Higher Dimensional Random Walks
$begingroup$
I've read on a paper that, in the two dimensional case, if you start from the origin and take steps of length one in an arbitrary direction (uniformely on the unit sphere $S^1$), the expected distance after $n$ steps from the starting point is approximated by $sqrt{npi}/2$.
(source: https://pdfs.semanticscholar.org/6685/1166d588821456477f2007a37bf0428a2cf2.pdf).
I was wondering if there was a similar formula for higher dimensional random walks, which means:
Starting from the origin in $mathbb{R}^d$, if i set $v_i = v_{i-1} + mu$, where $mu in S^{d-1}$ picked at random at every step and $v_0 = 0$, what is the expected value of $|v_n|$? e.g. how distant is the point from the origin after having taken $n$ steps in random directions?
I don't need a precise formula, everything that just gives an idea on how large the expected distance is works just fine.
(if you could add a reference to the answer as well it would be great! :D )
Thank you very much!
probability approximation random-walk probability-limit-theorems
asked Jan 10 at 23:05
AlfredAlfred
335
$endgroup$
add a comment |
$begingroup$
I've read on a paper that, in the two dimensional case, if you start from the origin and take steps of length one in an arbitrary direction (uniformely on the unit sphere $S^1$), the expected distance after $n$ steps from the starting point is approximated by $sqrt{npi}/2$.
(source: https://pdfs.semanticscholar.org/6685/1166d588821456477f2007a37bf0428a2cf2.pdf).
I was wondering if there was a similar formula for higher dimensional random walks, which means:
Starting from the origin in $mathbb{R}^d$, if i set $v_i = v_{i-1} + mu$, where $mu in S^{d-1}$ picked at random at every step and $v_0 = 0$, what is the expected value of $|v_n|$? e.g. how distant is the point from the origin after having taken $n$ steps in random directions?
I don't need a precise formula, everything that just gives an idea on how large the expected distance is works just fine.
(if you could add a reference to the answer as well it would be great! :D )
Thank you very much!
probability approximation random-walk probability-limit-theorems
asked Jan 10 at 23:05
AlfredAlfred
335
$endgroup$
add a comment |
$begingroup$
I've read on a paper that, in the two dimensional case, if you start from the origin and take steps of length one in an arbitrary direction (uniformely on the unit sphere $S^1$), the expected distance after $n$ steps from the starting point is approximated by $sqrt{npi}/2$.
(source: https://pdfs.semanticscholar.org/6685/1166d588821456477f2007a37bf0428a2cf2.pdf).
I was wondering if there was a similar formula for higher dimensional random walks, which means:
Starting from the origin in $mathbb{R}^d$, if i set $v_i = v_{i-1} + mu$, where $mu in S^{d-1}$ picked at random at every step and $v_0 = 0$, what is the expected value of $|v_n|$? e.g. how distant is the point from the origin after having taken $n$ steps in random directions?
I don't need a precise formula, everything that just gives an idea on how large the expected distance is works just fine.
(if you could add a reference to the answer as well it would be great! :D )
Thank you very much!
probability approximation random-walk probability-limit-theorems
asked Jan 10 at 23:05
AlfredAlfred
335
$endgroup$
I've read on a paper that, in the two dimensional case, if you start from the origin and take steps of length one in an arbitrary direction (uniformely on the unit sphere $S^1$), the expected distance after $n$ steps from the starting point is approximated by $sqrt{npi}/2$.
(source: https://pdfs.semanticscholar.org/6685/1166d588821456477f2007a37bf0428a2cf2.pdf).
I was wondering if there was a similar formula for higher dimensional random walks, which means:
Starting from the origin in $mathbb{R}^d$, if i set $v_i = v_{i-1} + mu$, where $mu in S^{d-1}$ picked at random at every step and $v_0 = 0$, what is the expected value of $|v_n|$? e.g. how distant is the point from the origin after having taken $n$ steps in random directions?
I don't need a precise formula, everything that just gives an idea on how large the expected distance is works just fine.
(if you could add a reference to the answer as well it would be great! :D )
Thank you very much!
probability approximation random-walk probability-limit-theorems
probability approximation random-walk probability-limit-theorems
asked Jan 10 at 23:05
AlfredAlfred
335
asked Jan 10 at 23:05
AlfredAlfred
335
asked Jan 10 at 23:05
AlfredAlfred
335
asked Jan 10 at 23:05
AlfredAlfred
335
asked Jan 10 at 23:05
AlfredAlfred
335
335
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
As described here:
$${displaystyle P(r)={frac {2r}{N}}e^{-r^{2}/N}}$$
answered Jan 10 at 23:08
David G. StorkDavid G. Stork
10.7k31332
$endgroup$
$begingroup$
Your answer is just for the case where the steps are all orthogonal between them ( 6 directions: (1,0,0)(0,1,0)(0,0,1) and the negatives for $R^3$ ). I'm looking for the general case, where the steps are allowed to be taken in any random direction. Thank you tho!
