responses to major issues below. 2 R#1. Re. correlations among noise. We totally agree with the reviewer that correlated noise is an important topic for 3

We thank all reviewers for their helpful comments and suggestions! We will address all minor issues. We will add references in the revision. UMG ( π) has cycles. We havn't thought about checking it from data, which is an interesting statistical Theorem 1 is effectively known.