Mixing
Convergence in probability, Y$X_n$ [closed]
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I'm struggling with the following problem
a) Show that if $X_n oversetpto X$ (convergence in probability) and $Y$ any random variable $X_nY oversetpto XY$
b) Find a case where the previous result isn't true if we change convergence in probability by convergence in distribution.
Thanks in advance for your help.
probability-theory convergence
asked Jul 26 at 2:25
MatÃas
11
closed as off-topic by spaceisdarkgreen, Taroccoesbrocco, Shailesh, amWhy, Adrian Keister Jul 27 at 0:19
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – spaceisdarkgreen, Taroccoesbrocco, Shailesh, amWhy, Adrian Keister
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up vote
-1
down vote
favorite
I'm struggling with the following problem
a) Show that if $X_n oversetpto X$ (convergence in probability) and $Y$ any random variable $X_nY oversetpto XY$
b) Find a case where the previous result isn't true if we change convergence in probability by convergence in distribution.
Thanks in advance for your help.
probability-theory convergence
asked Jul 26 at 2:25
MatÃas
11
closed as off-topic by spaceisdarkgreen, Taroccoesbrocco, Shailesh, amWhy, Adrian Keister Jul 27 at 0:19
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – spaceisdarkgreen, Taroccoesbrocco, Shailesh, amWhy, Adrian Keister
add a comment |Â
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
I'm struggling with the following problem
a) Show that if $X_n oversetpto X$ (convergence in probability) and $Y$ any random variable $X_nY oversetpto XY$
b) Find a case where the previous result isn't true if we change convergence in probability by convergence in distribution.
Thanks in advance for your help.
probability-theory convergence
asked Jul 26 at 2:25
MatÃas
11
I'm struggling with the following problem
a) Show that if $X_n oversetpto X$ (convergence in probability) and $Y$ any random variable $X_nY oversetpto XY$
b) Find a case where the previous result isn't true if we change convergence in probability by convergence in distribution.
Thanks in advance for your help.
probability-theory convergence
asked Jul 26 at 2:25
MatÃas
11
asked Jul 26 at 2:25
MatÃas
11
asked Jul 26 at 2:25
MatÃas
11
asked Jul 26 at 2:25
MatÃas
11
11
closed as off-topic by spaceisdarkgreen, Taroccoesbrocco, Shailesh, amWhy, Adrian Keister Jul 27 at 0:19
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – spaceisdarkgreen, Taroccoesbrocco, Shailesh, amWhy, Adrian Keister
closed as off-topic by spaceisdarkgreen, Taroccoesbrocco, Shailesh, amWhy, Adrian Keister Jul 27 at 0:19
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – spaceisdarkgreen, Taroccoesbrocco, Shailesh, amWhy, Adrian Keister
add a comment |Â
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1 Answer
1
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a), direct method:
For any $c > 0$, you can show $$P(|X_n Y - X Y| > epsilon, |Y| ge c) to 0.$$
Fix $delta > 0$. For an appropriately chosen $c > 0$, we have
$$P(|X_n Y - XY| > epsilon)
le P(|X_n Y - XY| > epsilon, |Y| ge c) + P(0 < |Y| < c)
le fracdelta2 + fracdelta2 = delta$$
for all large $n$.
a), alternate method:
The random vector $(X_n, Y)$ converges to $(X,Y)$ in probability. Thus by the continuous mapping theorem applied to this random vector, $X_n Y oversetpto XY$.
b): Let $X$ be a nonzero symmetric random variable, and let $X_n := -X$ and $Y := X$. We have $X_n oversetdto -X$ because $X$ is symmetric. However, $X_n Y = -X^2$ while $X Y = X^2$.
edited Jul 26 at 19:25
E-A
1,8051312
answered Jul 26 at 2:56
angryavian
34.6k12874
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
a), direct method:
For any $c > 0$, you can show $$P(|X_n Y - X Y| > epsilon, |Y| ge c) to 0.$$
Fix $delta > 0$. For an appropriately chosen $c > 0$, we have
$$P(|X_n Y - XY| > epsilon)
le P(|X_n Y - XY| > epsilon, |Y| ge c) + P(0 < |Y| < c)
le fracdelta2 + fracdelta2 = delta$$
for all large $n$.
