Advanced Probability Theory: Understanding Notation for Sequences of Random Variables
up vote
1
down vote
favorite
Let ${X_n}$ be a sequence of random variables, X a random variable, and let ${X_n} rightarrow X$ a.e.
Let $A_m(epsilon)$ be the event:
$A_m(epsilon) = bigcap_{n=m}^{infty} {|X_n-X| <epsilon}$
I see this type of notation alot, and I am a bit confused. Can we think of ${|X_n-X| <epsilon}$ as the sequence ${|X_n-X|: |X_n-X| <epsilon}$, which is the collection of random variables $|X_n-X|$ such that $|X_n-X|<epsilon $ for all $omega in Omega$.
Let me know. Help with advanced probability theory notation is greatly appreciated.
sequences-and-series probability-theory random-variables
asked Nov 18 at 1:50
kpr62
224
add a comment |
up vote
1
down vote
favorite
Let ${X_n}$ be a sequence of random variables, X a random variable, and let ${X_n} rightarrow X$ a.e.
Let $A_m(epsilon)$ be the event:
$A_m(epsilon) = bigcap_{n=m}^{infty} {|X_n-X| <epsilon}$
I see this type of notation alot, and I am a bit confused. Can we think of ${|X_n-X| <epsilon}$ as the sequence ${|X_n-X|: |X_n-X| <epsilon}$, which is the collection of random variables $|X_n-X|$ such that $|X_n-X|<epsilon $ for all $omega in Omega$.
Let me know. Help with advanced probability theory notation is greatly appreciated.
sequences-and-series probability-theory random-variables
asked Nov 18 at 1:50
kpr62
224
${ |X_n - X | < epsilon }$ is the event that $X_n$ is within $epsilon$ of $X$. It is a subset of the sample space.
– littleO
Nov 18 at 3:44
It was suggested by another source that I think of it as ${omega: |X_n (omega)-X(omega)|< epsilon, omega in Omega }$. If this incorrect, or if someone has a better more detailed example, please post :)
– kpr62
Nov 18 at 3:49
Yes, that's correct.
– littleO
Nov 18 at 3:56
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let ${X_n}$ be a sequence of random variables, X a random variable, and let ${X_n} rightarrow X$ a.e.
Let $A_m(epsilon)$ be the event:
$A_m(epsilon) = bigcap_{n=m}^{infty} {|X_n-X| <epsilon}$
I see this type of notation alot, and I am a bit confused. Can we think of ${|X_n-X| <epsilon}$ as the sequence ${|X_n-X|: |X_n-X| <epsilon}$, which is the collection of random variables $|X_n-X|$ such that $|X_n-X|<epsilon $ for all $omega in Omega$.
Let me know. Help with advanced probability theory notation is greatly appreciated.
sequences-and-series probability-theory random-variables
asked Nov 18 at 1:50
kpr62
224
Let ${X_n}$ be a sequence of random variables, X a random variable, and let ${X_n} rightarrow X$ a.e.
Let $A_m(epsilon)$ be the event:
$A_m(epsilon) = bigcap_{n=m}^{infty} {|X_n-X| <epsilon}$
I see this type of notation alot, and I am a bit confused. Can we think of ${|X_n-X| <epsilon}$ as the sequence ${|X_n-X|: |X_n-X| <epsilon}$, which is the collection of random variables $|X_n-X|$ such that $|X_n-X|<epsilon $ for all $omega in Omega$.
Let me know. Help with advanced probability theory notation is greatly appreciated.
sequences-and-series probability-theory random-variables
sequences-and-series probability-theory random-variables
asked Nov 18 at 1:50
kpr62
224
asked Nov 18 at 1:50
kpr62
224
edited Nov 18 at 3:38
asked Nov 18 at 1:50
kpr62
224
asked Nov 18 at 1:50
kpr62
224
asked Nov 18 at 1:50
kpr62
224
224
${ |X_n - X | < epsilon }$ is the event that $X_n$ is within $epsilon$ of $X$. It is a subset of the sample space.
– littleO
Nov 18 at 3:44
It was suggested by another source that I think of it as ${omega: |X_n (omega)-X(omega)|< epsilon, omega in Omega }$. If this incorrect, or if someone has a better more detailed example, please post :)
– kpr62
Nov 18 at 3:49
Yes, that's correct.
– littleO
Nov 18 at 3:56
add a comment |
${ |X_n - X | < epsilon }$ is the event that $X_n$ is within $epsilon$ of $X$. It is a subset of the sample space.
– littleO
Nov 18 at 3:44
It was suggested by another source that I think of it as ${omega: |X_n (omega)-X(omega)|< epsilon, omega in Omega }$. If this incorrect, or if someone has a better more detailed example, please post :)
– kpr62
Nov 18 at 3:49
Yes, that's correct.
– littleO
Nov 18 at 3:56
${ |X_n - X | < epsilon }$ is the event that $X_n$ is within $epsilon$ of $X$. It is a subset of the sample space.
– littleO
Nov 18 at 3:44
${ |X_n - X | < epsilon }$ is the event that $X_n$ is within $epsilon$ of $X$. It is a subset of the sample space.
– littleO
Nov 18 at 3:44
It was suggested by another source that I think of it as ${omega: |X_n (omega)-X(omega)|< epsilon, omega in Omega }$. If this incorrect, or if someone has a better more detailed example, please post :)
– kpr62
Nov 18 at 3:49
It was suggested by another source that I think of it as ${omega: |X_n (omega)-X(omega)|< epsilon, omega in Omega }$. If this incorrect, or if someone has a better more detailed example, please post :)
– kpr62
Nov 18 at 3:49
Yes, that's correct.
– littleO
Nov 18 at 3:56
Yes, that's correct.
– littleO
Nov 18 at 3:56
add a comment |
active
oldest
votes
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%2f3003054%2fadvanced-probability-theory-understanding-notation-for-sequences-of-random-vari%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
${ |X_n - X | < epsilon }$ is the event that $X_n$ is within $epsilon$ of $X$. It is a subset of the sample space.
– littleO
Nov 18 at 3:44
It was suggested by another source that I think of it as ${omega: |X_n (omega)-X(omega)|< epsilon, omega in Omega }$. If this incorrect, or if someone has a better more detailed example, please post :)
– kpr62
Nov 18 at 3:49
Yes, that's correct.
– littleO
Nov 18 at 3:56