Closed cshen6 closed 7 years ago
Main:
Proof forward: most of thm1 and thm2 holds directly without iid (in dcorr07), so the only step to justify is the SLLN for v-statistic in thm 2.
Carefully check through the SLLN for exchangeable random variables & the V-statistic. Need finite first moment in SLLN for conditional iid, I think.
Also need finite second moment above; and note that the adjacency matrix is 2-array joint exchangeable.
What about directed & weighted graph model?
@cshen6 Some useful properties valid for exchangeable variables are actually proven in http://www.sciencedirect.com/science/article/pii/S0047259X13000262.
@youjin1207 yes indeed! although it is for coordinate exchangeability (dimension-wise), rather than observation exchangeability (sample-wise).
@cshen6 On page 6, they actually mentioned sample exchangeability, even though they didn't specifically prove it. In my understanding, we apply conditional CLT for conditionally independence random variables. I don't know if we actually have to PROVE it, since the results are already there - I don't see the difference between Theorem in my current draft and conditional CLT - instead of sub-sigma-algebra notion, I used g function or eta.
@youjin1207 I listed the forward and backward directions above for thm X, which is the last theoretical piece. Either way, it will lead to proving the theorem and the corollary.
@cshen6 Now I don't think distance in conditional variable and that in un-conditional variable are the same so we cannot approach in that way.
ok I will think about it before our next meeting.
also, I think adjacency matrix is in fact exchangeable, if we look at the definition and let the exchangeability to be defined on both row and column of the adjacency matrix.
Important:
Tentative: