Convergence of expectation of hitting time of a symmetric random walk
$begingroup$
Let ${X_{n}: ngeq1}$ be i.i.d random variables with the common distribution $mathbb{P}(X_n=1)=p$ and $mathbb{P}(X_n=-1)=q=1-p$ where $0<p<1$. Define $S_0:=0$ and $S_n = sum_{j=1}^{n} X_j$ for $ngeq 1$. Then ${S_n:ngeq0}$ is a $(p-q)random;walk$ on $mathbb{Z}$ (When $p=q=frac{1}{2};,;{S_n:ngeq0};is;a;symmetric;random;walk$). Given two positive integers a and b, consider
$$tau:=inf{ngeq1:S_n=-a;or;S_n=b}$$
I want to show that $mathbb{E}[tau]<infty$.
What I know and I can use is that
Given a filtered space $left(S, mathcal{Sigma}, {mathcal{Sigma_n: nge 0}}, mathbb{P}right)$, and a stopping time $tau: S to {0, 1,2, ldots}$ with respect to ${mathcal{Sigma_n: nge 0}}$, suppose that there exist $minmathbb{N}$ and $epsilon$ such that, for every $nge 0$:
$$mathbb{E}left[mathbb{I}_{tau le n+m} | mathcal{Sigma_n}right] > epsilon quad text{a.s.}.$$
then $mathbb{E}[tau]<infty$.
And I also know that it's sufficient to show that there exist $minmathbb{N}$ and $epsilon$ such that, for every $nge 1$ and every $A={S_1=a_1,cdots , S_n=a_n}$ where $a_j in mathbb{Z}$ for $1leq j leq n$:
$$mathbb{P}(A cap{tauleq n+m})geq epsilon mathbb{P}(A)$$.
I'm trying to build a link between these two. Any hint is appreciated.
probability-theory random-walk stopping-times expected-value
asked Dec 3 '18 at 21:14
Weak NullstellensatzWeak Nullstellensatz
213
$endgroup$
add a comment |
$begingroup$
Let ${X_{n}: ngeq1}$ be i.i.d random variables with the common distribution $mathbb{P}(X_n=1)=p$ and $mathbb{P}(X_n=-1)=q=1-p$ where $0<p<1$. Define $S_0:=0$ and $S_n = sum_{j=1}^{n} X_j$ for $ngeq 1$. Then ${S_n:ngeq0}$ is a $(p-q)random;walk$ on $mathbb{Z}$ (When $p=q=frac{1}{2};,;{S_n:ngeq0};is;a;symmetric;random;walk$). Given two positive integers a and b, consider
$$tau:=inf{ngeq1:S_n=-a;or;S_n=b}$$
I want to show that $mathbb{E}[tau]<infty$.
What I know and I can use is that
Given a filtered space $left(S, mathcal{Sigma}, {mathcal{Sigma_n: nge 0}}, mathbb{P}right)$, and a stopping time $tau: S to {0, 1,2, ldots}$ with respect to ${mathcal{Sigma_n: nge 0}}$, suppose that there exist $minmathbb{N}$ and $epsilon$ such that, for every $nge 0$:
$$mathbb{E}left[mathbb{I}_{tau le n+m} | mathcal{Sigma_n}right] > epsilon quad text{a.s.}.$$
then $mathbb{E}[tau]<infty$.
And I also know that it's sufficient to show that there exist $minmathbb{N}$ and $epsilon$ such that, for every $nge 1$ and every $A={S_1=a_1,cdots , S_n=a_n}$ where $a_j in mathbb{Z}$ for $1leq j leq n$:
$$mathbb{P}(A cap{tauleq n+m})geq epsilon mathbb{P}(A)$$.
I'm trying to build a link between these two. Any hint is appreciated.
probability-theory random-walk stopping-times expected-value
asked Dec 3 '18 at 21:14
Weak NullstellensatzWeak Nullstellensatz
213
$endgroup$
$begingroup$
What's wrong about using martingale techniques? It's much easier... moreover, it is very likely that you are not the first to ask this question, so you might want to use the search
$endgroup$
– saz
Dec 3 '18 at 21:25
add a comment |
$begingroup$
Let ${X_{n}: ngeq1}$ be i.i.d random variables with the common distribution $mathbb{P}(X_n=1)=p$ and $mathbb{P}(X_n=-1)=q=1-p$ where $0<p<1$. Define $S_0:=0$ and $S_n = sum_{j=1}^{n} X_j$ for $ngeq 1$. Then ${S_n:ngeq0}$ is a $(p-q)random;walk$ on $mathbb{Z}$ (When $p=q=frac{1}{2};,;{S_n:ngeq0};is;a;symmetric;random;walk$). Given two positive integers a and b, consider
$$tau:=inf{ngeq1:S_n=-a;or;S_n=b}$$
I want to show that $mathbb{E}[tau]<infty$.
