Mixing
Limit in distribution of increasing variance normal random variable
$begingroup$
Suppose you have a sequence of normal random variables $X_n sim N(0, n)$. Is there any random variable $X$ such that $X_n to X$ in distribution.
Here, I use the following definition: $X_n to X$ in distribution if $P(X_n leq x) to P(X leq x) equiv F_X(x)$ for every $x$ for which $F_x$ is continuous.
functional-analysis stochastic-calculus
asked Jan 21 at 18:53
BrisãoBrisão
1799
$endgroup$
add a comment |
$begingroup$
Suppose you have a sequence of normal random variables $X_n sim N(0, n)$. Is there any random variable $X$ such that $X_n to X$ in distribution.
Here, I use the following definition: $X_n to X$ in distribution if $P(X_n leq x) to P(X leq x) equiv F_X(x)$ for every $x$ for which $F_x$ is continuous.
functional-analysis stochastic-calculus
asked Jan 21 at 18:53
BrisãoBrisão
1799
$endgroup$
add a comment |
$begingroup$
Suppose you have a sequence of normal random variables $X_n sim N(0, n)$. Is there any random variable $X$ such that $X_n to X$ in distribution.
Here, I use the following definition: $X_n to X$ in distribution if $P(X_n leq x) to P(X leq x) equiv F_X(x)$ for every $x$ for which $F_x$ is continuous.
functional-analysis stochastic-calculus
asked Jan 21 at 18:53
BrisãoBrisão
1799
$endgroup$
Suppose you have a sequence of normal random variables $X_n sim N(0, n)$. Is there any random variable $X$ such that $X_n to X$ in distribution.
Here, I use the following definition: $X_n to X$ in distribution if $P(X_n leq x) to P(X leq x) equiv F_X(x)$ for every $x$ for which $F_x$ is continuous.
functional-analysis stochastic-calculus
functional-analysis stochastic-calculus
asked Jan 21 at 18:53
BrisãoBrisão
1799
asked Jan 21 at 18:53
BrisãoBrisão
1799
asked Jan 21 at 18:53
BrisãoBrisão
1799
asked Jan 21 at 18:53
BrisãoBrisão
1799
asked Jan 21 at 18:53
BrisãoBrisão
1799
1799
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
$X_n overset{d}{=} sqrt{n} Z$ where $Z sim N(0, 1)$ so
$$P(X_n le x) = P(Z le x / sqrt{n}) to frac{1}{2}.$$
Now show that no CDF can equal $1/2$ wherever it is continuous.
answered Jan 21 at 19:02
angryavianangryavian
42k23381
$endgroup$
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%2f3082245%2flimit-in-distribution-of-increasing-variance-normal-random-variable%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
$begingroup$
$X_n overset{d}{=} sqrt{n} Z$ where $Z sim N(0, 1)$ so
$$P(X_n le x) = P(Z le x / sqrt{n}) to frac{1}{2}.$$
Now show that no CDF can equal $1/2$ wherever it is continuous.
answered Jan 21 at 19:02
angryavianangryavian
42k23381
$endgroup$
add a comment |
$begingroup$
$X_n overset{d}{=} sqrt{n} Z$ where $Z sim N(0, 1)$ so
$$P(X_n le x) = P(Z le x / sqrt{n}) to frac{1}{2}.$$
Now show that no CDF can equal $1/2$ wherever it is continuous.
answered Jan 21 at 19:02
angryavianangryavian
42k23381
$endgroup$
add a comment |
$begingroup$
$X_n overset{d}{=} sqrt{n} Z$ where $Z sim N(0, 1)$ so
$$P(X_n le x) = P(Z le x / sqrt{n}) to frac{1}{2}.$$
Now show that no CDF can equal $1/2$ wherever it is continuous.
answered Jan 21 at 19:02
angryavianangryavian
42k23381
$endgroup$
$X_n overset{d}{=} sqrt{n} Z$ where $Z sim N(0, 1)$ so
$$P(X_n le x) = P(Z le x / sqrt{n}) to frac{1}{2}.$$
Now show that no CDF can equal $1/2$ wherever it is continuous.
answered Jan 21 at 19:02
angryavianangryavian
42k23381
answered Jan 21 at 19:02
angryavianangryavian
42k23381
answered Jan 21 at 19:02
angryavianangryavian
42k23381
answered Jan 21 at 19:02
angryavianangryavian
42k23381
42k23381
add a comment |
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%2f3082245%2flimit-in-distribution-of-increasing-variance-normal-random-variable%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
