What does convergence in distribution “in the Gromov–Hausdorff” sense mean?












9












$begingroup$


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







share|cite|improve this question









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    9












    $begingroup$


    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







    share|cite|improve this question









    $endgroup$















      9












      9








      9


      1



      $begingroup$


      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







      share|cite|improve this question









      $endgroup$




      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






      share|cite|improve this question













      share|cite|improve this question











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      share|cite|improve this question










      asked Mar 29 at 16:00









      Matthew KahleMatthew Kahle

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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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            $begingroup$

            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.






            share|cite|improve this 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.






              share|cite|improve this 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.






                share|cite|improve this answer











                $endgroup$



                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.







                share|cite|improve this answer














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                share|cite|improve this answer








                edited Mar 29 at 19:44

























                answered Mar 29 at 16:34









                Nate EldredgeNate Eldredge

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                20.3k371117






























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