Equivalence of limit supremum of sequence of sets and sequence of functions
Let $X_1,X_2,dots$ be a sequence of real-valued random variables where $X_n:Omega to mathbb{R}$.
For any sequence of events $A_n subset Omega$ define $limsup_nA_n := bigcap_{n=1}^{infty}bigcup_{m=n}^{infty}A_m$.
Moreover, define the (extended-real) random variable $Y equiv (limsup_nX_n)$ by $Y(omega) := limsup_n{X_n(omega) mid n=1,2,dots}$.
How would you establish that
$$
limsup_{n}{omega mid X_n(omega) in B} = {omega mid (limsup_nX_n)(omega) in B }, qquad (*)
$$
for any Borel set $B$? Is the proposition $(*)$ even true?
probability probability-theory measure-theory limsup-and-liminf
asked Nov 25 at 23:27
jesterII
1,19621226
add a comment |
Let $X_1,X_2,dots$ be a sequence of real-valued random variables where $X_n:Omega to mathbb{R}$.
For any sequence of events $A_n subset Omega$ define $limsup_nA_n := bigcap_{n=1}^{infty}bigcup_{m=n}^{infty}A_m$.
Moreover, define the (extended-real) random variable $Y equiv (limsup_nX_n)$ by $Y(omega) := limsup_n{X_n(omega) mid n=1,2,dots}$.
How would you establish that
$$
limsup_{n}{omega mid X_n(omega) in B} = {omega mid (limsup_nX_n)(omega) in B }, qquad (*)
$$
for any Borel set $B$? Is the proposition $(*)$ even true?
probability probability-theory measure-theory limsup-and-liminf
asked Nov 25 at 23:27
jesterII
1,19621226
add a comment |
Let $X_1,X_2,dots$ be a sequence of real-valued random variables where $X_n:Omega to mathbb{R}$.
For any sequence of events $A_n subset Omega$ define $limsup_nA_n := bigcap_{n=1}^{infty}bigcup_{m=n}^{infty}A_m$.
Moreover, define the (extended-real) random variable $Y equiv (limsup_nX_n)$ by $Y(omega) := limsup_n{X_n(omega) mid n=1,2,dots}$.
How would you establish that
$$
limsup_{n}{omega mid X_n(omega) in B} = {omega mid (limsup_nX_n)(omega) in B }, qquad (*)
$$
for any Borel set $B$? Is the proposition $(*)$ even true?
probability probability-theory measure-theory limsup-and-liminf
asked Nov 25 at 23:27
jesterII
1,19621226
Let $X_1,X_2,dots$ be a sequence of real-valued random variables where $X_n:Omega to mathbb{R}$.
For any sequence of events $A_n subset Omega$ define $limsup_nA_n := bigcap_{n=1}^{infty}bigcup_{m=n}^{infty}A_m$.
Moreover, define the (extended-real) random variable $Y equiv (limsup_nX_n)$ by $Y(omega) := limsup_n{X_n(omega) mid n=1,2,dots}$.
How would you establish that
$$
limsup_{n}{omega mid X_n(omega) in B} = {omega mid (limsup_nX_n)(omega) in B }, qquad (*)
$$
for any Borel set $B$? Is the proposition $(*)$ even true?
probability probability-theory measure-theory limsup-and-liminf
probability probability-theory measure-theory limsup-and-liminf
asked Nov 25 at 23:27
jesterII
1,19621226
asked Nov 25 at 23:27
jesterII
1,19621226
edited Nov 27 at 1:48
asked Nov 25 at 23:27
jesterII
1,19621226
asked Nov 25 at 23:27
jesterII
1,19621226
asked Nov 25 at 23:27
jesterII
1,19621226
1,19621226
add a comment |
add a comment |
1 Answer
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After further thought I believe that $(*)$ is in general false.
Let $B = { b}$ be a singleton set. Observe that
$omega in { omega' mid (limsup_nX_n)(omega') = b }$ if and only if $limsup{X_1(omega), X_2(omega), dots } = b$.
