A clarification on Stable Convergence of Triangular Arrays of Random Variables
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Consider the following important theorem, due to Jacod (1997), on stable convergence of triangular arrays of random variables. In what follows $mathcal{F}_t$ indicates a filtration on $[0,1]$ and $t_{j,n}$ is a partition of the time-interval $[0,1]$.
My question: is it strictly necessary that the conditioning that appears in conditions from $1)$ to $5)$ is taken with respect to $mathcal{F}_{t_{j-1,n}}$ ? Would the theorem still apply if, for example, conditioning with respect to $mathcal{F}_{t_{j-2,n}}$ is considered?
random-variables weak-convergence
asked Jan 9 at 10:14
AlmostSureUserAlmostSureUser
331418
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add a comment |
$begingroup$
Consider the following important theorem, due to Jacod (1997), on stable convergence of triangular arrays of random variables. In what follows $mathcal{F}_t$ indicates a filtration on $[0,1]$ and $t_{j,n}$ is a partition of the time-interval $[0,1]$.
My question: is it strictly necessary that the conditioning that appears in conditions from $1)$ to $5)$ is taken with respect to $mathcal{F}_{t_{j-1,n}}$ ? Would the theorem still apply if, for example, conditioning with respect to $mathcal{F}_{t_{j-2,n}}$ is considered?
random-variables weak-convergence
asked Jan 9 at 10:14
AlmostSureUserAlmostSureUser
331418
$endgroup$
add a comment |
$begingroup$
Consider the following important theorem, due to Jacod (1997), on stable convergence of triangular arrays of random variables. In what follows $mathcal{F}_t$ indicates a filtration on $[0,1]$ and $t_{j,n}$ is a partition of the time-interval $[0,1]$.
My question: is it strictly necessary that the conditioning that appears in conditions from $1)$ to $5)$ is taken with respect to $mathcal{F}_{t_{j-1,n}}$ ? Would the theorem still apply if, for example, conditioning with respect to $mathcal{F}_{t_{j-2,n}}$ is considered?
random-variables weak-convergence
asked Jan 9 at 10:14
AlmostSureUserAlmostSureUser
331418
$endgroup$
Consider the following important theorem, due to Jacod (1997), on stable convergence of triangular arrays of random variables. In what follows $mathcal{F}_t$ indicates a filtration on $[0,1]$ and $t_{j,n}$ is a partition of the time-interval $[0,1]$.
My question: is it strictly necessary that the conditioning that appears in conditions from $1)$ to $5)$ is taken with respect to $mathcal{F}_{t_{j-1,n}}$ ? Would the theorem still apply if, for example, conditioning with respect to $mathcal{F}_{t_{j-2,n}}$ is considered?
random-variables weak-convergence
random-variables weak-convergence
asked Jan 9 at 10:14
AlmostSureUserAlmostSureUser
331418
asked Jan 9 at 10:14
AlmostSureUserAlmostSureUser
331418
asked Jan 9 at 10:14
AlmostSureUserAlmostSureUser
331418
asked Jan 9 at 10:14
AlmostSureUserAlmostSureUser
331418
asked Jan 9 at 10:14
AlmostSureUserAlmostSureUser
331418
331418
add a comment |
add a comment |
1 Answer
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Here are some ideas, for example for assumption 1. Let
$$
tag{1'}sum_{j=1}^nmathbb Eleft[Y_{t_{j,n}}midmathcal F_{t_{j-2},n}right]to 0 mbox{ in probability}.
$$
If (1') holds, in order to check (1), we have to prove that
$$
tag{1'}sum_{j=1}^nd_{n,j}to 0 mbox{ in probability},
$$
where $d_{n,j}=mathbb Eleft[Y_{t_{j,n}}midmathcal F_{t_{j-1},n}right]-mathbb Eleft[Y_{t_{j,n}}midmathcal F_{t_{j-2},n}right]$. The fact that $left(d_{n,j}right)$ is a martingale difference sequence can help.
