Mixing
Convergence of sequence of stochastic processes
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I want to show that for any $y_0>0$ $$sup_{0<y<y_0} y^{-1} X_n(y) overset{P}{to}0, hspace{25mm}(1)$$
Here $X_n(y)$ is to be regarded as a sequence of real-valued stochastic process defined on $[0, infty]$.
The situation is as follows, I can show that for any $0< epsilon <y_0$
$$sup_{epsilon<y<y_0} y^{-1} X_n(y) overset{P}{to}0$$
Also I can show that $$lim_{y to 0}y^{-1}X_n(y)=0 text{ a.s.}$$ Is there any way to connect these two so that we get the first statement (1)?
probability-theory convergence stochastic-processes
asked Jan 29 at 12:05
JoogsJoogs
22719
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add a comment |
$begingroup$
I want to show that for any $y_0>0$ $$sup_{0<y<y_0} y^{-1} X_n(y) overset{P}{to}0, hspace{25mm}(1)$$
Here $X_n(y)$ is to be regarded as a sequence of real-valued stochastic process defined on $[0, infty]$.
The situation is as follows, I can show that for any $0< epsilon <y_0$
$$sup_{epsilon<y<y_0} y^{-1} X_n(y) overset{P}{to}0$$
Also I can show that $$lim_{y to 0}y^{-1}X_n(y)=0 text{ a.s.}$$ Is there any way to connect these two so that we get the first statement (1)?
probability-theory convergence stochastic-processes
asked Jan 29 at 12:05
JoogsJoogs
22719
$endgroup$
add a comment |
$begingroup$
I want to show that for any $y_0>0$ $$sup_{0<y<y_0} y^{-1} X_n(y) overset{P}{to}0, hspace{25mm}(1)$$
Here $X_n(y)$ is to be regarded as a sequence of real-valued stochastic process defined on $[0, infty]$.
The situation is as follows, I can show that for any $0< epsilon <y_0$
$$sup_{epsilon<y<y_0} y^{-1} X_n(y) overset{P}{to}0$$
Also I can show that $$lim_{y to 0}y^{-1}X_n(y)=0 text{ a.s.}$$ Is there any way to connect these two so that we get the first statement (1)?
probability-theory convergence stochastic-processes
asked Jan 29 at 12:05
JoogsJoogs
22719
$endgroup$
I want to show that for any $y_0>0$ $$sup_{0<y<y_0} y^{-1} X_n(y) overset{P}{to}0, hspace{25mm}(1)$$
Here $X_n(y)$ is to be regarded as a sequence of real-valued stochastic process defined on $[0, infty]$.
The situation is as follows, I can show that for any $0< epsilon <y_0$
$$sup_{epsilon<y<y_0} y^{-1} X_n(y) overset{P}{to}0$$
Also I can show that $$lim_{y to 0}y^{-1}X_n(y)=0 text{ a.s.}$$ Is there any way to connect these two so that we get the first statement (1)?
probability-theory convergence stochastic-processes
probability-theory convergence stochastic-processes
asked Jan 29 at 12:05
JoogsJoogs
22719
asked Jan 29 at 12:05
JoogsJoogs
22719
asked Jan 29 at 12:05
JoogsJoogs
22719
asked Jan 29 at 12:05
JoogsJoogs
22719
asked Jan 29 at 12:05
JoogsJoogs
22719
22719
add a comment |
add a comment |
1 Answer
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$X_n(y)=frac {sin (n y)} {sqrt {ny}}$ is a counterexample.
answered Jan 29 at 12:35
Kavi Rama MurthyKavi Rama Murthy
71.4k53170
$endgroup$
$begingroup$
What more would we need to require for $X_n(y)$ for (1) to hold?
$endgroup$
– Joogs
Jan 29 at 12:43
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@Joogs Uniformity (w.r.t. $n$) in the second limit will be sufficient.
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– Kavi Rama Murthy
Jan 29 at 23:08
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$X_n(y)=frac {sin (n y)} {sqrt {ny}}$ is a counterexample.
answered Jan 29 at 12:35
Kavi Rama MurthyKavi Rama Murthy
71.4k53170
$endgroup$
$begingroup$
What more would we need to require for $X_n(y)$ for (1) to hold?
$endgroup$
– Joogs
Jan 29 at 12:43
$begingroup$
@Joogs Uniformity (w.r.t. $n$) in the second limit will be sufficient.
$endgroup$
– Kavi Rama Murthy
Jan 29 at 23:08
add a comment |
$begingroup$
$X_n(y)=frac {sin (n y)} {sqrt {ny}}$ is a counterexample.
answered Jan 29 at 12:35
Kavi Rama MurthyKavi Rama Murthy
71.4k53170
$endgroup$
$begingroup$
What more would we need to require for $X_n(y)$ for (1) to hold?
$endgroup$
– Joogs
Jan 29 at 12:43
$begingroup$
@Joogs Uniformity (w.r.t. $n$) in the second limit will be sufficient.
$endgroup$
– Kavi Rama Murthy
Jan 29 at 23:08
add a comment |
$begingroup$
$X_n(y)=frac {sin (n y)} {sqrt {ny}}$ is a counterexample.
answered Jan 29 at 12:35
Kavi Rama MurthyKavi Rama Murthy
71.4k53170
$endgroup$
$X_n(y)=frac {sin (n y)} {sqrt {ny}}$ is a counterexample.
answered Jan 29 at 12:35
Kavi Rama MurthyKavi Rama Murthy
71.4k53170
answered Jan 29 at 12:35
Kavi Rama MurthyKavi Rama Murthy
71.4k53170
answered Jan 29 at 12:35
Kavi Rama MurthyKavi Rama Murthy
71.4k53170
answered Jan 29 at 12:35
Kavi Rama MurthyKavi Rama Murthy
71.4k53170
71.4k53170
$begingroup$
What more would we need to require for $X_n(y)$ for (1) to hold?
$endgroup$
– Joogs
Jan 29 at 12:43
$begingroup$
@Joogs Uniformity (w.r.t. $n$) in the second limit will be sufficient.
$endgroup$
– Kavi Rama Murthy
Jan 29 at 23:08
add a comment |
$begingroup$
What more would we need to require for $X_n(y)$ for (1) to hold?
$endgroup$
– Joogs
Jan 29 at 12:43
$begingroup$
@Joogs Uniformity (w.r.t. $n$) in the second limit will be sufficient.
$endgroup$
– Kavi Rama Murthy
Jan 29 at 23:08
$begingroup$
What more would we need to require for $X_n(y)$ for (1) to hold?
$endgroup$
– Joogs
Jan 29 at 12:43
$begingroup$
What more would we need to require for $X_n(y)$ for (1) to hold?
$endgroup$
– Joogs
Jan 29 at 12:43
$begingroup$
@Joogs Uniformity (w.r.t. $n$) in the second limit will be sufficient.
$endgroup$
– Kavi Rama Murthy
Jan 29 at 23:08
$begingroup$
@Joogs Uniformity (w.r.t. $n$) in the second limit will be sufficient.
$endgroup$
– Kavi Rama Murthy
Jan 29 at 23:08
add a comment |
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