Birkhoff average of $x mapsto x+1$ in $mathbb R$ with $L^p$ observable
Let $f in mathcal L^p(mathbb R, lambda)$, where $lambda$ is Lebesgue measure and $p in (1,infty)$. And let $T : mathbb R to mathbb R$ be the map $T(x) = x+1$. I want to show:
$$
frac 1 n sum_{k=0}^{n-1} f circ T^k xrightarrow{L^p} 0 quad textrm{as } n to infty.
$$
This exercise is coming from a probability theory textbook, so I want to avoid advanced tools from ergodic theory. I can show the $n$th Birkhoff average is in $mathcal L^p$, but I'm having trouble proving anything about this sequence of $mathcal L^p$ functions (Cauchy, a.e. convergent, etc.). Any suggestions?
real-analysis probability-theory dynamical-systems ergodic-theory
asked Dec 27 '18 at 0:10
D Ford
549213
add a comment |
Let $f in mathcal L^p(mathbb R, lambda)$, where $lambda$ is Lebesgue measure and $p in (1,infty)$. And let $T : mathbb R to mathbb R$ be the map $T(x) = x+1$. I want to show:
$$
frac 1 n sum_{k=0}^{n-1} f circ T^k xrightarrow{L^p} 0 quad textrm{as } n to infty.
$$
This exercise is coming from a probability theory textbook, so I want to avoid advanced tools from ergodic theory. I can show the $n$th Birkhoff average is in $mathcal L^p$, but I'm having trouble proving anything about this sequence of $mathcal L^p$ functions (Cauchy, a.e. convergent, etc.). Any suggestions?
real-analysis probability-theory dynamical-systems ergodic-theory
asked Dec 27 '18 at 0:10
D Ford
549213
add a comment |
Let $f in mathcal L^p(mathbb R, lambda)$, where $lambda$ is Lebesgue measure and $p in (1,infty)$. And let $T : mathbb R to mathbb R$ be the map $T(x) = x+1$. I want to show:
$$
frac 1 n sum_{k=0}^{n-1} f circ T^k xrightarrow{L^p} 0 quad textrm{as } n to infty.
$$
This exercise is coming from a probability theory textbook, so I want to avoid advanced tools from ergodic theory. I can show the $n$th Birkhoff average is in $mathcal L^p$, but I'm having trouble proving anything about this sequence of $mathcal L^p$ functions (Cauchy, a.e. convergent, etc.). Any suggestions?
real-analysis probability-theory dynamical-systems ergodic-theory
asked Dec 27 '18 at 0:10
D Ford
549213
Let $f in mathcal L^p(mathbb R, lambda)$, where $lambda$ is Lebesgue measure and $p in (1,infty)$. And let $T : mathbb R to mathbb R$ be the map $T(x) = x+1$. I want to show:
$$
frac 1 n sum_{k=0}^{n-1} f circ T^k xrightarrow{L^p} 0 quad textrm{as } n to infty.
$$
This exercise is coming from a probability theory textbook, so I want to avoid advanced tools from ergodic theory. I can show the $n$th Birkhoff average is in $mathcal L^p$, but I'm having trouble proving anything about this sequence of $mathcal L^p$ functions (Cauchy, a.e. convergent, etc.). Any suggestions?
real-analysis probability-theory dynamical-systems ergodic-theory
real-analysis probability-theory dynamical-systems ergodic-theory
asked Dec 27 '18 at 0:10
D Ford
549213
asked Dec 27 '18 at 0:10
D Ford
549213
asked Dec 27 '18 at 0:10
D Ford
549213
asked Dec 27 '18 at 0:10
D Ford
549213
asked Dec 27 '18 at 0:10
D Ford
549213
549213
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
First note the result is obvious if $f$ is compactly supported. Then, since compactly supported functions are dense, we are done if we can get a bound of the form $||frac{1}{N}sum_{n le N} f(cdot+n)||_p le C ||f||_p$ with $C$ independent of $f$ and $N$. But we can get $C=1$ by triangle inequality.
answered Dec 27 '18 at 0:18
mathworker21
8,6371928
Thanks for the tip! I might try using the simple function argument without using Minkowski's inequality, only because that hasn't appeared yet in the textbook I'm referencing. Fingers crossed.
– D Ford
Dec 27 '18 at 0:29
@DFord do you mean proving the result is true for $f$ simple?
