Mixing
real sequence and convergence in probability
$begingroup$
$X_n$ is a sequence of random variables.$X_n equiv a_n$, $a_n $ is a real sequence.
Then prove that $X_n $ converges in probability iff $a_n$ converges and then $X_n to lim_{ntoinfty} a_n$ in probability.
I have a feeling that the above statement is trivially true but I am not so sure. If anyone can prove otherwise,please do so.
convergence
asked May 30 '14 at 11:43
kris91kris91
18612
$endgroup$
add a comment |
$begingroup$
$X_n$ is a sequence of random variables.$X_n equiv a_n$, $a_n $ is a real sequence.
Then prove that $X_n $ converges in probability iff $a_n$ converges and then $X_n to lim_{ntoinfty} a_n$ in probability.
I have a feeling that the above statement is trivially true but I am not so sure. If anyone can prove otherwise,please do so.
convergence
asked May 30 '14 at 11:43
kris91kris91
18612
$endgroup$
1
$begingroup$
write down the definition of convergence in probability and see what you can do
$endgroup$
– mm-aops
May 30 '14 at 11:45
$begingroup$
Hrmm if you're working with a general measure (which is a generalization of a probability measure) then convergence in measure is not equivalent to convergence almost everywhere. Is there something I'm missing about this specific question that allows for those two things to be equivalent?
$endgroup$
– DanZimm
May 30 '14 at 11:55
$begingroup$
no.This was all there was to the question.I did write down the definition of convergence in probability and I think that this statement is trivially true but the thing is nothing is said about the nature of convergence of real sequence.
$endgroup$
– kris91
May 30 '14 at 14:07
add a comment |
$begingroup$
$X_n$ is a sequence of random variables.$X_n equiv a_n$, $a_n $ is a real sequence.
Then prove that $X_n $ converges in probability iff $a_n$ converges and then $X_n to lim_{ntoinfty} a_n$ in probability.
I have a feeling that the above statement is trivially true but I am not so sure. If anyone can prove otherwise,please do so.
convergence
asked May 30 '14 at 11:43
kris91kris91
18612
$endgroup$
$X_n$ is a sequence of random variables.$X_n equiv a_n$, $a_n $ is a real sequence.
Then prove that $X_n $ converges in probability iff $a_n$ converges and then $X_n to lim_{ntoinfty} a_n$ in probability.
I have a feeling that the above statement is trivially true but I am not so sure. If anyone can prove otherwise,please do so.
convergence
convergence
asked May 30 '14 at 11:43
kris91kris91
18612
asked May 30 '14 at 11:43
kris91kris91
18612
asked May 30 '14 at 11:43
kris91kris91
18612
asked May 30 '14 at 11:43
kris91kris91
18612
asked May 30 '14 at 11:43
kris91kris91
18612
18612
1
$begingroup$
write down the definition of convergence in probability and see what you can do
$endgroup$
– mm-aops
May 30 '14 at 11:45
$begingroup$
Hrmm if you're working with a general measure (which is a generalization of a probability measure) then convergence in measure is not equivalent to convergence almost everywhere. Is there something I'm missing about this specific question that allows for those two things to be equivalent?
$endgroup$
– DanZimm
May 30 '14 at 11:55
$begingroup$
no.This was all there was to the question.I did write down the definition of convergence in probability and I think that this statement is trivially true but the thing is nothing is said about the nature of convergence of real sequence.
$endgroup$
– kris91
May 30 '14 at 14:07
add a comment |
1
$begingroup$
write down the definition of convergence in probability and see what you can do
$endgroup$
– mm-aops
May 30 '14 at 11:45
$begingroup$
Hrmm if you're working with a general measure (which is a generalization of a probability measure) then convergence in measure is not equivalent to convergence almost everywhere. Is there something I'm missing about this specific question that allows for those two things to be equivalent?
$endgroup$
– DanZimm
May 30 '14 at 11:55
$begingroup$
no.This was all there was to the question.I did write down the definition of convergence in probability and I think that this statement is trivially true but the thing is nothing is said about the nature of convergence of real sequence.
$endgroup$
– kris91
May 30 '14 at 14:07
1
1
$begingroup$
write down the definition of convergence in probability and see what you can do
$endgroup$
– mm-aops
May 30 '14 at 11:45
$begingroup$
write down the definition of convergence in probability and see what you can do
$endgroup$
– mm-aops
May 30 '14 at 11:45
$begingroup$
Hrmm if you're working with a general measure (which is a generalization of a probability measure) then convergence in measure is not equivalent to convergence almost everywhere. Is there something I'm missing about this specific question that allows for those two things to be equivalent?
