What does convergence in distribution “in the Gromov–Hausdorff” sense mean?
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I am trying to understand this survey article by Le Gall on Brownian geometry, especially the statement of Theorem 1.
The basic statement of the theorem is
$$(m_n,d_n) to (m_{infty}, d_{infty})$$
"in the Gromov–Hausdorff sense" as $n to infty$, where the convergence is in distribution.
Here $(m_n,d_n)$ and $(m_{infty},d_{infty})$ are both random compact metric spaces. So how do we interpret this? We might hope for a statement along the lines of the following.
For every compact metric space $(X,d)$ and $R > 0$, we have
$$(*) , , , mathbb{P} left[ d_{GH}[ (m_n,d_n), (X,d) ] < R right] to mathbb{P} left[ d_{GH}[ (m_{infty},d_{infty}), (X,d) ] < R right]$$
as $n to infty$.
But even when we talk about convergence in distribution for real random variables (instead of compact-metric-space-valued random variables), we have to be careful to restrict our attention to points where the cumulative distribution function is continuous. So I wonder if (*) is too strong?
pr.probability mg.metric-geometry
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I am trying to understand this survey article by Le Gall on Brownian geometry, especially the statement of Theorem 1.
The basic statement of the theorem is
$$(m_n,d_n) to (m_{infty}, d_{infty})$$
"in the Gromov–Hausdorff sense" as $n to infty$, where the convergence is in distribution.
Here $(m_n,d_n)$ and $(m_{infty},d_{infty})$ are both random compact metric spaces. So how do we interpret this? We might hope for a statement along the lines of the following.
For every compact metric space $(X,d)$ and $R > 0$, we have
$$(*) , , , mathbb{P} left[ d_{GH}[ (m_n,d_n), (X,d) ] < R right] to mathbb{P} left[ d_{GH}[ (m_{infty},d_{infty}), (X,d) ] < R right]$$
as $n to infty$.
But even when we talk about convergence in distribution for real random variables (instead of compact-metric-space-valued random variables), we have to be careful to restrict our attention to points where the cumulative distribution function is continuous. So I wonder if (*) is too strong?
pr.probability mg.metric-geometry
$endgroup$
add a comment |
$begingroup$
I am trying to understand this survey article by Le Gall on Brownian geometry, especially the statement of Theorem 1.
The basic statement of the theorem is
$$(m_n,d_n) to (m_{infty}, d_{infty})$$
"in the Gromov–Hausdorff sense" as $n to infty$, where the convergence is in distribution.
Here $(m_n,d_n)$ and $(m_{infty},d_{infty})$ are both random compact metric spaces. So how do we interpret this? We might hope for a statement along the lines of the following.
For every compact metric space $(X,d)$ and $R > 0$, we have
$$(*) , , , mathbb{P} left[ d_{GH}[ (m_n,d_n), (X,d) ] < R right] to mathbb{P} left[ d_{GH}[ (m_{infty},d_{infty}), (X,d) ] < R right]$$
as $n to infty$.
But even when we talk about convergence in distribution for real random variables (instead of compact-metric-space-valued random variables), we have to be careful to restrict our attention to points where the cumulative distribution function is continuous. So I wonder if (*) is too strong?
pr.probability mg.metric-geometry
$endgroup$
I am trying to understand this survey article by Le Gall on Brownian geometry, especially the statement of Theorem 1.
The basic statement of the theorem is
$$(m_n,d_n) to (m_{infty}, d_{infty})$$
"in the Gromov–Hausdorff sense" as $n to infty$, where the convergence is in distribution.
Here $(m_n,d_n)$ and $(m_{infty},d_{infty})$ are both random compact metric spaces. So how do we interpret this? We might hope for a statement along the lines of the following.
For every compact metric space $(X,d)$ and $R > 0$, we have
$$(*) , , , mathbb{P} left[ d_{GH}[ (m_n,d_n), (X,d) ] < R right] to mathbb{P} left[ d_{GH}[ (m_{infty},d_{infty}), (X,d) ] < R right]$$
as $n to infty$.
But even when we talk about convergence in distribution for real random variables (instead of compact-metric-space-valued random variables), we have to be careful to restrict our attention to points where the cumulative distribution function is continuous. So I wonder if (*) is too strong?
pr.probability mg.metric-geometry
pr.probability mg.metric-geometry
asked Mar 29 at 16:00
Matthew KahleMatthew Kahle
4,5722950
4,5722950
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Following the notation of the paper, let $mathbb{K}$ be the metric space of all compact metric spaces, equipped with the Gromov-Hausdorff metric $mathrm{d_{GH}}$. Then we can express convergence in distribution in the usual way: for every bounded continuous $F : mathbb{K} to mathbb{R}$, we have $mathbb{E}[F((m_n, d_n))] to mathbb{E}[F((m_infty, d_infty))]$. The portmanteau theorem gives you several other equivalent statements.
In other words, this is just the usual notion of convergence in distribution for random variables taking their values in a metric space $S$, where that metric space happens to be $S = (mathbb{K}, mathrm{d_{GH}})$, the metric space of all compact metric spaces.
In particular, if $(X,d)$ is a fixed compact metric space, the function $mathrm{d_{GH}}(cdot, (X,d)) : mathbb{K} to mathbb{R}$ is a continuous function. So if we let $Y_n = mathrm{d_{GH}}((m_n, d_n), (X, d))$, then the scalar-valued random variables $Y_n$ converge in distribution to $Y$. So your formula (*) holds, but as you say, only for values of $R$ at which the function $R mapsto mathbb{P}[mathrm{d_{GH}}((m_infty, d_infty), (X,d)) < R]$ is continuous.