a), alternate method:
The random vector $(X_n, Y)$ converges to $(X,Y)$ in probability. Thus by the continuous mapping theorem applied to this random vector, $X_n Y oversetpto XY$.
b): Let $X$ be a nonzero symmetric random variable, and let $X_n := -X$ and $Y := X$. We have $X_n oversetdto -X$ because $X$ is symmetric. However, $X_n Y = -X^2$ while $X Y = X^2$.
edited Jul 26 at 19:25
E-A
1,8051312
answered Jul 26 at 2:56
angryavian
34.6k12874
add a comment |Â
up vote
0
down vote
accepted
a), direct method:
For any $c > 0$, you can show $$P(|X_n Y - X Y| > epsilon, |Y| ge c) to 0.$$
Fix $delta > 0$. For an appropriately chosen $c > 0$, we have
$$P(|X_n Y - XY| > epsilon)
le P(|X_n Y - XY| > epsilon, |Y| ge c) + P(0 < |Y| < c)
le fracdelta2 + fracdelta2 = delta$$
for all large $n$.
a), alternate method:
The random vector $(X_n, Y)$ converges to $(X,Y)$ in probability. Thus by the continuous mapping theorem applied to this random vector, $X_n Y oversetpto XY$.
b): Let $X$ be a nonzero symmetric random variable, and let $X_n := -X$ and $Y := X$. We have $X_n oversetdto -X$ because $X$ is symmetric. However, $X_n Y = -X^2$ while $X Y = X^2$.
edited Jul 26 at 19:25
E-A
1,8051312
answered Jul 26 at 2:56
angryavian
34.6k12874
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
a), direct method:
For any $c > 0$, you can show $$P(|X_n Y - X Y| > epsilon, |Y| ge c) to 0.$$
Fix $delta > 0$. For an appropriately chosen $c > 0$, we have
$$P(|X_n Y - XY| > epsilon)
le P(|X_n Y - XY| > epsilon, |Y| ge c) + P(0 < |Y| < c)
le fracdelta2 + fracdelta2 = delta$$
for all large $n$.
a), alternate method:
The random vector $(X_n, Y)$ converges to $(X,Y)$ in probability. Thus by the continuous mapping theorem applied to this random vector, $X_n Y oversetpto XY$.
b): Let $X$ be a nonzero symmetric random variable, and let $X_n := -X$ and $Y := X$. We have $X_n oversetdto -X$ because $X$ is symmetric. However, $X_n Y = -X^2$ while $X Y = X^2$.
edited Jul 26 at 19:25
E-A
1,8051312
answered Jul 26 at 2:56
angryavian
34.6k12874
a), direct method:
For any $c > 0$, you can show $$P(|X_n Y - X Y| > epsilon, |Y| ge c) to 0.$$
Fix $delta > 0$. For an appropriately chosen $c > 0$, we have
$$P(|X_n Y - XY| > epsilon)
le P(|X_n Y - XY| > epsilon, |Y| ge c) + P(0 < |Y| < c)
le fracdelta2 + fracdelta2 = delta$$
for all large $n$.
a), alternate method:
The random vector $(X_n, Y)$ converges to $(X,Y)$ in probability. Thus by the continuous mapping theorem applied to this random vector, $X_n Y oversetpto XY$.
b): Let $X$ be a nonzero symmetric random variable, and let $X_n := -X$ and $Y := X$. We have $X_n oversetdto -X$ because $X$ is symmetric. However, $X_n Y = -X^2$ while $X Y = X^2$.
edited Jul 26 at 19:25
E-A
1,8051312
answered Jul 26 at 2:56
angryavian
34.6k12874
edited Jul 26 at 19:25
E-A
1,8051312
edited Jul 26 at 19:25
E-A
1,8051312
edited Jul 26 at 19:25
E-A
1,8051312
1,8051312
answered Jul 26 at 2:56
angryavian
34.6k12874
answered Jul 26 at 2:56
angryavian
34.6k12874
answered Jul 26 at 2:56
angryavian
34.6k12874
34.6k12874
add a comment |Â
add a comment |Â