answered Jan 19 at 11:05
Davide GiraudoDavide Giraudo
128k17156268
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Here are some ideas, for example for assumption 1. Let
$$
tag{1'}sum_{j=1}^nmathbb Eleft[Y_{t_{j,n}}midmathcal F_{t_{j-2},n}right]to 0 mbox{ in probability}.
$$
If (1') holds, in order to check (1), we have to prove that
$$
tag{1'}sum_{j=1}^nd_{n,j}to 0 mbox{ in probability},
$$
where $d_{n,j}=mathbb Eleft[Y_{t_{j,n}}midmathcal F_{t_{j-1},n}right]-mathbb Eleft[Y_{t_{j,n}}midmathcal F_{t_{j-2},n}right]$. The fact that $left(d_{n,j}right)$ is a martingale difference sequence can help.
answered Jan 19 at 11:05
Davide GiraudoDavide Giraudo
128k17156268
$endgroup$
add a comment |
$begingroup$
Here are some ideas, for example for assumption 1. Let
$$
tag{1'}sum_{j=1}^nmathbb Eleft[Y_{t_{j,n}}midmathcal F_{t_{j-2},n}right]to 0 mbox{ in probability}.
$$
If (1') holds, in order to check (1), we have to prove that
$$
tag{1'}sum_{j=1}^nd_{n,j}to 0 mbox{ in probability},
$$
where $d_{n,j}=mathbb Eleft[Y_{t_{j,n}}midmathcal F_{t_{j-1},n}right]-mathbb Eleft[Y_{t_{j,n}}midmathcal F_{t_{j-2},n}right]$. The fact that $left(d_{n,j}right)$ is a martingale difference sequence can help.
answered Jan 19 at 11:05
Davide GiraudoDavide Giraudo
128k17156268
$endgroup$
add a comment |
$begingroup$
Here are some ideas, for example for assumption 1. Let
$$
tag{1'}sum_{j=1}^nmathbb Eleft[Y_{t_{j,n}}midmathcal F_{t_{j-2},n}right]to 0 mbox{ in probability}.
$$
If (1') holds, in order to check (1), we have to prove that
$$
tag{1'}sum_{j=1}^nd_{n,j}to 0 mbox{ in probability},
$$
where $d_{n,j}=mathbb Eleft[Y_{t_{j,n}}midmathcal F_{t_{j-1},n}right]-mathbb Eleft[Y_{t_{j,n}}midmathcal F_{t_{j-2},n}right]$. The fact that $left(d_{n,j}right)$ is a martingale difference sequence can help.
answered Jan 19 at 11:05
Davide GiraudoDavide Giraudo
128k17156268
$endgroup$
Here are some ideas, for example for assumption 1. Let
$$
tag{1'}sum_{j=1}^nmathbb Eleft[Y_{t_{j,n}}midmathcal F_{t_{j-2},n}right]to 0 mbox{ in probability}.
$$
If (1') holds, in order to check (1), we have to prove that
$$
tag{1'}sum_{j=1}^nd_{n,j}to 0 mbox{ in probability},
$$
where $d_{n,j}=mathbb Eleft[Y_{t_{j,n}}midmathcal F_{t_{j-1},n}right]-mathbb Eleft[Y_{t_{j,n}}midmathcal F_{t_{j-2},n}right]$. The fact that $left(d_{n,j}right)$ is a martingale difference sequence can help.
answered Jan 19 at 11:05
Davide GiraudoDavide Giraudo
128k17156268
answered Jan 19 at 11:05
Davide GiraudoDavide Giraudo
128k17156268
answered Jan 19 at 11:05
Davide GiraudoDavide Giraudo
128k17156268
answered Jan 19 at 11:05
Davide GiraudoDavide Giraudo
128k17156268
128k17156268
add a comment |
add a comment |
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