$endgroup$
– Alfred
Jan 10 at 23:17
$begingroup$
Actually, if the formula you posted is an upper bound of the one I need, would be great as well, but is it the case?
$endgroup$
– Alfred
Jan 10 at 23:20
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: "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%2f3069286%2fhigher-dimensional-random-walks%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
$begingroup$
As described here:
$${displaystyle P(r)={frac {2r}{N}}e^{-r^{2}/N}}$$
answered Jan 10 at 23:08
David G. StorkDavid G. Stork
10.7k31332
$endgroup$
$begingroup$
Your answer is just for the case where the steps are all orthogonal between them ( 6 directions: (1,0,0)(0,1,0)(0,0,1) and the negatives for $R^3$ ). I'm looking for the general case, where the steps are allowed to be taken in any random direction. Thank you tho!
$endgroup$
– Alfred
Jan 10 at 23:17
$begingroup$
Actually, if the formula you posted is an upper bound of the one I need, would be great as well, but is it the case?
$endgroup$
– Alfred
Jan 10 at 23:20
add a comment |
$begingroup$
As described here:
$${displaystyle P(r)={frac {2r}{N}}e^{-r^{2}/N}}$$
answered Jan 10 at 23:08
David G. StorkDavid G. Stork
10.7k31332
$endgroup$
$begingroup$
Your answer is just for the case where the steps are all orthogonal between them ( 6 directions: (1,0,0)(0,1,0)(0,0,1) and the negatives for $R^3$ ). I'm looking for the general case, where the steps are allowed to be taken in any random direction. Thank you tho!
$endgroup$
– Alfred
Jan 10 at 23:17
$begingroup$
Actually, if the formula you posted is an upper bound of the one I need, would be great as well, but is it the case?
$endgroup$
– Alfred
Jan 10 at 23:20
add a comment |
$begingroup$
As described here:
$${displaystyle P(r)={frac {2r}{N}}e^{-r^{2}/N}}$$
answered Jan 10 at 23:08
David G. StorkDavid G. Stork
10.7k31332
$endgroup$
As described here:
$${displaystyle P(r)={frac {2r}{N}}e^{-r^{2}/N}}$$
answered Jan 10 at 23:08
David G. StorkDavid G. Stork
10.7k31332
answered Jan 10 at 23:08
David G. StorkDavid G. Stork
10.7k31332
answered Jan 10 at 23:08
David G. StorkDavid G. Stork
10.7k31332
answered Jan 10 at 23:08
David G. StorkDavid G. Stork
10.7k31332
10.7k31332
$begingroup$
Your answer is just for the case where the steps are all orthogonal between them ( 6 directions: (1,0,0)(0,1,0)(0,0,1) and the negatives for $R^3$ ). I'm looking for the general case, where the steps are allowed to be taken in any random direction. Thank you tho!
$endgroup$
– Alfred
Jan 10 at 23:17
$begingroup$
Actually, if the formula you posted is an upper bound of the one I need, would be great as well, but is it the case?
$endgroup$
– Alfred
Jan 10 at 23:20
add a comment |
$begingroup$
Your answer is just for the case where the steps are all orthogonal between them ( 6 directions: (1,0,0)(0,1,0)(0,0,1) and the negatives for $R^3$ ). I'm looking for the general case, where the steps are allowed to be taken in any random direction. Thank you tho!
$endgroup$
– Alfred
Jan 10 at 23:17
$begingroup$
Actually, if the formula you posted is an upper bound of the one I need, would be great as well, but is it the case?
$endgroup$
– Alfred
Jan 10 at 23:20
$begingroup$
Your answer is just for the case where the steps are all orthogonal between them ( 6 directions: (1,0,0)(0,1,0)(0,0,1) and the negatives for $R^3$ ). I'm looking for the general case, where the steps are allowed to be taken in any random direction. Thank you tho!
$endgroup$
– Alfred
Jan 10 at 23:17
$begingroup$
Your answer is just for the case where the steps are all orthogonal between them ( 6 directions: (1,0,0)(0,1,0)(0,0,1) and the negatives for $R^3$ ). I'm looking for the general case, where the steps are allowed to be taken in any random direction. Thank you tho!
$endgroup$
– Alfred
Jan 10 at 23:17
$begingroup$
Actually, if the formula you posted is an upper bound of the one I need, would be great as well, but is it the case?
$endgroup$
– Alfred
Jan 10 at 23:20
$begingroup$
Actually, if the formula you posted is an upper bound of the one I need, would be great as well, but is it the case?
$endgroup$
– Alfred
Jan 10 at 23:20
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%2f3069286%2fhigher-dimensional-random-walks%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
