Mixing
Reference request: Continuous Mapping Theorem
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I'm looking for a reference (preferably, a textbook) for the Continuous Mapping Theorem, as stated in the Wikipedia page. In particular, I'm interested in point 3., concerning a.s. convergence.
I've looked into the reference suggested in the Wikipedia page, namely P. Billingsley, "Convergence of Probability Measures" (1999), but it seems to me that the statement (Theorem 2.7, p. 27) concerns weak convergence only.
Any help is very welcome! Thanks!
probability probability-theory reference-request convergence
asked Feb 2 at 19:31
LudwigLudwig
818715
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|
show 1 more comment
$begingroup$
I'm looking for a reference (preferably, a textbook) for the Continuous Mapping Theorem, as stated in the Wikipedia page. In particular, I'm interested in point 3., concerning a.s. convergence.
I've looked into the reference suggested in the Wikipedia page, namely P. Billingsley, "Convergence of Probability Measures" (1999), but it seems to me that the statement (Theorem 2.7, p. 27) concerns weak convergence only.
Any help is very welcome! Thanks!
probability probability-theory reference-request convergence
asked Feb 2 at 19:31
LudwigLudwig
818715
$endgroup$
$begingroup$
What is the continuous mapping theorem?
$endgroup$
– user370967
Feb 2 at 19:46
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@Math_QED You can check the Wikipedia link for the statement.
$endgroup$
– Ludwig
Feb 2 at 19:49
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Almost sure convergence is trivial to prove. The other ones are the difficult ones.
$endgroup$
– user370967
Feb 2 at 19:51
$begingroup$
@Math_QED: Yes, I agree. However, I would like to find a textbook with a statement including all kind of convergence.
$endgroup$
– Ludwig
Feb 2 at 19:54
$begingroup$
I think this should be in all serious probability/measure theoretic references. Maybe check Cohn's measure theory.
$endgroup$
– user370967
Feb 2 at 19:56
|
show 1 more comment
$begingroup$
I'm looking for a reference (preferably, a textbook) for the Continuous Mapping Theorem, as stated in the Wikipedia page. In particular, I'm interested in point 3., concerning a.s. convergence.
I've looked into the reference suggested in the Wikipedia page, namely P. Billingsley, "Convergence of Probability Measures" (1999), but it seems to me that the statement (Theorem 2.7, p. 27) concerns weak convergence only.
Any help is very welcome! Thanks!
probability probability-theory reference-request convergence
asked Feb 2 at 19:31
LudwigLudwig
818715
$endgroup$
I'm looking for a reference (preferably, a textbook) for the Continuous Mapping Theorem, as stated in the Wikipedia page. In particular, I'm interested in point 3., concerning a.s. convergence.
I've looked into the reference suggested in the Wikipedia page, namely P. Billingsley, "Convergence of Probability Measures" (1999), but it seems to me that the statement (Theorem 2.7, p. 27) concerns weak convergence only.
Any help is very welcome! Thanks!
probability probability-theory reference-request convergence
probability probability-theory reference-request convergence
asked Feb 2 at 19:31
LudwigLudwig
818715
asked Feb 2 at 19:31
LudwigLudwig
818715
asked Feb 2 at 19:31
LudwigLudwig
818715
asked Feb 2 at 19:31
LudwigLudwig
818715
asked Feb 2 at 19:31
LudwigLudwig
818715
818715
$begingroup$
What is the continuous mapping theorem?
$endgroup$
– user370967
Feb 2 at 19:46
$begingroup$
@Math_QED You can check the Wikipedia link for the statement.
$endgroup$
– Ludwig
Feb 2 at 19:49
$begingroup$
Almost sure convergence is trivial to prove. The other ones are the difficult ones.
$endgroup$
– user370967
Feb 2 at 19:51
$begingroup$
@Math_QED: Yes, I agree. However, I would like to find a textbook with a statement including all kind of convergence.
$endgroup$
– Ludwig
Feb 2 at 19:54
$begingroup$
I think this should be in all serious probability/measure theoretic references. Maybe check Cohn's measure theory.
$endgroup$
– user370967
Feb 2 at 19:56
|
show 1 more comment
$begingroup$
What is the continuous mapping theorem?
$endgroup$
– user370967
Feb 2 at 19:46
$begingroup$
@Math_QED You can check the Wikipedia link for the statement.
$endgroup$
– Ludwig
Feb 2 at 19:49
$begingroup$
Almost sure convergence is trivial to prove. The other ones are the difficult ones.
