Convergence of discrete-time Markov chain to Feller processes
Let
$(Omega,mathcal A,operatorname P)$ be a probability space
$(X_t)_{tge0}$ be a Feller process on $(Omega,mathcal A,operatorname P)$
$(h_d)_{dinmathbb N}subseteq(0,infty)$ with $$h_dxrightarrow{ntoinfty}0$$
$left(Y^{(d)}_nright)_{ninmathbb N_0}$ be a time-homogeneous Markov chain on $(Omega,mathcal A,operatorname P)$ and $$X^{(d)}_t:=Y^{(d)}_{lfloorfrac t{h_d}rfloor};;;text{for }tge0$$ for $dinmathbb N$
$N$ be a Poisson process on $(Omega,mathcal A,operatorname P)$ with parameter $1$ independent of $Y^{(d)}$ for all $dinmathbb N$ and $$N^{(d)}_t:=N_{frac t{h_d}};;;text{for }tge0$$ as well as $$tilde X^{(d)}_t:=Y^{(d)}_{N^{(d)}_t};;;text{for }tge0$$ for $dinmathbb N$
Note that $N^{(d)}$ is a Poisson process with parameter $h_d^{-1}$ for all $dinmathbb N$.
How can we show that (in probability with respect to the Skorohod topology) $X^{(d)}xrightarrow{dtoinfty}X$ iff $tilde X^{(d)}xrightarrow{dtoinfty}X$?
In the book of Kallenberg, the author is mentioning that the claim follows from the following two theorems:
I don't get how we need to apply them. Clearly, for fixed $tge0$, we can consider $$frac1dsum_{i=1}^dleft(N^{(i)}_t-N^{(i-1)}_tright)$$ with $N^{(0)}_t:=0$. However, while independent, the $N^{(i)}_t-N^{(i-1)}$ are not identically distributed ...
If it's hard to prove in the general setting, it's okay for me to assume $h_d^{-1}=d$ for all $dinmathbb N$. In that case, the strong law of large numbers yields $$sup_{tin[0,:T]}left|frac1d N^{(d)}_t-tright|xrightarrow{dtoinfty}0;;;text{almost surely for all }T>0tag1.$$ Now, let $tau^{(d)}_0:=0$, $$tau_n^{(d)}:=infleft{t>tau^{(d)}_{n-1}:N^{(d)}_t-N^{(d)}_{tau^{(d)}_{n-1}}>0right};;;text{for }dinmathbb N$$ and $$lambda^{(d)}_t:=sum_{n=0}^infty1_{left[frac nd,:frac{n+1}dright)}(t)left(tau^{(d)}_n+(dt-n)left(tau^{(d)}_{n+1}-tau^{(d)}_nright)right);;;text{for }tge0$$ for $dinmathbb N$. Moreoverr, let $T>0$ and $rho_T$ denote the metric inducing the Skorohod $J_1$-topology on the space of càdlàg functions $[0,T]tomathbb R$. We should obtain $$rho_Tleft(X^{(d)},tilde X^{(d)}right)lesup_{tin[0,:T]}left|lambda^{(d)}_t-tright|+sup_{tin[0,:T]}left|X^{(d)}_t-tilde X^{(d)}_{lambda^{(d)}_t}right|tag2,$$ where the last term should be $0$. So, if we could show that the first term converges in probability to $0$ as $dtoinfty$, we should be able to conclude (since $T$ was arbitrary).
probability-theory stochastic-processes markov-chains markov-process stochastic-analysis
asked Nov 29 at 18:46
0xbadf00d
1,80341430
add a comment |
Let
$(Omega,mathcal A,operatorname P)$ be a probability space
$(X_t)_{tge0}$ be a Feller process on $(Omega,mathcal A,operatorname P)$
$(h_d)_{dinmathbb N}subseteq(0,infty)$ with $$h_dxrightarrow{ntoinfty}0$$
$left(Y^{(d)}_nright)_{ninmathbb N_0}$ be a time-homogeneous Markov chain on $(Omega,mathcal A,operatorname P)$ and $$X^{(d)}_t:=Y^{(d)}_{lfloorfrac t{h_d}rfloor};;;text{for }tge0$$ for $dinmathbb N$
$N$ be a Poisson process on $(Omega,mathcal A,operatorname P)$ with parameter $1$ independent of $Y^{(d)}$ for all $dinmathbb N$ and $$N^{(d)}_t:=N_{frac t{h_d}};;;text{for }tge0$$ as well as $$tilde X^{(d)}_t:=Y^{(d)}_{N^{(d)}_t};;;text{for }tge0$$ for $dinmathbb N$
Note that $N^{(d)}$ is a Poisson process with parameter $h_d^{-1}$ for all $dinmathbb N$.
How can we show that (in probability with respect to the Skorohod topology) $X^{(d)}xrightarrow{dtoinfty}X$ iff $tilde X^{(d)}xrightarrow{dtoinfty}X$?
