What does convergence in distribution “in the Gromov–Hausdorff” sense mean? The Next CEO of Stack OverflowEssentially one random metric on $mathbbS^2$?Gromov-Hausdorff convergence for locally finite metric spacesEstimating the variance of error in empirical approximation to a distributionError for the convergence by distributionHausdorff convergence of submanifolds in Riemannian manifoldsGromov-Hausdorff limits of 2-dimensional Riemannian surfaces“Local” functional central limit theorem for the empirical distribution functionCovering numbers of uniformly bounded subsets of Gromov-Hausdorff spaceDoes rate of convergence in probability come from a metric?Generalizing Gromov Hausdorff distance using Vietoris topology

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



The Next CEO of Stack OverflowEssentially one random metric on $mathbbS^2$?Gromov-Hausdorff convergence for locally finite metric spacesEstimating the variance of error in empirical approximation to a distributionError for the convergence by distributionHausdorff convergence of submanifolds in Riemannian manifoldsGromov-Hausdorff limits of 2-dimensional Riemannian surfaces“Local” functional central limit theorem for the empirical distribution functionCovering numbers of uniformly bounded subsets of Gromov-Hausdorff spaceDoes rate of convergence in probability come from a metric?Generalizing Gromov Hausdorff distance using Vietoris topology










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
$$(*) , , , mathbbP left[ d_GH[ (m_n,d_n), (X,d) ] < R right] to mathbbP left[ d_GH[ (m_infty,d_infty), (X,d) ] < R right]$$
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?










share|cite|improve this question









$endgroup$
















    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
    $$(*) , , , mathbbP left[ d_GH[ (m_n,d_n), (X,d) ] < R right] to mathbbP left[ d_GH[ (m_infty,d_infty), (X,d) ] < R right]$$
    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?










    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
      $$(*) , , , mathbbP left[ d_GH[ (m_n,d_n), (X,d) ] < R right] to mathbbP left[ d_GH[ (m_infty,d_infty), (X,d) ] < R right]$$
      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?










      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
      $$(*) , , , mathbbP left[ d_GH[ (m_n,d_n), (X,d) ] < R right] to mathbbP left[ d_GH[ (m_infty,d_infty), (X,d) ] < R right]$$
      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













      share|cite|improve this question











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      asked 2 days ago









      Matthew KahleMatthew Kahle

      4,5722950




      4,5722950




















          1 Answer
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          11












          $begingroup$

          Following the notation of the paper, let $mathbbK$ be the metric space of all compact metric spaces, equipped with the Gromov-Hausdorff metric $mathrmd_GH$. Then we can express convergence in distribution in the usual way: for every bounded continuous $F : mathbbK to mathbbR$, we have $mathbbE[F((m_n, d_n))] to mathbbE[F((m_infty, d_infty))]$. 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 $S = (mathbbK, mathrmd_GH)$, the metric space of all compact metric spaces.



          In particular, if $(X,d)$ is a fixed compact metric space, the function $mathrmd_GH(cdot, (X,d)) : mathbbK to mathbbR$ is a continuous function. So if we let $Y_n = mathrmd_GH((m_n, d_n), (X, d))$, 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 $R mapsto mathbbP[mathrmd_GH((m_infty, d_infty), (X,d)) < R]$ is continuous.






          share|cite|improve this answer











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            11












            $begingroup$

            Following the notation of the paper, let $mathbbK$ be the metric space of all compact metric spaces, equipped with the Gromov-Hausdorff metric $mathrmd_GH$. Then we can express convergence in distribution in the usual way: for every bounded continuous $F : mathbbK to mathbbR$, we have $mathbbE[F((m_n, d_n))] to mathbbE[F((m_infty, d_infty))]$. 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 $S = (mathbbK, mathrmd_GH)$, the metric space of all compact metric spaces.



            In particular, if $(X,d)$ is a fixed compact metric space, the function $mathrmd_GH(cdot, (X,d)) : mathbbK to mathbbR$ is a continuous function. So if we let $Y_n = mathrmd_GH((m_n, d_n), (X, d))$, 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 $R mapsto mathbbP[mathrmd_GH((m_infty, d_infty), (X,d)) < R]$ is continuous.






            share|cite|improve this answer











            $endgroup$

















              11












              $begingroup$

              Following the notation of the paper, let $mathbbK$ be the metric space of all compact metric spaces, equipped with the Gromov-Hausdorff metric $mathrmd_GH$. Then we can express convergence in distribution in the usual way: for every bounded continuous $F : mathbbK to mathbbR$, we have $mathbbE[F((m_n, d_n))] to mathbbE[F((m_infty, d_infty))]$. 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 $S = (mathbbK, mathrmd_GH)$, the metric space of all compact metric spaces.



              In particular, if $(X,d)$ is a fixed compact metric space, the function $mathrmd_GH(cdot, (X,d)) : mathbbK to mathbbR$ is a continuous function. So if we let $Y_n = mathrmd_GH((m_n, d_n), (X, d))$, 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 $R mapsto mathbbP[mathrmd_GH((m_infty, d_infty), (X,d)) < R]$ is continuous.






              share|cite|improve this answer











              $endgroup$















                11












                11








                11





                $begingroup$

                Following the notation of the paper, let $mathbbK$ be the metric space of all compact metric spaces, equipped with the Gromov-Hausdorff metric $mathrmd_GH$. Then we can express convergence in distribution in the usual way: for every bounded continuous $F : mathbbK to mathbbR$, we have $mathbbE[F((m_n, d_n))] to mathbbE[F((m_infty, d_infty))]$. 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 $S = (mathbbK, mathrmd_GH)$, the metric space of all compact metric spaces.



                In particular, if $(X,d)$ is a fixed compact metric space, the function $mathrmd_GH(cdot, (X,d)) : mathbbK to mathbbR$ is a continuous function. So if we let $Y_n = mathrmd_GH((m_n, d_n), (X, d))$, 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 $R mapsto mathbbP[mathrmd_GH((m_infty, d_infty), (X,d)) < R]$ is continuous.






                share|cite|improve this answer











                $endgroup$



                Following the notation of the paper, let $mathbbK$ be the metric space of all compact metric spaces, equipped with the Gromov-Hausdorff metric $mathrmd_GH$. Then we can express convergence in distribution in the usual way: for every bounded continuous $F : mathbbK to mathbbR$, we have $mathbbE[F((m_n, d_n))] to mathbbE[F((m_infty, d_infty))]$. 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 $S = (mathbbK, mathrmd_GH)$, the metric space of all compact metric spaces.



                In particular, if $(X,d)$ is a fixed compact metric space, the function $mathrmd_GH(cdot, (X,d)) : mathbbK to mathbbR$ is a continuous function. So if we let $Y_n = mathrmd_GH((m_n, d_n), (X, d))$, 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 $R mapsto mathbbP[mathrmd_GH((m_infty, d_infty), (X,d)) < R]$ is continuous.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 2 days ago

























                answered 2 days ago









                Nate EldredgeNate Eldredge

                20.2k371117




                20.2k371117



























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