$omega in limsup_n{ omega' mid X_n(omega') = b }$ if and only if for all $n$ there exists $m>n$ such that $X_m(omega)=b$, or in other words $X_n(omega) = b$ infinitely often.
Let $omega$ be such that the sequence $(X_n(omega))_{n=1}^{infty} = (0,1,0,1,0,1,0,1,dots)$ and put $b=0$. Then $(limsup_nX_n)(omega) = 1$, however $X_n(omega) = 0$ occurs infinitely often.
answered Nov 29 at 19:36
jesterII
1,19621226
add a comment |
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1 Answer
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1 Answer
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After further thought I believe that $(*)$ is in general false.
Let $B = { b}$ be a singleton set. Observe that
$omega in { omega' mid (limsup_nX_n)(omega') = b }$ if and only if $limsup{X_1(omega), X_2(omega), dots } = b$.
$omega in limsup_n{ omega' mid X_n(omega') = b }$ if and only if for all $n$ there exists $m>n$ such that $X_m(omega)=b$, or in other words $X_n(omega) = b$ infinitely often.
Let $omega$ be such that the sequence $(X_n(omega))_{n=1}^{infty} = (0,1,0,1,0,1,0,1,dots)$ and put $b=0$. Then $(limsup_nX_n)(omega) = 1$, however $X_n(omega) = 0$ occurs infinitely often.
answered Nov 29 at 19:36
jesterII
1,19621226
add a comment |
After further thought I believe that $(*)$ is in general false.
Let $B = { b}$ be a singleton set. Observe that
$omega in { omega' mid (limsup_nX_n)(omega') = b }$ if and only if $limsup{X_1(omega), X_2(omega), dots } = b$.
$omega in limsup_n{ omega' mid X_n(omega') = b }$ if and only if for all $n$ there exists $m>n$ such that $X_m(omega)=b$, or in other words $X_n(omega) = b$ infinitely often.
Let $omega$ be such that the sequence $(X_n(omega))_{n=1}^{infty} = (0,1,0,1,0,1,0,1,dots)$ and put $b=0$. Then $(limsup_nX_n)(omega) = 1$, however $X_n(omega) = 0$ occurs infinitely often.
answered Nov 29 at 19:36
jesterII
1,19621226
add a comment |
After further thought I believe that $(*)$ is in general false.
Let $B = { b}$ be a singleton set. Observe that
$omega in { omega' mid (limsup_nX_n)(omega') = b }$ if and only if $limsup{X_1(omega), X_2(omega), dots } = b$.
$omega in limsup_n{ omega' mid X_n(omega') = b }$ if and only if for all $n$ there exists $m>n$ such that $X_m(omega)=b$, or in other words $X_n(omega) = b$ infinitely often.
Let $omega$ be such that the sequence $(X_n(omega))_{n=1}^{infty} = (0,1,0,1,0,1,0,1,dots)$ and put $b=0$. Then $(limsup_nX_n)(omega) = 1$, however $X_n(omega) = 0$ occurs infinitely often.
answered Nov 29 at 19:36
jesterII
1,19621226
After further thought I believe that $(*)$ is in general false.
Let $B = { b}$ be a singleton set. Observe that
$omega in { omega' mid (limsup_nX_n)(omega') = b }$ if and only if $limsup{X_1(omega), X_2(omega), dots } = b$.
$omega in limsup_n{ omega' mid X_n(omega') = b }$ if and only if for all $n$ there exists $m>n$ such that $X_m(omega)=b$, or in other words $X_n(omega) = b$ infinitely often.
Let $omega$ be such that the sequence $(X_n(omega))_{n=1}^{infty} = (0,1,0,1,0,1,0,1,dots)$ and put $b=0$. Then $(limsup_nX_n)(omega) = 1$, however $X_n(omega) = 0$ occurs infinitely often.
answered Nov 29 at 19:36
jesterII
1,19621226
answered Nov 29 at 19:36
jesterII
1,19621226
answered Nov 29 at 19:36
jesterII
1,19621226
answered Nov 29 at 19:36
jesterII
1,19621226
1,19621226
add a comment |
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