$endgroup$
– DanZimm
May 30 '14 at 11:55
$begingroup$
Hrmm if you're working with a general measure (which is a generalization of a probability measure) then convergence in measure is not equivalent to convergence almost everywhere. Is there something I'm missing about this specific question that allows for those two things to be equivalent?
$endgroup$
– DanZimm
May 30 '14 at 11:55
$begingroup$
no.This was all there was to the question.I did write down the definition of convergence in probability and I think that this statement is trivially true but the thing is nothing is said about the nature of convergence of real sequence.
$endgroup$
– kris91
May 30 '14 at 14:07
$begingroup$
no.This was all there was to the question.I did write down the definition of convergence in probability and I think that this statement is trivially true but the thing is nothing is said about the nature of convergence of real sequence.
$endgroup$
– kris91
May 30 '14 at 14:07
add a comment |
1 Answer
1
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oldest
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$begingroup$
For any real number $a$ and $varepsilon > 0$,
$$
Pr{leftlvert X_n-arightrvertgt varepsilon }=
begin{cases}
1&mbox{ if }leftlvert a_n-arightrvertgt varepsilon;\
0&mbox{ if }leftlvert a_n-arightrvertleq varepsilon.
end{cases}
$$
answered Jan 5 at 14:57
Davide GiraudoDavide Giraudo
126k16150261
$endgroup$
add a comment |
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1 Answer
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active
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1 Answer
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active
oldest
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active
oldest
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active
oldest
votes
$begingroup$
For any real number $a$ and $varepsilon > 0$,
$$
Pr{leftlvert X_n-arightrvertgt varepsilon }=
begin{cases}
1&mbox{ if }leftlvert a_n-arightrvertgt varepsilon;\
0&mbox{ if }leftlvert a_n-arightrvertleq varepsilon.
end{cases}
$$
answered Jan 5 at 14:57
Davide GiraudoDavide Giraudo
126k16150261
$endgroup$
add a comment |
$begingroup$
For any real number $a$ and $varepsilon > 0$,
$$
Pr{leftlvert X_n-arightrvertgt varepsilon }=
begin{cases}
1&mbox{ if }leftlvert a_n-arightrvertgt varepsilon;\
0&mbox{ if }leftlvert a_n-arightrvertleq varepsilon.
end{cases}
$$
answered Jan 5 at 14:57
Davide GiraudoDavide Giraudo
126k16150261
$endgroup$
add a comment |
$begingroup$
For any real number $a$ and $varepsilon > 0$,
$$
Pr{leftlvert X_n-arightrvertgt varepsilon }=
begin{cases}
1&mbox{ if }leftlvert a_n-arightrvertgt varepsilon;\
0&mbox{ if }leftlvert a_n-arightrvertleq varepsilon.
end{cases}
$$
answered Jan 5 at 14:57
Davide GiraudoDavide Giraudo
126k16150261
$endgroup$
For any real number $a$ and $varepsilon > 0$,
$$
Pr{leftlvert X_n-arightrvertgt varepsilon }=
begin{cases}
1&mbox{ if }leftlvert a_n-arightrvertgt varepsilon;\
0&mbox{ if }leftlvert a_n-arightrvertleq varepsilon.
end{cases}
$$
answered Jan 5 at 14:57
Davide GiraudoDavide Giraudo
126k16150261
answered Jan 5 at 14:57
Davide GiraudoDavide Giraudo
126k16150261
answered Jan 5 at 14:57
Davide GiraudoDavide Giraudo
126k16150261
answered Jan 5 at 14:57
Davide GiraudoDavide Giraudo
126k16150261
126k16150261
add a comment |
add a comment |
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$begingroup$
write down the definition of convergence in probability and see what you can do
$endgroup$
– mm-aops
May 30 '14 at 11:45
$begingroup$
Hrmm if you're working with a general measure (which is a generalization of a probability measure) then convergence in measure is not equivalent to convergence almost everywhere. Is there something I'm missing about this specific question that allows for those two things to be equivalent?
$endgroup$
– DanZimm
May 30 '14 at 11:55
$begingroup$
no.This was all there was to the question.I did write down the definition of convergence in probability and I think that this statement is trivially true but the thing is nothing is said about the nature of convergence of real sequence.
$endgroup$
– kris91
May 30 '14 at 14:07