In the book of Kallenberg, the author is mentioning that the claim follows from the following two theorems:
I don't get how we need to apply them. Clearly, for fixed $tge0$, we can consider $$frac1dsum_{i=1}^dleft(N^{(i)}_t-N^{(i-1)}_tright)$$ with $N^{(0)}_t:=0$. However, while independent, the $N^{(i)}_t-N^{(i-1)}$ are not identically distributed ...
If it's hard to prove in the general setting, it's okay for me to assume $h_d^{-1}=d$ for all $dinmathbb N$. In that case, the strong law of large numbers yields $$sup_{tin[0,:T]}left|frac1d N^{(d)}_t-tright|xrightarrow{dtoinfty}0;;;text{almost surely for all }T>0tag1.$$ Now, let $tau^{(d)}_0:=0$, $$tau_n^{(d)}:=infleft{t>tau^{(d)}_{n-1}:N^{(d)}_t-N^{(d)}_{tau^{(d)}_{n-1}}>0right};;;text{for }dinmathbb N$$ and $$lambda^{(d)}_t:=sum_{n=0}^infty1_{left[frac nd,:frac{n+1}dright)}(t)left(tau^{(d)}_n+(dt-n)left(tau^{(d)}_{n+1}-tau^{(d)}_nright)right);;;text{for }tge0$$ for $dinmathbb N$. Moreoverr, let $T>0$ and $rho_T$ denote the metric inducing the Skorohod $J_1$-topology on the space of càdlàg functions $[0,T]tomathbb R$. We should obtain $$rho_Tleft(X^{(d)},tilde X^{(d)}right)lesup_{tin[0,:T]}left|lambda^{(d)}_t-tright|+sup_{tin[0,:T]}left|X^{(d)}_t-tilde X^{(d)}_{lambda^{(d)}_t}right|tag2,$$ where the last term should be $0$. So, if we could show that the first term converges in probability to $0$ as $dtoinfty$, we should be able to conclude (since $T$ was arbitrary).
probability-theory stochastic-processes markov-chains markov-process stochastic-analysis
asked Nov 29 at 18:46
0xbadf00d
1,80341430
add a comment |
Let
$(Omega,mathcal A,operatorname P)$ be a probability space
$(X_t)_{tge0}$ be a Feller process on $(Omega,mathcal A,operatorname P)$
$(h_d)_{dinmathbb N}subseteq(0,infty)$ with $$h_dxrightarrow{ntoinfty}0$$
$left(Y^{(d)}_nright)_{ninmathbb N_0}$ be a time-homogeneous Markov chain on $(Omega,mathcal A,operatorname P)$ and $$X^{(d)}_t:=Y^{(d)}_{lfloorfrac t{h_d}rfloor};;;text{for }tge0$$ for $dinmathbb N$
$N$ be a Poisson process on $(Omega,mathcal A,operatorname P)$ with parameter $1$ independent of $Y^{(d)}$ for all $dinmathbb N$ and $$N^{(d)}_t:=N_{frac t{h_d}};;;text{for }tge0$$ as well as $$tilde X^{(d)}_t:=Y^{(d)}_{N^{(d)}_t};;;text{for }tge0$$ for $dinmathbb N$
Note that $N^{(d)}$ is a Poisson process with parameter $h_d^{-1}$ for all $dinmathbb N$.
How can we show that (in probability with respect to the Skorohod topology) $X^{(d)}xrightarrow{dtoinfty}X$ iff $tilde X^{(d)}xrightarrow{dtoinfty}X$?
In the book of Kallenberg, the author is mentioning that the claim follows from the following two theorems:
I don't get how we need to apply them. Clearly, for fixed $tge0$, we can consider $$frac1dsum_{i=1}^dleft(N^{(i)}_t-N^{(i-1)}_tright)$$ with $N^{(0)}_t:=0$. However, while independent, the $N^{(i)}_t-N^{(i-1)}$ are not identically distributed ...
If it's hard to prove in the general setting, it's okay for me to assume $h_d^{-1}=d$ for all $dinmathbb N$. In that case, the strong law of large numbers yields $$sup_{tin[0,:T]}left|frac1d N^{(d)}_t-tright|xrightarrow{dtoinfty}0;;;text{almost surely for all }T>0tag1.$$ Now, let $tau^{(d)}_0:=0$, $$tau_n^{(d)}:=infleft{t>tau^{(d)}_{n-1}:N^{(d)}_t-N^{(d)}_{tau^{(d)}_{n-1}}>0right};;;text{for }dinmathbb N$$ and $$lambda^{(d)}_t:=sum_{n=0}^infty1_{left[frac nd,:frac{n+1}dright)}(t)left(tau^{(d)}_n+(dt-n)left(tau^{(d)}_{n+1}-tau^{(d)}_nright)right);;;text{for }tge0$$ for $dinmathbb N$. Moreoverr, let $T>0$ and $rho_T$ denote the metric inducing the Skorohod $J_1$-topology on the space of càdlàg functions $[0,T]tomathbb R$. We should obtain $$rho_Tleft(X^{(d)},tilde X^{(d)}right)lesup_{tin[0,:T]}left|lambda^{(d)}_t-tright|+sup_{tin[0,:T]}left|X^{(d)}_t-tilde X^{(d)}_{lambda^{(d)}_t}right|tag2,$$ where the last term should be $0$. So, if we could show that the first term converges in probability to $0$ as $dtoinfty$, we should be able to conclude (since $T$ was arbitrary).
