Timeline for Uniqueness of the limit sequence of discrete probability measures
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 26, 2019 at 20:14 | answer | added | Dieter Kadelka | timeline score: 2 | |
| Apr 23, 2019 at 9:47 | comment | added | Dieter Kadelka | Starting with the comment of Remling if $\rho$ is a limit point, then $d\rho = fd\lambda$ for some $f \geq c > 0$. Put $F(t) := \int_0^t f(x)dx$, then $F$ continuous, strictly increasing with $F(0) = 0, F(1) = 1$ and image measure $\rho^T = \lambda$. Apply $F$ to the double sequence $x_i^n$. Then for this transformed sequence $\rho_n \to \lambda$. For this case f.i. the condition $\liminf_{n \to \infty} N_n r_n = 1$ is sufficient (not necessary). Then apply the backtransformation $F^{-1}$ and translate the conditions for the special case. | |
| Apr 23, 2019 at 1:20 | comment | added | Ben Ciotti | @ChristianRemling Excellent point re absolute continuity with respect to Lebesgue measure. I was aware of this, but I'm impressed you figured it out as fast as you did. | |
| Apr 21, 2019 at 12:07 | history | edited | Pietro Majer | CC BY-SA 4.0 |
spelling
|
| Apr 21, 2019 at 2:07 | answer | added | user95282 | timeline score: 4 | |
| Apr 21, 2019 at 0:20 | comment | added | Christian Remling | @MargaretFriedland: Except that this has unbounded density, so can't be the limit when the points $x_n$ satisfy the OP's assumptions. | |
| Apr 20, 2019 at 21:21 | comment | added | Margaret Friedland | If the measure $\rho$ is the arcsine measure, then such conditions can be formulated. Let $\hat I_n=\prod_{x,y \in X_n, x \ne y}{|x-y|}^{\frac{1}{N_n(N_n-1)}}$. If $\lim_{n \to \infty} \hat I_n = e^{-1/4}$, then $\rho_n \to \rho$ vaguely. This is because $\rho$ is the (unique) measure with minimal logarithmic energy on $[0,1]$ (viewed as a subset of the complex plane) and $1/4$ is the value of this minimal energy. The numbers $I_n$ can be thought of as ``discrete energies". | |
| Apr 20, 2019 at 16:32 | comment | added | Christian Remling | If the inf equals $c>0$, then your condition says that any limit point $\rho$ is ac with respect to Lebesgue measure with density $\le 1/c$. Conversely, all such prob measures are possible as limit points. I don't think there will be a useful criterion how to read off convergence from the points that is not near tautological. | |
| Apr 20, 2019 at 6:15 | comment | added | Pierre PC | At some point, you'll need to rule out the situation where $X_{2n}$ consists of $n$ points uniformly distributed in $[0;1/2]$ and $X_{2n+1}$ consists of $n$ points in $[1/2;1]$. More generally, if two sequences converge to two different limits and satisfy your assumptions, a mixture as described above will have two limit points and still satisfy your criteria. | |
| Apr 20, 2019 at 3:09 | history | asked | Ben Ciotti | CC BY-SA 4.0 |