Lemma 18 ([Posterior covariance and outlier subspaces]). Use the critical sequence and notation of Proposition 17 ([The prediction curve]), with I→0 in the gap case. The following assertions hold. For every fixed c<∞ , uniformly over …

Lemma 18 ([Posterior covariance and outlier subspaces]). Use the critical sequence and notation of Proposition 17 ([The prediction curve]), with I→0 in the gap case. The following assertions hold. For every fixed c<∞ , uniformly over deterministic orthogonal projections Π with Π≤cI , |Πw|^2=o(I). For every ε>0 , for all sufficiently far members of the sequence there is a deterministic subspace V_ε of dimension at most C_εI such that, for every v⊥V_ε , v^*( ww^*)v≤ε|v|^2, v^*( b_sb_s^*)v≤ε|v|^2, v^*( M_s)v≤ε|v|^2. In the gap case V_ε may be taken to be zero, and \| b_sb_s^*\|_ +\| M_s\|_ =o(1) . In the strict case, for every fixed r∈(0,1/4) and ε>0 , the conditional covariance C_r=(w(Y_0) U_r) satisfies (C_r\1_ C_r>ε )=o(I). The indicator denotes the spectral projection. No assertion about additional Gaussian probes or Equation is needed for the gap case.