– mathworker21
Dec 27 '18 at 0:31
@DFord if so, I edited my answer to make life easier for you
– mathworker21
Dec 27 '18 at 0:32
Well, the argument via simple functions or compactly supported ones still uses the $L^p$ triangle inequality, a.k.a. Minkowski's inequality. Since this theorem isn't proven in this textbook until the following section, I'd like to avoid it if possible.
– D Ford
Dec 27 '18 at 4:10
@DFord the textbook hasn't proven that $||cdot||_p$ is a norm?? also, you should be precise with language. you should say "the argument showing that the result for simple functions implies the result for all $L^p$ functions" rather than just saying "the argument via simple functions"
– mathworker21
Dec 27 '18 at 4:23
|
show 2 more comments
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%2f3053452%2fbirkhoff-average-of-x-mapsto-x1-in-mathbb-r-with-lp-observable%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
First note the result is obvious if $f$ is compactly supported. Then, since compactly supported functions are dense, we are done if we can get a bound of the form $||frac{1}{N}sum_{n le N} f(cdot+n)||_p le C ||f||_p$ with $C$ independent of $f$ and $N$. But we can get $C=1$ by triangle inequality.
answered Dec 27 '18 at 0:18
mathworker21
8,6371928
Thanks for the tip! I might try using the simple function argument without using Minkowski's inequality, only because that hasn't appeared yet in the textbook I'm referencing. Fingers crossed.
– D Ford
Dec 27 '18 at 0:29
@DFord do you mean proving the result is true for $f$ simple?
– mathworker21
Dec 27 '18 at 0:31
@DFord if so, I edited my answer to make life easier for you
– mathworker21
Dec 27 '18 at 0:32
Well, the argument via simple functions or compactly supported ones still uses the $L^p$ triangle inequality, a.k.a. Minkowski's inequality. Since this theorem isn't proven in this textbook until the following section, I'd like to avoid it if possible.
– D Ford
Dec 27 '18 at 4:10
@DFord the textbook hasn't proven that $||cdot||_p$ is a norm?? also, you should be precise with language. you should say "the argument showing that the result for simple functions implies the result for all $L^p$ functions" rather than just saying "the argument via simple functions"
– mathworker21
Dec 27 '18 at 4:23
|
show 2 more comments
First note the result is obvious if $f$ is compactly supported. Then, since compactly supported functions are dense, we are done if we can get a bound of the form $||frac{1}{N}sum_{n le N} f(cdot+n)||_p le C ||f||_p$ with $C$ independent of $f$ and $N$. But we can get $C=1$ by triangle inequality.
answered Dec 27 '18 at 0:18
mathworker21
8,6371928
Thanks for the tip! I might try using the simple function argument without using Minkowski's inequality, only because that hasn't appeared yet in the textbook I'm referencing. Fingers crossed.
– D Ford
Dec 27 '18 at 0:29
@DFord do you mean proving the result is true for $f$ simple?
– mathworker21
Dec 27 '18 at 0:31
@DFord if so, I edited my answer to make life easier for you
– mathworker21
Dec 27 '18 at 0:32
Well, the argument via simple functions or compactly supported ones still uses the $L^p$ triangle inequality, a.k.a. Minkowski's inequality. Since this theorem isn't proven in this textbook until the following section, I'd like to avoid it if possible.
– D Ford
Dec 27 '18 at 4:10
@DFord the textbook hasn't proven that $||cdot||_p$ is a norm?? also, you should be precise with language. you should say "the argument showing that the result for simple functions implies the result for all $L^p$ functions" rather than just saying "the argument via simple functions"
– mathworker21
Dec 27 '18 at 4:23
|
show 2 more comments
First note the result is obvious if $f$ is compactly supported. Then, since compactly supported functions are dense, we are done if we can get a bound of the form $||frac{1}{N}sum_{n le N} f(cdot+n)||_p le C ||f||_p$ with $C$ independent of $f$ and $N$. But we can get $C=1$ by triangle inequality.
answered Dec 27 '18 at 0:18
mathworker21
8,6371928
First note the result is obvious if $f$ is compactly supported. Then, since compactly supported functions are dense, we are done if we can get a bound of the form $||frac{1}{N}sum_{n le N} f(cdot+n)||_p le C ||f||_p$ with $C$ independent of $f$ and $N$. But we can get $C=1$ by triangle inequality.
answered Dec 27 '18 at 0:18
mathworker21
8,6371928
edited Dec 27 '18 at 0:32
answered Dec 27 '18 at 0:18
mathworker21
8,6371928
answered Dec 27 '18 at 0:18
mathworker21
8,6371928
answered Dec 27 '18 at 0:18
mathworker21
8,6371928
8,6371928
Thanks for the tip! I might try using the simple function argument without using Minkowski's inequality, only because that hasn't appeared yet in the textbook I'm referencing. Fingers crossed.
