Elementary proof for convergence in probability for square of random variable
up vote
1
down vote
favorite
It is known from the continuous-mapping theorem that if $X_n to X$ in probability then $g(X_n) to g(X)$.
I wonder if there is an elementary proof for the case that $g(x) = x^2$, i.e. showing that $X^2_n to X^2$.
One approach is to use the fact that $X_n to X$ in probability if and only for even subsequence ${n_k}$ there is a subsubequence ${n_{k_j}}$ such that $X_{n_{k_j}} to X$ almost surely, so we can conclude that for this subsubsequence we have $X^2_{n_{k_j}} to X^2$ almost surely and so $X^2_n to X^2$ in probability.
Is there an even more elementary proof (using e.g. Markov, Cauchy-Schwartz , etc.) that directly shows that $mathbb{P}[|X_n^2 - X^2| > epsilon] to 0$?
probability-theory convergence random-variables
asked Nov 28 at 1:08
jesterII
1,19121226
add a comment |
up vote
1
down vote
favorite
It is known from the continuous-mapping theorem that if $X_n to X$ in probability then $g(X_n) to g(X)$.
I wonder if there is an elementary proof for the case that $g(x) = x^2$, i.e. showing that $X^2_n to X^2$.
One approach is to use the fact that $X_n to X$ in probability if and only for even subsequence ${n_k}$ there is a subsubequence ${n_{k_j}}$ such that $X_{n_{k_j}} to X$ almost surely, so we can conclude that for this subsubsequence we have $X^2_{n_{k_j}} to X^2$ almost surely and so $X^2_n to X^2$ in probability.
Is there an even more elementary proof (using e.g. Markov, Cauchy-Schwartz , etc.) that directly shows that $mathbb{P}[|X_n^2 - X^2| > epsilon] to 0$?
probability-theory convergence random-variables
asked Nov 28 at 1:08
jesterII
1,19121226
1
math.stackexchange.com/questions/339124/…
– d.k.o.
Nov 28 at 1:29
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
It is known from the continuous-mapping theorem that if $X_n to X$ in probability then $g(X_n) to g(X)$.
I wonder if there is an elementary proof for the case that $g(x) = x^2$, i.e. showing that $X^2_n to X^2$.
One approach is to use the fact that $X_n to X$ in probability if and only for even subsequence ${n_k}$ there is a subsubequence ${n_{k_j}}$ such that $X_{n_{k_j}} to X$ almost surely, so we can conclude that for this subsubsequence we have $X^2_{n_{k_j}} to X^2$ almost surely and so $X^2_n to X^2$ in probability.
Is there an even more elementary proof (using e.g. Markov, Cauchy-Schwartz , etc.) that directly shows that $mathbb{P}[|X_n^2 - X^2| > epsilon] to 0$?
probability-theory convergence random-variables
asked Nov 28 at 1:08
jesterII
1,19121226
It is known from the continuous-mapping theorem that if $X_n to X$ in probability then $g(X_n) to g(X)$.
I wonder if there is an elementary proof for the case that $g(x) = x^2$, i.e. showing that $X^2_n to X^2$.
One approach is to use the fact that $X_n to X$ in probability if and only for even subsequence ${n_k}$ there is a subsubequence ${n_{k_j}}$ such that $X_{n_{k_j}} to X$ almost surely, so we can conclude that for this subsubsequence we have $X^2_{n_{k_j}} to X^2$ almost surely and so $X^2_n to X^2$ in probability.
Is there an even more elementary proof (using e.g. Markov, Cauchy-Schwartz , etc.) that directly shows that $mathbb{P}[|X_n^2 - X^2| > epsilon] to 0$?
probability-theory convergence random-variables
probability-theory convergence random-variables
asked Nov 28 at 1:08
jesterII
1,19121226
asked Nov 28 at 1:08
jesterII
1,19121226
asked Nov 28 at 1:08
jesterII
1,19121226
asked Nov 28 at 1:08
jesterII
1,19121226
asked Nov 28 at 1:08
jesterII
1,19121226
1,19121226
1
math.stackexchange.com/questions/339124/…
– d.k.o.
Nov 28 at 1:29
add a comment |
1
math.stackexchange.com/questions/339124/…
– d.k.o.
Nov 28 at 1:29
1
1
math.stackexchange.com/questions/339124/…
– d.k.o.
Nov 28 at 1:29
math.stackexchange.com/questions/339124/…
– d.k.o.