What I know and I can use is that
Given a filtered space $left(S, mathcal{Sigma}, {mathcal{Sigma_n: nge 0}}, mathbb{P}right)$, and a stopping time $tau: S to {0, 1,2, ldots}$ with respect to ${mathcal{Sigma_n: nge 0}}$, suppose that there exist $minmathbb{N}$ and $epsilon$ such that, for every $nge 0$:
$$mathbb{E}left[mathbb{I}_{tau le n+m} | mathcal{Sigma_n}right] > epsilon quad text{a.s.}.$$
then $mathbb{E}[tau]<infty$.
And I also know that it's sufficient to show that there exist $minmathbb{N}$ and $epsilon$ such that, for every $nge 1$ and every $A={S_1=a_1,cdots , S_n=a_n}$ where $a_j in mathbb{Z}$ for $1leq j leq n$:
$$mathbb{P}(A cap{tauleq n+m})geq epsilon mathbb{P}(A)$$.
I'm trying to build a link between these two. Any hint is appreciated.
probability-theory random-walk stopping-times expected-value
asked Dec 3 '18 at 21:14
Weak NullstellensatzWeak Nullstellensatz
213
$endgroup$
Let ${X_{n}: ngeq1}$ be i.i.d random variables with the common distribution $mathbb{P}(X_n=1)=p$ and $mathbb{P}(X_n=-1)=q=1-p$ where $0<p<1$. Define $S_0:=0$ and $S_n = sum_{j=1}^{n} X_j$ for $ngeq 1$. Then ${S_n:ngeq0}$ is a $(p-q)random;walk$ on $mathbb{Z}$ (When $p=q=frac{1}{2};,;{S_n:ngeq0};is;a;symmetric;random;walk$). Given two positive integers a and b, consider
$$tau:=inf{ngeq1:S_n=-a;or;S_n=b}$$
I want to show that $mathbb{E}[tau]<infty$.
What I know and I can use is that
Given a filtered space $left(S, mathcal{Sigma}, {mathcal{Sigma_n: nge 0}}, mathbb{P}right)$, and a stopping time $tau: S to {0, 1,2, ldots}$ with respect to ${mathcal{Sigma_n: nge 0}}$, suppose that there exist $minmathbb{N}$ and $epsilon$ such that, for every $nge 0$:
$$mathbb{E}left[mathbb{I}_{tau le n+m} | mathcal{Sigma_n}right] > epsilon quad text{a.s.}.$$
then $mathbb{E}[tau]<infty$.
And I also know that it's sufficient to show that there exist $minmathbb{N}$ and $epsilon$ such that, for every $nge 1$ and every $A={S_1=a_1,cdots , S_n=a_n}$ where $a_j in mathbb{Z}$ for $1leq j leq n$:
$$mathbb{P}(A cap{tauleq n+m})geq epsilon mathbb{P}(A)$$.
I'm trying to build a link between these two. Any hint is appreciated.
probability-theory random-walk stopping-times expected-value
probability-theory random-walk stopping-times expected-value
asked Dec 3 '18 at 21:14
Weak NullstellensatzWeak Nullstellensatz
213
asked Dec 3 '18 at 21:14
Weak NullstellensatzWeak Nullstellensatz
213
asked Dec 3 '18 at 21:14
Weak NullstellensatzWeak Nullstellensatz
213
asked Dec 3 '18 at 21:14
Weak NullstellensatzWeak Nullstellensatz
213
asked Dec 3 '18 at 21:14
Weak NullstellensatzWeak Nullstellensatz
213
213
$begingroup$
What's wrong about using martingale techniques? It's much easier... moreover, it is very likely that you are not the first to ask this question, so you might want to use the search
$endgroup$
– saz
Dec 3 '18 at 21:25
add a comment |
$begingroup$
What's wrong about using martingale techniques? It's much easier... moreover, it is very likely that you are not the first to ask this question, so you might want to use the search
$endgroup$
– saz
Dec 3 '18 at 21:25
$begingroup$
What's wrong about using martingale techniques? It's much easier... moreover, it is very likely that you are not the first to ask this question, so you might want to use the search
$endgroup$
– saz
Dec 3 '18 at 21:25
$begingroup$
What's wrong about using martingale techniques? It's much easier... moreover, it is very likely that you are not the first to ask this question, so you might want to use the search
$endgroup$
– saz
Dec 3 '18 at 21:25
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%2f3024685%2fconvergence-of-expectation-of-hitting-time-of-a-symmetric-random-walk%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%2f3024685%2fconvergence-of-expectation-of-hitting-time-of-a-symmetric-random-walk%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
$begingroup$
What's wrong about using martingale techniques? It's much easier... moreover, it is very likely that you are not the first to ask this question, so you might want to use the search
$endgroup$
– saz
Dec 3 '18 at 21:25