Characteristic function and point masses
Let $X$ be some random variable with characteristic function $hat{X}$. Show that $X$ has no point masses if and only if
$$
lim_{Tto +infty}frac1{2T}int_{-T}^Tleftlvert hat{X}(t)rightrvert^2mathrm dt=0.
$$
I think I have to consider the probability $P(X − Y = 0)$ where $Y$ is independent of $X$ and $X=Y$ in distribution, but I don't know how proceed.
random-variables characteristic-functions
edited Dec 10 at 11:13
Davide Giraudo
125k16150259
asked Nov 28 at 17:54
Francesca Ballatore
186
add a comment |
Let $X$ be some random variable with characteristic function $hat{X}$. Show that $X$ has no point masses if and only if
$$
lim_{Tto +infty}frac1{2T}int_{-T}^Tleftlvert hat{X}(t)rightrvert^2mathrm dt=0.
$$
I think I have to consider the probability $P(X − Y = 0)$ where $Y$ is independent of $X$ and $X=Y$ in distribution, but I don't know how proceed.
random-variables characteristic-functions
edited Dec 10 at 11:13
Davide Giraudo
125k16150259
asked Nov 28 at 17:54
Francesca Ballatore
186
What is your definition of a point mass?
– Paul
Nov 28 at 17:58
A probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. So I think that the point masses that I have to consider in the exercise are those that have the probability exactly equal to some value.
– Francesca Ballatore
Nov 28 at 18:05
Did you try to work with the inversion formula for characteristic functions?
– Davide Giraudo
Dec 12 at 10:16
add a comment |
Let $X$ be some random variable with characteristic function $hat{X}$. Show that $X$ has no point masses if and only if
$$
lim_{Tto +infty}frac1{2T}int_{-T}^Tleftlvert hat{X}(t)rightrvert^2mathrm dt=0.
$$
I think I have to consider the probability $P(X − Y = 0)$ where $Y$ is independent of $X$ and $X=Y$ in distribution, but I don't know how proceed.
random-variables characteristic-functions
edited Dec 10 at 11:13
Davide Giraudo
125k16150259
asked Nov 28 at 17:54
Francesca Ballatore
186
Let $X$ be some random variable with characteristic function $hat{X}$. Show that $X$ has no point masses if and only if
$$
lim_{Tto +infty}frac1{2T}int_{-T}^Tleftlvert hat{X}(t)rightrvert^2mathrm dt=0.
$$
I think I have to consider the probability $P(X − Y = 0)$ where $Y$ is independent of $X$ and $X=Y$ in distribution, but I don't know how proceed.
random-variables characteristic-functions
random-variables characteristic-functions
edited Dec 10 at 11:13
Davide Giraudo
125k16150259
asked Nov 28 at 17:54
Francesca Ballatore
186
edited Dec 10 at 11:13
Davide Giraudo
125k16150259
asked Nov 28 at 17:54
Francesca Ballatore
186
edited Dec 10 at 11:13
Davide Giraudo
125k16150259
edited Dec 10 at 11:13
Davide Giraudo
125k16150259
edited Dec 10 at 11:13
Davide Giraudo
125k16150259
125k16150259
asked Nov 28 at 17:54
Francesca Ballatore
186
asked Nov 28 at 17:54
Francesca Ballatore
186
asked Nov 28 at 17:54
Francesca Ballatore
186
186
What is your definition of a point mass?
– Paul
Nov 28 at 17:58
A probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. So I think that the point masses that I have to consider in the exercise are those that have the probability exactly equal to some value.
– Francesca Ballatore
Nov 28 at 18:05
Did you try to work with the inversion formula for characteristic functions?
– Davide Giraudo
Dec 12 at 10:16
add a comment |
What is your definition of a point mass?
– Paul
Nov 28 at 17:58
A probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. So I think that the point masses that I have to consider in the exercise are those that have the probability exactly equal to some value.
– Francesca Ballatore
Nov 28 at 18:05
Did you try to work with the inversion formula for characteristic functions?
– Davide Giraudo
Dec 12 at 10:16
What is your definition of a point mass?
– Paul
Nov 28 at 17:58
What is your definition of a point mass?
– Paul
Nov 28 at 17:58
A probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. So I think that the point masses that I have to consider in the exercise are those that have the probability exactly equal to some value.
– Francesca Ballatore
Nov 28 at 18:05
A probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. So I think that the point masses that I have to consider in the exercise are those that have the probability exactly equal to some value.
– Francesca Ballatore
Nov 28 at 18:05
Did you try to work with the inversion formula for characteristic functions?
– Davide Giraudo
Dec 12 at 10:16
Did you try to work with the inversion formula for characteristic functions?
– Davide Giraudo
Dec 12 at 10:16
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%2f3017469%2fcharacteristic-function-and-point-masses%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%2f3017469%2fcharacteristic-function-and-point-masses%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
What is your definition of a point mass?
– Paul
Nov 28 at 17:58
A probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. So I think that the point masses that I have to consider in the exercise are those that have the probability exactly equal to some value.
– Francesca Ballatore
Nov 28 at 18:05
Did you try to work with the inversion formula for characteristic functions?
– Davide Giraudo
Dec 12 at 10:16