– D Ford
Dec 27 '18 at 0:29
@DFord do you mean proving the result is true for $f$ simple?
– mathworker21
Dec 27 '18 at 0:31
@DFord if so, I edited my answer to make life easier for you
– mathworker21
Dec 27 '18 at 0:32
Well, the argument via simple functions or compactly supported ones still uses the $L^p$ triangle inequality, a.k.a. Minkowski's inequality. Since this theorem isn't proven in this textbook until the following section, I'd like to avoid it if possible.
– D Ford
Dec 27 '18 at 4:10
@DFord the textbook hasn't proven that $||cdot||_p$ is a norm?? also, you should be precise with language. you should say "the argument showing that the result for simple functions implies the result for all $L^p$ functions" rather than just saying "the argument via simple functions"
– mathworker21
Dec 27 '18 at 4:23
|
show 2 more comments
Thanks for the tip! I might try using the simple function argument without using Minkowski's inequality, only because that hasn't appeared yet in the textbook I'm referencing. Fingers crossed.
– D Ford
Dec 27 '18 at 0:29
@DFord do you mean proving the result is true for $f$ simple?
– mathworker21
Dec 27 '18 at 0:31
@DFord if so, I edited my answer to make life easier for you
– mathworker21
Dec 27 '18 at 0:32
Well, the argument via simple functions or compactly supported ones still uses the $L^p$ triangle inequality, a.k.a. Minkowski's inequality. Since this theorem isn't proven in this textbook until the following section, I'd like to avoid it if possible.
– D Ford
Dec 27 '18 at 4:10
@DFord the textbook hasn't proven that $||cdot||_p$ is a norm?? also, you should be precise with language. you should say "the argument showing that the result for simple functions implies the result for all $L^p$ functions" rather than just saying "the argument via simple functions"
– mathworker21
Dec 27 '18 at 4:23
Thanks for the tip! I might try using the simple function argument without using Minkowski's inequality, only because that hasn't appeared yet in the textbook I'm referencing. Fingers crossed.
– D Ford
Dec 27 '18 at 0:29
Thanks for the tip! I might try using the simple function argument without using Minkowski's inequality, only because that hasn't appeared yet in the textbook I'm referencing. Fingers crossed.
– D Ford
Dec 27 '18 at 0:29
@DFord do you mean proving the result is true for $f$ simple?
– mathworker21
Dec 27 '18 at 0:31
@DFord do you mean proving the result is true for $f$ simple?
– mathworker21
Dec 27 '18 at 0:31
@DFord if so, I edited my answer to make life easier for you
– mathworker21
Dec 27 '18 at 0:32
@DFord if so, I edited my answer to make life easier for you
– mathworker21
Dec 27 '18 at 0:32
Well, the argument via simple functions or compactly supported ones still uses the $L^p$ triangle inequality, a.k.a. Minkowski's inequality. Since this theorem isn't proven in this textbook until the following section, I'd like to avoid it if possible.
– D Ford
Dec 27 '18 at 4:10
Well, the argument via simple functions or compactly supported ones still uses the $L^p$ triangle inequality, a.k.a. Minkowski's inequality. Since this theorem isn't proven in this textbook until the following section, I'd like to avoid it if possible.
– D Ford
Dec 27 '18 at 4:10
@DFord the textbook hasn't proven that $||cdot||_p$ is a norm?? also, you should be precise with language. you should say "the argument showing that the result for simple functions implies the result for all $L^p$ functions" rather than just saying "the argument via simple functions"
– mathworker21
Dec 27 '18 at 4:23
@DFord the textbook hasn't proven that $||cdot||_p$ is a norm?? also, you should be precise with language. you should say "the argument showing that the result for simple functions implies the result for all $L^p$ functions" rather than just saying "the argument via simple functions"
– mathworker21
Dec 27 '18 at 4:23
|
show 2 more comments
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%2f3053452%2fbirkhoff-average-of-x-mapsto-x1-in-mathbb-r-with-lp-observable%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
