Showing $mathbb{E}S_{tau}^2=mathbb{E}tau$.
$begingroup$
Suppose that $x_1, x_2,...x_n$ are independent copies of random variable $x$ having distribution $$P(x=1)=P(x=-1)=frac{1}{2}.$$ In addition, suppose that $mathcal{D}=mathcal{D}_{x_1,...,x_k}(k=1,...,n),$ $S_k=x_1+x_2+...+x_k$ for $(k=1,...,n)$ and $tau$ is a stopping time with respect to the decomposition sequence $mathcal{D}_1 preceq mathcal{D}_2 preceq ... preceq mathcal{D}_n.$ Show that $mathbb{E}S_{tau}^2=mathbb{E}tau$.
How should I start and what to use. I have no idea... I know that $mathbb{E}S_k=mathbb{E}x_1 mathbb{E}k$. Do I need to show that $S_k$ is a martingale?
probability probability-theory random-variables martingales random-walk
asked Jan 15 at 15:14
AtstovasAtstovas
1089
$endgroup$
add a comment |
$begingroup$
Suppose that $x_1, x_2,...x_n$ are independent copies of random variable $x$ having distribution $$P(x=1)=P(x=-1)=frac{1}{2}.$$ In addition, suppose that $mathcal{D}=mathcal{D}_{x_1,...,x_k}(k=1,...,n),$ $S_k=x_1+x_2+...+x_k$ for $(k=1,...,n)$ and $tau$ is a stopping time with respect to the decomposition sequence $mathcal{D}_1 preceq mathcal{D}_2 preceq ... preceq mathcal{D}_n.$ Show that $mathbb{E}S_{tau}^2=mathbb{E}tau$.
How should I start and what to use. I have no idea... I know that $mathbb{E}S_k=mathbb{E}x_1 mathbb{E}k$. Do I need to show that $S_k$ is a martingale?
probability probability-theory random-variables martingales random-walk
asked Jan 15 at 15:14
AtstovasAtstovas
1089
$endgroup$
$begingroup$
I just fixed it. What should I do next?
$endgroup$
– Atstovas
Jan 15 at 15:37
$begingroup$
So I have no idea what I have to do. I've never had any exercise like this one.
$endgroup$
– Atstovas
Jan 15 at 16:14
$begingroup$
@saz, $n$ appears to be fixed, here, so your $tau$ is not a stopping time with respect to the filtration mentioned in the problem. This gets around the issue you're talking about.
$endgroup$
– Marcus M
Jan 15 at 17:13
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@MarcusM Ah, I see, thanks for the explanation.
$endgroup$
– saz
Jan 15 at 17:45
$begingroup$
Possible duplicate of math.stackexchange.com/questions/378463/…
$endgroup$
– E-A
Jan 16 at 7:20
add a comment |
$begingroup$
Suppose that $x_1, x_2,...x_n$ are independent copies of random variable $x$ having distribution $$P(x=1)=P(x=-1)=frac{1}{2}.$$ In addition, suppose that $mathcal{D}=mathcal{D}_{x_1,...,x_k}(k=1,...,n),$ $S_k=x_1+x_2+...+x_k$ for $(k=1,...,n)$ and $tau$ is a stopping time with respect to the decomposition sequence $mathcal{D}_1 preceq mathcal{D}_2 preceq ... preceq mathcal{D}_n.$ Show that $mathbb{E}S_{tau}^2=mathbb{E}tau$.
How should I start and what to use. I have no idea... I know that $mathbb{E}S_k=mathbb{E}x_1 mathbb{E}k$. Do I need to show that $S_k$ is a martingale?
probability probability-theory random-variables martingales random-walk
asked Jan 15 at 15:14
AtstovasAtstovas
1089
$endgroup$
Suppose that $x_1, x_2,...x_n$ are independent copies of random variable $x$ having distribution $$P(x=1)=P(x=-1)=frac{1}{2}.$$ In addition, suppose that $mathcal{D}=mathcal{D}_{x_1,...,x_k}(k=1,...,n),$ $S_k=x_1+x_2+...+x_k$ for $(k=1,...,n)$ and $tau$ is a stopping time with respect to the decomposition sequence $mathcal{D}_1 preceq mathcal{D}_2 preceq ... preceq mathcal{D}_n.$ Show that $mathbb{E}S_{tau}^2=mathbb{E}tau$.
