Timeline for Weak convergence of $\mathcal{L}^2$ valued random variables
Current License: CC BY-SA 4.0
12 events
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| Jan 13, 2020 at 2:15 | comment | added | Iosif Pinelis | @esner1994 : What will work in the compact case is your assumption that $g$ is continuous and hence $g(x,\cdot)$ is continuous for each $x$. | |
| Jan 12, 2020 at 21:35 | comment | added | esner1994 | And how would you proceed in the compact case? The prooves of the three Theorems mentioned above are all similar, but adjusting the arguments there doesn't work, because with $L^2$ we have to deal with a space of equivalence classes of functions and so the projections $f \rightarrow f(t)$ are not even well-defined. I forgot to mention: There is also such a result for reproducing kernel hilbert spaces (see Theroem 93 in Thomas-Agnan "Reproducing kernel hilbert spaces in probability and statistics"), but the proof is also the same as for $C[0,1]$. | |
| Jan 12, 2020 at 20:38 | comment | added | Iosif Pinelis | The difficulty here is that we have to deal with $L^2(\mathbb R)$, and $\mathbb R$ is not compact. | |
| Jan 12, 2020 at 19:39 | history | edited | esner1994 | CC BY-SA 4.0 |
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| Jan 12, 2020 at 19:38 | comment | added | esner1994 | $C[0,1]$: Billingsley, "Convergence of probability measures", Theorem 8.1 $D[0,1]$: Billingsley, "Convergence of probability measures", Theorem 15.1 $C[0,\infty)$ Whitt, "Weak Convergence of Probability Measures on the Function Space $C[ 0, \infty)$", Theorem 3 Sorry, I thought the last one was about $C(\mathbb{R})$, but it probably holds as well. I corrected it. | |
| Jan 12, 2020 at 18:50 | comment | added | Iosif Pinelis | Can you give references to those "similar results [that] hold for the spaces $C[0,1]$, $D[0,1]$ and the Frechet space $C(\mathbb{R})$"? | |
| Jan 12, 2020 at 18:43 | history | edited | esner1994 |
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| Jan 12, 2020 at 15:25 | history | edited | esner1994 | CC BY-SA 4.0 |
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| Jan 12, 2020 at 15:20 | comment | added | esner1994 | Yes, I mean the convergence in the Levy-Prokhorov metric. | |
| Jan 12, 2020 at 15:12 | comment | added | fedja | And how do you define convergence in distribution for $L^2$-valued random variables? (Is it just the convergence of $\mathbb P_n$ in the Levy-Prokhorov metric or something weaker or stronger than that?) | |
| Jan 12, 2020 at 14:15 | review | First posts | |||
| Jan 12, 2020 at 16:43 | |||||
| Jan 12, 2020 at 14:11 | history | asked | esner1994 | CC BY-SA 4.0 |