$endgroup$
– user370967
Feb 2 at 19:51
$begingroup$
@Math_QED: Yes, I agree. However, I would like to find a textbook with a statement including all kind of convergence.
$endgroup$
– Ludwig
Feb 2 at 19:54
$begingroup$
I think this should be in all serious probability/measure theoretic references. Maybe check Cohn's measure theory.
$endgroup$
– user370967
Feb 2 at 19:56
$begingroup$
What is the continuous mapping theorem?
$endgroup$
– user370967
Feb 2 at 19:46
$begingroup$
What is the continuous mapping theorem?
$endgroup$
– user370967
Feb 2 at 19:46
$begingroup$
@Math_QED You can check the Wikipedia link for the statement.
$endgroup$
– Ludwig
Feb 2 at 19:49
$begingroup$
@Math_QED You can check the Wikipedia link for the statement.
$endgroup$
– Ludwig
Feb 2 at 19:49
$begingroup$
Almost sure convergence is trivial to prove. The other ones are the difficult ones.
$endgroup$
– user370967
Feb 2 at 19:51
$begingroup$
Almost sure convergence is trivial to prove. The other ones are the difficult ones.
$endgroup$
– user370967
Feb 2 at 19:51
$begingroup$
@Math_QED: Yes, I agree. However, I would like to find a textbook with a statement including all kind of convergence.
$endgroup$
– Ludwig
Feb 2 at 19:54
$begingroup$
@Math_QED: Yes, I agree. However, I would like to find a textbook with a statement including all kind of convergence.
$endgroup$
– Ludwig
Feb 2 at 19:54
$begingroup$
I think this should be in all serious probability/measure theoretic references. Maybe check Cohn's measure theory.
$endgroup$
– user370967
Feb 2 at 19:56
$begingroup$
I think this should be in all serious probability/measure theoretic references. Maybe check Cohn's measure theory.
$endgroup$
– user370967
Feb 2 at 19:56
|
show 1 more comment
1 Answer
1
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votes
$begingroup$
Almost sure convergence is trivial.
Note that $${omegamid X_n(omega) to X(omega)} subset {omega mid g(X_n(omega)) to g(X(omega))}$$
where $X_n,X$ are random variables and $g$ is continuous. Then, take probabilities of both sides of the inequality.
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1 Answer
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1 Answer
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$begingroup$
Almost sure convergence is trivial.
Note that $${omegamid X_n(omega) to X(omega)} subset {omega mid g(X_n(omega)) to g(X(omega))}$$
where $X_n,X$ are random variables and $g$ is continuous. Then, take probabilities of both sides of the inequality.
$endgroup$
add a comment |
$begingroup$
Almost sure convergence is trivial.
Note that $${omegamid X_n(omega) to X(omega)} subset {omega mid g(X_n(omega)) to g(X(omega))}$$
where $X_n,X$ are random variables and $g$ is continuous. Then, take probabilities of both sides of the inequality.
$endgroup$
add a comment |
$begingroup$
Almost sure convergence is trivial.
Note that $${omegamid X_n(omega) to X(omega)} subset {omega mid g(X_n(omega)) to g(X(omega))}$$
where $X_n,X$ are random variables and $g$ is continuous. Then, take probabilities of both sides of the inequality.
$endgroup$
Almost sure convergence is trivial.
Note that $${omegamid X_n(omega) to X(omega)} subset {omega mid g(X_n(omega)) to g(X(omega))}$$
where $X_n,X$ are random variables and $g$ is continuous. Then, take probabilities of both sides of the inequality.
answered Feb 2 at 19:55
user370967
add a comment |
add a comment |
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$begingroup$
What is the continuous mapping theorem?
$endgroup$
– user370967
Feb 2 at 19:46
$begingroup$
@Math_QED You can check the Wikipedia link for the statement.
$endgroup$
– Ludwig
Feb 2 at 19:49
$begingroup$
Almost sure convergence is trivial to prove. The other ones are the difficult ones.
$endgroup$
– user370967
Feb 2 at 19:51
$begingroup$
@Math_QED: Yes, I agree. However, I would like to find a textbook with a statement including all kind of convergence.
$endgroup$
– Ludwig
Feb 2 at 19:54
$begingroup$
I think this should be in all serious probability/measure theoretic references. Maybe check Cohn's measure theory.
$endgroup$
– user370967
Feb 2 at 19:56