How should I start and what to use. I have no idea... I know that $mathbb{E}S_k=mathbb{E}x_1 mathbb{E}k$. Do I need to show that $S_k$ is a martingale?
probability probability-theory random-variables martingales random-walk
probability probability-theory random-variables martingales random-walk
asked Jan 15 at 15:14
AtstovasAtstovas
1089
asked Jan 15 at 15:14
AtstovasAtstovas
1089
edited Jan 15 at 16:12
Atstovas
asked Jan 15 at 15:14
AtstovasAtstovas
1089
asked Jan 15 at 15:14
AtstovasAtstovas
1089
asked Jan 15 at 15:14
AtstovasAtstovas
1089
1089
$begingroup$
I just fixed it. What should I do next?
$endgroup$
– Atstovas
Jan 15 at 15:37
$begingroup$
So I have no idea what I have to do. I've never had any exercise like this one.
$endgroup$
– Atstovas
Jan 15 at 16:14
$begingroup$
@saz, $n$ appears to be fixed, here, so your $tau$ is not a stopping time with respect to the filtration mentioned in the problem. This gets around the issue you're talking about.
$endgroup$
– Marcus M
Jan 15 at 17:13
$begingroup$
@MarcusM Ah, I see, thanks for the explanation.
$endgroup$
– saz
Jan 15 at 17:45
$begingroup$
Possible duplicate of math.stackexchange.com/questions/378463/…
$endgroup$
– E-A
Jan 16 at 7:20
add a comment |
$begingroup$
I just fixed it. What should I do next?
$endgroup$
– Atstovas
Jan 15 at 15:37
$begingroup$
So I have no idea what I have to do. I've never had any exercise like this one.
$endgroup$
– Atstovas
Jan 15 at 16:14
$begingroup$
@saz, $n$ appears to be fixed, here, so your $tau$ is not a stopping time with respect to the filtration mentioned in the problem. This gets around the issue you're talking about.
$endgroup$
– Marcus M
Jan 15 at 17:13
$begingroup$
@MarcusM Ah, I see, thanks for the explanation.
$endgroup$
– saz
Jan 15 at 17:45
$begingroup$
Possible duplicate of math.stackexchange.com/questions/378463/…
$endgroup$
– E-A
Jan 16 at 7:20
$begingroup$
I just fixed it. What should I do next?
$endgroup$
– Atstovas
Jan 15 at 15:37
$begingroup$
I just fixed it. What should I do next?
$endgroup$
– Atstovas
Jan 15 at 15:37
$begingroup$
So I have no idea what I have to do. I've never had any exercise like this one.
$endgroup$
– Atstovas
Jan 15 at 16:14
$begingroup$
So I have no idea what I have to do. I've never had any exercise like this one.
$endgroup$
– Atstovas
Jan 15 at 16:14
$begingroup$
@saz, $n$ appears to be fixed, here, so your $tau$ is not a stopping time with respect to the filtration mentioned in the problem. This gets around the issue you're talking about.
$endgroup$
– Marcus M
Jan 15 at 17:13
$begingroup$
@saz, $n$ appears to be fixed, here, so your $tau$ is not a stopping time with respect to the filtration mentioned in the problem. This gets around the issue you're talking about.
$endgroup$
– Marcus M
Jan 15 at 17:13
$begingroup$
@MarcusM Ah, I see, thanks for the explanation.
$endgroup$
– saz
Jan 15 at 17:45
$begingroup$
@MarcusM Ah, I see, thanks for the explanation.
$endgroup$
– saz
Jan 15 at 17:45
$begingroup$
Possible duplicate of math.stackexchange.com/questions/378463/…
$endgroup$
– E-A
Jan 16 at 7:20
$begingroup$
Possible duplicate of math.stackexchange.com/questions/378463/…
$endgroup$
– E-A
Jan 16 at 7:20
add a comment |
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$begingroup$
I just fixed it. What should I do next?
$endgroup$
– Atstovas
Jan 15 at 15:37
$begingroup$
So I have no idea what I have to do. I've never had any exercise like this one.
$endgroup$
– Atstovas
Jan 15 at 16:14
$begingroup$
@saz, $n$ appears to be fixed, here, so your $tau$ is not a stopping time with respect to the filtration mentioned in the problem. This gets around the issue you're talking about.
$endgroup$
– Marcus M
Jan 15 at 17:13
$begingroup$
@MarcusM Ah, I see, thanks for the explanation.
$endgroup$
– saz
Jan 15 at 17:45
$begingroup$
Possible duplicate of math.stackexchange.com/questions/378463/…
$endgroup$
– E-A
Jan 16 at 7:20