Nov 28 at 1:29
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
Let $epsilon >0, eta >0$ ($epsilon <1$) and choose $M>1$ such that $P{|X| >M} <eta$. Then $|X_n-X| leq frac {epsilon} M$ and $|X| leq M$ together imply that $|X_n^{2}-X^{2}|=|X_n-X|(|X_n-X|+2|X|) leq frac {epsilon} M (1+2M)<3epsilon$. Hence $P{|X_n^{2}-X^{2}| >3epsilon} leq P{|X_n-X| > frac {epsilon} M}+P{|X| >M}$. For $n$ sufficiently large we get $P{|X_n^{2}-X^{2}| >3epsilon} <2eta$. This proves that $X_n^{2} to X^{2}$ in probability.
answered Nov 28 at 6:15
Kavi Rama Murthy
47.1k31854
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',
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%2f3016564%2felementary-proof-for-convergence-in-probability-for-square-of-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
up vote
1
down vote
accepted
Let $epsilon >0, eta >0$ ($epsilon <1$) and choose $M>1$ such that $P{|X| >M} <eta$. Then $|X_n-X| leq frac {epsilon} M$ and $|X| leq M$ together imply that $|X_n^{2}-X^{2}|=|X_n-X|(|X_n-X|+2|X|) leq frac {epsilon} M (1+2M)<3epsilon$. Hence $P{|X_n^{2}-X^{2}| >3epsilon} leq P{|X_n-X| > frac {epsilon} M}+P{|X| >M}$. For $n$ sufficiently large we get $P{|X_n^{2}-X^{2}| >3epsilon} <2eta$. This proves that $X_n^{2} to X^{2}$ in probability.
answered Nov 28 at 6:15
Kavi Rama Murthy
47.1k31854
add a comment |
up vote
1
down vote
accepted
Let $epsilon >0, eta >0$ ($epsilon <1$) and choose $M>1$ such that $P{|X| >M} <eta$. Then $|X_n-X| leq frac {epsilon} M$ and $|X| leq M$ together imply that $|X_n^{2}-X^{2}|=|X_n-X|(|X_n-X|+2|X|) leq frac {epsilon} M (1+2M)<3epsilon$. Hence $P{|X_n^{2}-X^{2}| >3epsilon} leq P{|X_n-X| > frac {epsilon} M}+P{|X| >M}$. For $n$ sufficiently large we get $P{|X_n^{2}-X^{2}| >3epsilon} <2eta$. This proves that $X_n^{2} to X^{2}$ in probability.
answered Nov 28 at 6:15
Kavi Rama Murthy
47.1k31854
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Let $epsilon >0, eta >0$ ($epsilon <1$) and choose $M>1$ such that $P{|X| >M} <eta$. Then $|X_n-X| leq frac {epsilon} M$ and $|X| leq M$ together imply that $|X_n^{2}-X^{2}|=|X_n-X|(|X_n-X|+2|X|) leq frac {epsilon} M (1+2M)<3epsilon$. Hence $P{|X_n^{2}-X^{2}| >3epsilon} leq P{|X_n-X| > frac {epsilon} M}+P{|X| >M}$. For $n$ sufficiently large we get $P{|X_n^{2}-X^{2}| >3epsilon} <2eta$. This proves that $X_n^{2} to X^{2}$ in probability.
answered Nov 28 at 6:15
Kavi Rama Murthy
47.1k31854
Let $epsilon >0, eta >0$ ($epsilon <1$) and choose $M>1$ such that $P{|X| >M} <eta$. Then $|X_n-X| leq frac {epsilon} M$ and $|X| leq M$ together imply that $|X_n^{2}-X^{2}|=|X_n-X|(|X_n-X|+2|X|) leq frac {epsilon} M (1+2M)<3epsilon$. Hence $P{|X_n^{2}-X^{2}| >3epsilon} leq P{|X_n-X| > frac {epsilon} M}+P{|X| >M}$. For $n$ sufficiently large we get $P{|X_n^{2}-X^{2}| >3epsilon} <2eta$. This proves that $X_n^{2} to X^{2}$ in probability.
answered Nov 28 at 6:15
Kavi Rama Murthy
47.1k31854
answered Nov 28 at 6:15
Kavi Rama Murthy
47.1k31854
answered Nov 28 at 6:15
Kavi Rama Murthy
47.1k31854
answered Nov 28 at 6:15
Kavi Rama Murthy
47.1k31854
47.1k31854
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.
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%2f3016564%2felementary-proof-for-convergence-in-probability-for-square-of-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
1
math.stackexchange.com/questions/339124/…
– d.k.o.
Nov 28 at 1:29