probability-theory stochastic-processes markov-chains markov-process stochastic-analysis
asked Nov 29 at 18:46
0xbadf00d
1,80341430
Let
$(Omega,mathcal A,operatorname P)$ be a probability space
$(X_t)_{tge0}$ be a Feller process on $(Omega,mathcal A,operatorname P)$
$(h_d)_{dinmathbb N}subseteq(0,infty)$ with $$h_dxrightarrow{ntoinfty}0$$
$left(Y^{(d)}_nright)_{ninmathbb N_0}$ be a time-homogeneous Markov chain on $(Omega,mathcal A,operatorname P)$ and $$X^{(d)}_t:=Y^{(d)}_{lfloorfrac t{h_d}rfloor};;;text{for }tge0$$ for $dinmathbb N$
$N$ be a Poisson process on $(Omega,mathcal A,operatorname P)$ with parameter $1$ independent of $Y^{(d)}$ for all $dinmathbb N$ and $$N^{(d)}_t:=N_{frac t{h_d}};;;text{for }tge0$$ as well as $$tilde X^{(d)}_t:=Y^{(d)}_{N^{(d)}_t};;;text{for }tge0$$ for $dinmathbb N$
Note that $N^{(d)}$ is a Poisson process with parameter $h_d^{-1}$ for all $dinmathbb N$.
How can we show that (in probability with respect to the Skorohod topology) $X^{(d)}xrightarrow{dtoinfty}X$ iff $tilde X^{(d)}xrightarrow{dtoinfty}X$?
In the book of Kallenberg, the author is mentioning that the claim follows from the following two theorems:
I don't get how we need to apply them. Clearly, for fixed $tge0$, we can consider $$frac1dsum_{i=1}^dleft(N^{(i)}_t-N^{(i-1)}_tright)$$ with $N^{(0)}_t:=0$. However, while independent, the $N^{(i)}_t-N^{(i-1)}$ are not identically distributed ...
If it's hard to prove in the general setting, it's okay for me to assume $h_d^{-1}=d$ for all $dinmathbb N$. In that case, the strong law of large numbers yields $$sup_{tin[0,:T]}left|frac1d N^{(d)}_t-tright|xrightarrow{dtoinfty}0;;;text{almost surely for all }T>0tag1.$$ Now, let $tau^{(d)}_0:=0$, $$tau_n^{(d)}:=infleft{t>tau^{(d)}_{n-1}:N^{(d)}_t-N^{(d)}_{tau^{(d)}_{n-1}}>0right};;;text{for }dinmathbb N$$ and $$lambda^{(d)}_t:=sum_{n=0}^infty1_{left[frac nd,:frac{n+1}dright)}(t)left(tau^{(d)}_n+(dt-n)left(tau^{(d)}_{n+1}-tau^{(d)}_nright)right);;;text{for }tge0$$ for $dinmathbb N$. Moreoverr, let $T>0$ and $rho_T$ denote the metric inducing the Skorohod $J_1$-topology on the space of càdlàg functions $[0,T]tomathbb R$. We should obtain $$rho_Tleft(X^{(d)},tilde X^{(d)}right)lesup_{tin[0,:T]}left|lambda^{(d)}_t-tright|+sup_{tin[0,:T]}left|X^{(d)}_t-tilde X^{(d)}_{lambda^{(d)}_t}right|tag2,$$ where the last term should be $0$. So, if we could show that the first term converges in probability to $0$ as $dtoinfty$, we should be able to conclude (since $T$ was arbitrary).
probability-theory stochastic-processes markov-chains markov-process stochastic-analysis
probability-theory stochastic-processes markov-chains markov-process stochastic-analysis
asked Nov 29 at 18:46
0xbadf00d
1,80341430
asked Nov 29 at 18:46
0xbadf00d
1,80341430
edited Dec 4 at 14:22
asked Nov 29 at 18:46
0xbadf00d
1,80341430
asked Nov 29 at 18:46
0xbadf00d
1,80341430
asked Nov 29 at 18:46
0xbadf00d
1,80341430
1,80341430
add a comment |
add a comment |
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%2f3019020%2fconvergence-of-discrete-time-markov-chain-to-feller-processes%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
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%2f3019020%2fconvergence-of-discrete-time-markov-chain-to-feller-processes%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
