27 '18 at 16:53
ParsevalParseval
3,0741719
asked Dec 27 '18 at 16:53
ParsevalParseval
3,0741719
asked Dec 27 '18 at 16:53
ParsevalParseval
3,0741719
3,0741719
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
They're using the uniqueness theorem so that, when they show that the uniform distribution is a valid stationary distribution for the doubly stochastic transition matrix, it must be the stationary distribution.
The uniqueness of the stationary distribution comes directly out of the irreducibility of $P$ + aperiodicity of $P$ + finite state space of the system, and the sums are just matrix multiplications over the state space (from $i=0$ to $M$).
answered Dec 28 '18 at 1:05
aghostinthefiguresaghostinthefigures
1,2891217
$endgroup$
add a comment |
Your Answer
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%2f3054153%2fquestion-about-proof-with-doubly-stochastic-matrix-why-do-we-need-the-uniquenes%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
$begingroup$
They're using the uniqueness theorem so that, when they show that the uniform distribution is a valid stationary distribution for the doubly stochastic transition matrix, it must be the stationary distribution.
The uniqueness of the stationary distribution comes directly out of the irreducibility of $P$ + aperiodicity of $P$ + finite state space of the system, and the sums are just matrix multiplications over the state space (from $i=0$ to $M$).
answered Dec 28 '18 at 1:05
aghostinthefiguresaghostinthefigures
1,2891217
$endgroup$
add a comment |
$begingroup$
They're using the uniqueness theorem so that, when they show that the uniform distribution is a valid stationary distribution for the doubly stochastic transition matrix, it must be the stationary distribution.
The uniqueness of the stationary distribution comes directly out of the irreducibility of $P$ + aperiodicity of $P$ + finite state space of the system, and the sums are just matrix multiplications over the state space (from $i=0$ to $M$).
answered Dec 28 '18 at 1:05
aghostinthefiguresaghostinthefigures
1,2891217
$endgroup$
add a comment |
$begingroup$
They're using the uniqueness theorem so that, when they show that the uniform distribution is a valid stationary distribution for the doubly stochastic transition matrix, it must be the stationary distribution.
The uniqueness of the stationary distribution comes directly out of the irreducibility of $P$ + aperiodicity of $P$ + finite state space of the system, and the sums are just matrix multiplications over the state space (from $i=0$ to $M$).
answered Dec 28 '18 at 1:05
aghostinthefiguresaghostinthefigures
1,2891217
$endgroup$
They're using the uniqueness theorem so that, when they show that the uniform distribution is a valid stationary distribution for the doubly stochastic transition matrix, it must be the stationary distribution.
The uniqueness of the stationary distribution comes directly out of the irreducibility of $P$ + aperiodicity of $P$ + finite state space of the system, and the sums are just matrix multiplications over the state space (from $i=0$ to $M$).
answered Dec 28 '18 at 1:05
aghostinthefiguresaghostinthefigures
1,2891217
answered Dec 28 '18 at 1:05
aghostinthefiguresaghostinthefigures
1,2891217
answered Dec 28 '18 at 1:05
aghostinthefiguresaghostinthefigures
1,2891217
answered Dec 28 '18 at 1:05
aghostinthefiguresaghostinthefigures
1,2891217
1,2891217
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.
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%2f3054153%2fquestion-about-proof-with-doubly-stochastic-matrix-why-do-we-need-the-uniquenes%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
