Are all transient states positive recurrent?
Clash Royale CLAN TAG#URR8PPP
$begingroup$
In my Stochastic process lecture notes, I noticed that in problems related to a finite Markov chain, the mean recurrence time are calculated only for recurrent states. I calculated the mean recurrence time of all transient states and found them to be positive recurrent.
Is mean recurrence time calculated only for recurrent states? And is it true that all transient states are positive recurrent?
stochastic-processes
asked Jan 3 at 7:19
user46697user46697
345212
$endgroup$
add a comment |
$begingroup$
In my Stochastic process lecture notes, I noticed that in problems related to a finite Markov chain, the mean recurrence time are calculated only for recurrent states. I calculated the mean recurrence time of all transient states and found them to be positive recurrent.
Is mean recurrence time calculated only for recurrent states? And is it true that all transient states are positive recurrent?
stochastic-processes
asked Jan 3 at 7:19
user46697user46697
345212
$endgroup$
add a comment |
$begingroup$
In my Stochastic process lecture notes, I noticed that in problems related to a finite Markov chain, the mean recurrence time are calculated only for recurrent states. I calculated the mean recurrence time of all transient states and found them to be positive recurrent.
Is mean recurrence time calculated only for recurrent states? And is it true that all transient states are positive recurrent?
stochastic-processes
asked Jan 3 at 7:19
user46697user46697
345212
$endgroup$
In my Stochastic process lecture notes, I noticed that in problems related to a finite Markov chain, the mean recurrence time are calculated only for recurrent states. I calculated the mean recurrence time of all transient states and found them to be positive recurrent.
Is mean recurrence time calculated only for recurrent states? And is it true that all transient states are positive recurrent?
stochastic-processes
stochastic-processes
asked Jan 3 at 7:19
user46697user46697
345212
asked Jan 3 at 7:19
user46697user46697
345212
asked Jan 3 at 7:19
user46697user46697
345212
asked Jan 3 at 7:19
user46697user46697
345212
asked Jan 3 at 7:19
user46697user46697
345212
345212
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Transient states are those which may never be visited again. Positive recurrent states will always be revisited within some (expected) finite time. Therefore, no transient state is positive recurrent.
answered Jan 3 at 7:32
conjecturesconjectures
3,1931531
$endgroup$
add a comment |
$begingroup$
To quote from Meyn & Tweedie's Markov Chains & Stochastic Stability, on page 175, a state $alpha$ is recurrent for a Markov chain $(Phi_t)_t$ if $mathbbE_alpha[eta_alpha]=infty$ (infinite expected number of visits) and transient if $mathbbE_alpha[eta_alpha]<infty$ (finite expected number of visits), when$$eta_alpha=sum_t=1^infty mathbbI_alpha(Phi_t)$$is the number of visits. This definition does not involve nullity or positivity. To further quote from Meyn & Tweedie's Markov Chains & Stochastic Stability, on page 453, a state $alpha$ is null if$$lim_t to infty P^t(alpha,alpha) = 0$$and positive if$$limsup_t to infty P^t(alpha,alpha) > 0$$If a state $alpha$ is transient then it is necessary null, while if a state $alpha$ is positive, it is necessary recurrent. An important result (see Theorems 10.0.1 & 10.4.9 in Meyn & Tweedie's Markov Chains & Stochastic Stability) is that an irreducible recurrent Markov chain $(Phi_t)_t$ admits a unique (up to multiplicative constants) invariant measure. When the measure is finite the chain is necessarily positive recurrent, while in the opposite case, it is null recurrent.
answered Jan 3 at 8:07
Xi'anXi'an
54.6k692351
$endgroup$
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: "65"
;
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: false,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: null,
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
,
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%2fstats.stackexchange.com%2fquestions%2f385407%2fare-all-transient-states-positive-recurrent%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Transient states are those which may never be visited again. Positive recurrent states will always be revisited within some (expected) finite time. Therefore, no transient state is positive recurrent.
answered Jan 3 at 7:32
conjecturesconjectures
3,1931531
$endgroup$
add a comment |
$begingroup$
Transient states are those which may never be visited again. Positive recurrent states will always be revisited within some (expected) finite time. Therefore, no transient state is positive recurrent.
answered Jan 3 at 7:32
conjecturesconjectures
3,1931531
$endgroup$
add a comment |
$begingroup$
Transient states are those which may never be visited again. Positive recurrent states will always be revisited within some (expected) finite time. Therefore, no transient state is positive recurrent.
answered Jan 3 at 7:32
conjecturesconjectures
3,1931531
$endgroup$
Transient states are those which may never be visited again. Positive recurrent states will always be revisited within some (expected) finite time. Therefore, no transient state is positive recurrent.