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1 Answer
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$begingroup$
Following the notation of the paper, let $mathbb{K}$ be the metric space of all compact metric spaces, equipped with the Gromov-Hausdorff metric $mathrm{d_{GH}}$. Then we can express convergence in distribution in the usual way: for every bounded continuous $F : mathbb{K} to mathbb{R}$, we have $mathbb{E}[F((m_n, d_n))] to mathbb{E}[F((m_infty, d_infty))]$. The portmanteau theorem gives you several other equivalent statements.
In other words, this is just the usual notion of convergence in distribution for random variables taking their values in a metric space $S$, where that metric space happens to be $S = (mathbb{K}, mathrm{d_{GH}})$, the metric space of all compact metric spaces.
In particular, if $(X,d)$ is a fixed compact metric space, the function $mathrm{d_{GH}}(cdot, (X,d)) : mathbb{K} to mathbb{R}$ is a continuous function. So if we let $Y_n = mathrm{d_{GH}}((m_n, d_n), (X, d))$, then the scalar-valued random variables $Y_n$ converge in distribution to $Y$. So your formula (*) holds, but as you say, only for values of $R$ at which the function $R mapsto mathbb{P}[mathrm{d_{GH}}((m_infty, d_infty), (X,d)) < R]$ is continuous.
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add a comment |
$begingroup$
Following the notation of the paper, let $mathbb{K}$ be the metric space of all compact metric spaces, equipped with the Gromov-Hausdorff metric $mathrm{d_{GH}}$. Then we can express convergence in distribution in the usual way: for every bounded continuous $F : mathbb{K} to mathbb{R}$, we have $mathbb{E}[F((m_n, d_n))] to mathbb{E}[F((m_infty, d_infty))]$. The portmanteau theorem gives you several other equivalent statements.
In other words, this is just the usual notion of convergence in distribution for random variables taking their values in a metric space $S$, where that metric space happens to be $S = (mathbb{K}, mathrm{d_{GH}})$, the metric space of all compact metric spaces.
In particular, if $(X,d)$ is a fixed compact metric space, the function $mathrm{d_{GH}}(cdot, (X,d)) : mathbb{K} to mathbb{R}$ is a continuous function. So if we let $Y_n = mathrm{d_{GH}}((m_n, d_n), (X, d))$, then the scalar-valued random variables $Y_n$ converge in distribution to $Y$. So your formula (*) holds, but as you say, only for values of $R$ at which the function $R mapsto mathbb{P}[mathrm{d_{GH}}((m_infty, d_infty), (X,d)) < R]$ is continuous.
$endgroup$
add a comment |
$begingroup$
Following the notation of the paper, let $mathbb{K}$ be the metric space of all compact metric spaces, equipped with the Gromov-Hausdorff metric $mathrm{d_{GH}}$. Then we can express convergence in distribution in the usual way: for every bounded continuous $F : mathbb{K} to mathbb{R}$, we have $mathbb{E}[F((m_n, d_n))] to mathbb{E}[F((m_infty, d_infty))]$. The portmanteau theorem gives you several other equivalent statements.
In other words, this is just the usual notion of convergence in distribution for random variables taking their values in a metric space $S$, where that metric space happens to be $S = (mathbb{K}, mathrm{d_{GH}})$, the metric space of all compact metric spaces.
In particular, if $(X,d)$ is a fixed compact metric space, the function $mathrm{d_{GH}}(cdot, (X,d)) : mathbb{K} to mathbb{R}$ is a continuous function. So if we let $Y_n = mathrm{d_{GH}}((m_n, d_n), (X, d))$, then the scalar-valued random variables $Y_n$ converge in distribution to $Y$. So your formula (*) holds, but as you say, only for values of $R$ at which the function $R mapsto mathbb{P}[mathrm{d_{GH}}((m_infty, d_infty), (X,d)) < R]$ is continuous.
$endgroup$
Following the notation of the paper, let $mathbb{K}$ be the metric space of all compact metric spaces, equipped with the Gromov-Hausdorff metric $mathrm{d_{GH}}$. Then we can express convergence in distribution in the usual way: for every bounded continuous $F : mathbb{K} to mathbb{R}$, we have $mathbb{E}[F((m_n, d_n))] to mathbb{E}[F((m_infty, d_infty))]$. The portmanteau theorem gives you several other equivalent statements.
In other words, this is just the usual notion of convergence in distribution for random variables taking their values in a metric space $S$, where that metric space happens to be $S = (mathbb{K}, mathrm{d_{GH}})$, the metric space of all compact metric spaces.
In particular, if $(X,d)$ is a fixed compact metric space, the function $mathrm{d_{GH}}(cdot, (X,d)) : mathbb{K} to mathbb{R}$ is a continuous function. So if we let $Y_n = mathrm{d_{GH}}((m_n, d_n), (X, d))$, then the scalar-valued random variables $Y_n$ converge in distribution to $Y$. So your formula (*) holds, but as you say, only for values of $R$ at which the function $R mapsto mathbb{P}[mathrm{d_{GH}}((m_infty, d_infty), (X,d)) < R]$ is continuous.
edited Mar 29 at 19:44
answered Mar 29 at 16:34
Nate EldredgeNate Eldredge
20.2k371117
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