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Equivalence over convergence in distribution
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Let $X$ be a metric space and $mu_1,mu_2,...,mu$ be Borel probability measures on $X$.
The Portmanteau theorem says the following are equivalent:
(a) $int_X gdmu_n to int_X gdmu$ for each $gin C_b(X)$
(b) For every $mu$-continuity set $A$, $mu_n(A)to mu(A)$
If one of these conditions holds, then we say $mu_nto mu$ weakly.
Now, let’s take $X=mathbbR^m$ and $F_n,F$ be the c.d.f’s of $mu_n,mu$ respectively.
If $F_n(x)to F(x)$ for every point $x$ at which $F$ is continuous, then does $mu_n to mu$ weakly?
If $m=1$, this is true and it can be found in many probability theory texts. Is it still true for $m>1$? I cannot find any reference for this. Is there any text having this proof? Or could someone show me how prove this directly? Thank you in advance.
probability-theory probability-distributions weak-convergence
asked Jul 16 at 19:38
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5,3942721
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up vote
4
down vote
favorite
Let $X$ be a metric space and $mu_1,mu_2,...,mu$ be Borel probability measures on $X$.
The Portmanteau theorem says the following are equivalent:
(a) $int_X gdmu_n to int_X gdmu$ for each $gin C_b(X)$
(b) For every $mu$-continuity set $A$, $mu_n(A)to mu(A)$
If one of these conditions holds, then we say $mu_nto mu$ weakly.
Now, let’s take $X=mathbbR^m$ and $F_n,F$ be the c.d.f’s of $mu_n,mu$ respectively.
If $F_n(x)to F(x)$ for every point $x$ at which $F$ is continuous, then does $mu_n to mu$ weakly?
If $m=1$, this is true and it can be found in many probability theory texts. Is it still true for $m>1$? I cannot find any reference for this. Is there any text having this proof? Or could someone show me how prove this directly? Thank you in advance.
probability-theory probability-distributions weak-convergence
asked Jul 16 at 19:38
Rubertos
5,3942721
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
Let $X$ be a metric space and $mu_1,mu_2,...,mu$ be Borel probability measures on $X$.
The Portmanteau theorem says the following are equivalent:
(a) $int_X gdmu_n to int_X gdmu$ for each $gin C_b(X)$
(b) For every $mu$-continuity set $A$, $mu_n(A)to mu(A)$
If one of these conditions holds, then we say $mu_nto mu$ weakly.
Now, let’s take $X=mathbbR^m$ and $F_n,F$ be the c.d.f’s of $mu_n,mu$ respectively.
If $F_n(x)to F(x)$ for every point $x$ at which $F$ is continuous, then does $mu_n to mu$ weakly?
If $m=1$, this is true and it can be found in many probability theory texts. Is it still true for $m>1$? I cannot find any reference for this. Is there any text having this proof? Or could someone show me how prove this directly? Thank you in advance.
probability-theory probability-distributions weak-convergence
asked Jul 16 at 19:38
Rubertos
5,3942721
Let $X$ be a metric space and $mu_1,mu_2,...,mu$ be Borel probability measures on $X$.
The Portmanteau theorem says the following are equivalent:
(a) $int_X gdmu_n to int_X gdmu$ for each $gin C_b(X)$
(b) For every $mu$-continuity set $A$, $mu_n(A)to mu(A)$
If one of these conditions holds, then we say $mu_nto mu$ weakly.
Now, let’s take $X=mathbbR^m$ and $F_n,F$ be the c.d.f’s of $mu_n,mu$ respectively.
If $F_n(x)to F(x)$ for every point $x$ at which $F$ is continuous, then does $mu_n to mu$ weakly?
If $m=1$, this is true and it can be found in many probability theory texts. Is it still true for $m>1$? I cannot find any reference for this. Is there any text having this proof? Or could someone show me how prove this directly? Thank you in advance.
probability-theory probability-distributions weak-convergence
asked Jul 16 at 19:38
Rubertos
5,3942721
asked Jul 16 at 19:38
Rubertos
5,3942721
asked Jul 16 at 19:38
Rubertos
5,3942721
asked Jul 16 at 19:38
Rubertos
5,3942721
5,3942721
add a comment |Â
add a comment |Â
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