Timeline for Intersection of projection of sets
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9 events
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| Oct 17, 2020 at 22:11 | comment | added | qp212223 | I understand why Villani uses the projection after creating a set that has measure 1 on the product measure. But this is unnecessary. Let $\Gamma = \{(x,y): \phi_k(y) - \psi_k(x) \rightarrow c(x,y)\}$. $\pi(\Gamma) = 1$ assuming we have chosen the subsequence out of the $L^1$ converging sequence for which convergence holds $\pi -$a.s. $\Gamma$ is c-cyclically monotone by the argument he gives precisely. We do not need the projection doing it this way. We do need the projection if we do it his way. | |
| Oct 16, 2020 at 4:05 | comment | added | Gabriel Clara | Upon rereading there seems to be a quite subtle reason why the projection argument comes up. Notice at the bottom of p. 68 how $\phi_k(y_i) - \psi_k(x_i) \to c(x_i, y_i)$ converges in expectation with respect to $\pi(\mathrm{d}x_i, \mathrm{d}y_i)$, for each $(x_i, y_i)$. Picking a sub-sequence yields almost sure convergence under $$\bigotimes_{i = 1}^{N} \pi(\mathrm{d} x_i, \mathrm{d} y_{i+1}),$$ but the latter is written simply as $\pi^{\otimes N}$. The projection argument is then natural to get the desired result for $\pi$. It's maybe a little unusual to write it out though. | |
| Oct 13, 2020 at 17:47 | comment | added | qp212223 | Well the proof for my $\Gamma$ is given on p. 68. Suppose we have $N$ points from $\Gamma$ defined in my comment, $\{(x_i, y_i)\}_{i=1}^N$. Then we obtain, by definition of $\phi_k, \psi_k$, $$\sum_{i=1}^N c(x_i, y_{i+1}) \ge \sum_{i=1}^N \phi_k(y_{i+1}) - \psi_k(x_i) = \sum_{i=1}^N \phi_k(y_{i}) - \psi_k(x_i) \rightarrow \sum_{i=1}^N c(x_i, y_i)$$ where we obtain the equality by just rearranging terms in a finite sum, and we may take the limit as $k \rightarrow \infty$ since again, we deal with a finite $N$.So $\Gamma$ is c-cyclically monotone by definition. | |
| Oct 12, 2020 at 22:58 | comment | added | Gabriel Clara | (2/2) Since $\pi$ has to be concentrated on any such $\Gamma$, the symmetric difference between the $\Gamma$ in the book and yours would have to be negligible with respect to $\pi$ though. The point about permutations is maybe clearer in the references above, where cyclical monotonicity is defined in terms of general permutations of $N$, not just via shifting each $y_i$ to $y_{i+1}$. | |
| Oct 12, 2020 at 22:53 | comment | added | Gabriel Clara | The problem is that you need to show that every possible $N$-tuple of $\Gamma$ satisfies the cyclical monotonicity condition. A priori, we know that all elements of $\Gamma_N$ satisfy the condition, but what if we could permute an element to produce a counterexample? The projection argument essentially discards points that would allow us to do this. Then we show that $\pi$ is concentrated on $\Gamma$ to conclude. If you can show that the same holds true for your $\Gamma$, then you found an alternative argument. | |
| Oct 12, 2020 at 6:45 | vote | accept | qp212223 | ||
| Oct 12, 2020 at 6:45 | comment | added | qp212223 | Thanks for the answer. I'm not even sure why Villani makes the projection argument. If you just let $\Gamma = \{(x,y): \phi_k(y) - \psi_k(x) \rightarrow c(x,y)\}$ for $\phi_k, \psi_k$ referenced on the previous page, this is a c-cyclically monotone set for which $\pi$ has measure $1$ so we'd already be done. | |
| Oct 12, 2020 at 4:09 | review | First posts | |||
| Oct 12, 2020 at 5:40 | |||||
| Oct 12, 2020 at 4:06 | history | answered | Gabriel Clara | CC BY-SA 4.0 |