answered Jan 3 at 7:32
conjecturesconjectures
3,1931531
answered Jan 3 at 7:32
conjecturesconjectures
3,1931531
answered Jan 3 at 7:32
conjecturesconjectures
3,1931531
answered Jan 3 at 7:32
conjecturesconjectures
3,1931531
3,1931531
add a comment |
add a comment |
$begingroup$
To quote from Meyn & Tweedie's Markov Chains & Stochastic Stability, on page 175, a state $alpha$ is recurrent for a Markov chain $(Phi_t)_t$ if $mathbbE_alpha[eta_alpha]=infty$ (infinite expected number of visits) and transient if $mathbbE_alpha[eta_alpha]<infty$ (finite expected number of visits), when$$eta_alpha=sum_t=1^infty mathbbI_alpha(Phi_t)$$is the number of visits. This definition does not involve nullity or positivity. To further quote from Meyn & Tweedie's Markov Chains & Stochastic Stability, on page 453, a state $alpha$ is null if$$lim_t to infty P^t(alpha,alpha) = 0$$and positive if$$limsup_t to infty P^t(alpha,alpha) > 0$$If a state $alpha$ is transient then it is necessary null, while if a state $alpha$ is positive, it is necessary recurrent. An important result (see Theorems 10.0.1 & 10.4.9 in Meyn & Tweedie's Markov Chains & Stochastic Stability) is that an irreducible recurrent Markov chain $(Phi_t)_t$ admits a unique (up to multiplicative constants) invariant measure. When the measure is finite the chain is necessarily positive recurrent, while in the opposite case, it is null recurrent.
answered Jan 3 at 8:07
Xi'anXi'an
54.6k692351
$endgroup$
add a comment |
$begingroup$
To quote from Meyn & Tweedie's Markov Chains & Stochastic Stability, on page 175, a state $alpha$ is recurrent for a Markov chain $(Phi_t)_t$ if $mathbbE_alpha[eta_alpha]=infty$ (infinite expected number of visits) and transient if $mathbbE_alpha[eta_alpha]<infty$ (finite expected number of visits), when$$eta_alpha=sum_t=1^infty mathbbI_alpha(Phi_t)$$is the number of visits. This definition does not involve nullity or positivity. To further quote from Meyn & Tweedie's Markov Chains & Stochastic Stability, on page 453, a state $alpha$ is null if$$lim_t to infty P^t(alpha,alpha) = 0$$and positive if$$limsup_t to infty P^t(alpha,alpha) > 0$$If a state $alpha$ is transient then it is necessary null, while if a state $alpha$ is positive, it is necessary recurrent. An important result (see Theorems 10.0.1 & 10.4.9 in Meyn & Tweedie's Markov Chains & Stochastic Stability) is that an irreducible recurrent Markov chain $(Phi_t)_t$ admits a unique (up to multiplicative constants) invariant measure. When the measure is finite the chain is necessarily positive recurrent, while in the opposite case, it is null recurrent.
answered Jan 3 at 8:07
Xi'anXi'an
54.6k692351
$endgroup$
add a comment |
$begingroup$
To quote from Meyn & Tweedie's Markov Chains & Stochastic Stability, on page 175, a state $alpha$ is recurrent for a Markov chain $(Phi_t)_t$ if $mathbbE_alpha[eta_alpha]=infty$ (infinite expected number of visits) and transient if $mathbbE_alpha[eta_alpha]<infty$ (finite expected number of visits), when$$eta_alpha=sum_t=1^infty mathbbI_alpha(Phi_t)$$is the number of visits. This definition does not involve nullity or positivity. To further quote from Meyn & Tweedie's Markov Chains & Stochastic Stability, on page 453, a state $alpha$ is null if$$lim_t to infty P^t(alpha,alpha) = 0$$and positive if$$limsup_t to infty P^t(alpha,alpha) > 0$$If a state $alpha$ is transient then it is necessary null, while if a state $alpha$ is positive, it is necessary recurrent. An important result (see Theorems 10.0.1 & 10.4.9 in Meyn & Tweedie's Markov Chains & Stochastic Stability) is that an irreducible recurrent Markov chain $(Phi_t)_t$ admits a unique (up to multiplicative constants) invariant measure. When the measure is finite the chain is necessarily positive recurrent, while in the opposite case, it is null recurrent.
answered Jan 3 at 8:07
Xi'anXi'an
54.6k692351
$endgroup$
To quote from Meyn & Tweedie's Markov Chains & Stochastic Stability, on page 175, a state $alpha$ is recurrent for a Markov chain $(Phi_t)_t$ if $mathbbE_alpha[eta_alpha]=infty$ (infinite expected number of visits) and transient if $mathbbE_alpha[eta_alpha]<infty$ (finite expected number of visits), when$$eta_alpha=sum_t=1^infty mathbbI_alpha(Phi_t)$$is the number of visits. This definition does not involve nullity or positivity. To further quote from Meyn & Tweedie's Markov Chains & Stochastic Stability, on page 453, a state $alpha$ is null if$$lim_t to infty P^t(alpha,alpha) = 0$$and positive if$$limsup_t to infty P^t(alpha,alpha) > 0$$If a state $alpha$ is transient then it is necessary null, while if a state $alpha$ is positive, it is necessary recurrent. An important result (see Theorems 10.0.1 & 10.4.9 in Meyn & Tweedie's Markov Chains & Stochastic Stability) is that an irreducible recurrent Markov chain $(Phi_t)_t$ admits a unique (up to multiplicative constants) invariant measure. When the measure is finite the chain is necessarily positive recurrent, while in the opposite case, it is null recurrent.
answered Jan 3 at 8:07
Xi'anXi'an
54.6k692351
edited Jan 3 at 8:17
answered Jan 3 at 8:07
Xi'anXi'an
54.6k692351
answered Jan 3 at 8:07
Xi'anXi'an
54.6k692351
answered Jan 3 at 8:07
Xi'anXi'an
54.6k692351
54.6k692351
add a comment |
add a comment |
Thanks for contributing an answer to Cross Validated!
- 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%2fstats.stackexchange.com%2fquestions%2f385407%2fare-all-transient-states-positive-recurrent%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
