Sample-covariance resolvent deterministic equivalents

Let $\xi$ have the common entry law. For integer $K$, set ${\xi }^{(K)}=\xi {\mathbf{1}}_{\{\mathrm{\mid }\xi \mathrm{\mid }\le K\}}-\mathbb{E}[\xi {\mathbf{1}}_{\{\mathrm{\mid }\xi \mathrm{\mid }\le K\}}]$, ${\sigma }_K^2=\mathbb{E}\mathrm{\mid }{\xi }^{(K)}{\mathrm{\mid }}^2$, and ${\mathrm{\Delta }}^{(K)}=\xi -{\xi }^{(K)}$. Then ${\sigma }_K\to 1$, and ${\tau }_K^2:=\mathbb{E}\mathrm{\mid }{\mathrm{\Delta }}^{(K)}{\mathrm{\mid }}^2\to 0$ because $\mathbb{E}\xi =0$ and $\mathbb{E}\mathrm{\mid }\xi {\mathrm{\mid }}^2<\mathrm{\infty }$. For all sufficiently large $K$, ${\sigma }_K>0$; put ${\zeta }^{(K)}={\xi }^{(K)}\mathrm{/}{\sigma }_K$. These entries are centered, bounded, iid and have unit second absolute moment, while ${\mathrm{\Delta }}^{(K)}$ is centered iid with finite fourth moment. At matrix level, $X_n={\sigma }_K{Z}_n^{(K)}+D_n^{(K)}$, where $Z_n^{(K)}$ and $D_n^{(K)}$ are the common upper-left blocks formed from ${\zeta }^{(K)}$ and ${\mathrm{\Delta }}^{(K)}$.

For each fixed sufficiently large integer $K$, the original, standardized, and residual arrays have second absolute moments $1$, $1$, and ${\tau }_K^2$, respectively. Applying the variance-scaled bound [F2] to each gives almost surely ${\mathrm{lim\hspace{0.17em}sup}}_n\frac{\mathrm{\parallel }{X}_n\mathrm{\parallel }}{\sqrt{N_n}}\le C_c,{\mathrm{lim\hspace{0.17em}sup}}_n\frac{\mathrm{\parallel }{Z}_n^{(K)}\mathrm{\parallel }}{\sqrt{N_n}}\le C_c,{\mathrm{lim\hspace{0.17em}sup}}_n\frac{\mathrm{\parallel }{D}_n^{(K)}\mathrm{\parallel }}{\sqrt{N_n}}\le C_c{\tau }_K,$ where $C_c=\sqrt{2}(1+\sqrt{c})$. If ${\tau }_K=0$, the centered residual entries are zero almost surely; countability of the double array gives $D_n^{(K)}=0$ for every $n$ on one probability-one event, so the residual bound still holds. Intersect these events over the countable set of such $K$. Writing $A_n^{(K)}=N_n^{-1}{T}_n^{1\mathrm{/}2}{Z}_n^{(K)}{Z}_n^{(K)\ast }{T}_n^{1\mathrm{/}2}$, the identity $U{U}^{\ast }-V{V}^{\ast }=(U-V)U^{\ast }+V(U-V{)}^{\ast }$ yields ${\mathrm{lim\hspace{0.17em}sup}}_n\mathrm{\parallel }{A}_n-{\sigma }_K^2{A}_n^{(K)}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le M{C}_c^2(1+{\sigma }_K){\tau }_K,{\mathrm{lim\hspace{0.17em}sup}}_n\mathrm{\parallel }{\sigma }_K^2{A}_n^{(K)}-A_n^{(K)}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le M{C}_c^2\mathrm{\mid }{\sigma }_K^2-1\mathrm{\mid },$ where $M={\sup }_n\mathrm{\parallel }{T}_n{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$. For positive semidefinite $B,C$ and $z\in {\mathbb{C}}^{+}$, the resolvent identity gives $\mathrm{\parallel }(B-zI{)}^{-1}-(C-zI{)}^{-1}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le \mathrm{\parallel }B-C{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\mathrm{/}(\mathrm{Im}z{)}^2$.

Fix $z\in {\mathbb{C}}^{+}$ and a deterministic unit direction sequence $x_n$. For each such $K$, [F1] applied to $Z_n^{(K)}$ gives $x_n^{\ast }(A_n^{(K)}-zI{)}^{-1}{x}_n\to m_{c,H}(z)$ almost surely, with the transform convention [F9]. By step 2.1, ${\mathrm{lim\hspace{0.17em}sup}}_n\mid x_n^{\ast }(A_n-zI{)}^{-1}{x}_n-m_{c,H}(z)\mid \le \frac{M{C}_c^2}{(\mathrm{Im}z{)}^2}((1+{\sigma }_K){\tau }_K+\mathrm{\mid }{\sigma }_K^2-1\mathrm{\mid })$ on the probability-one event obtained by intersecting the events from step 2.1 with the almost-sure convergence event for this fixed truncation level. The right side tends to zero as $K\to \mathrm{\infty }$, so the directional Stieltjes transforms converge almost surely at every fixed $z\in {\mathbb{C}}^{+}$. Intersecting over a countable dense set gives simultaneous convergence there.

By [F2], $\mathrm{\parallel }{A}_n{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le M\mathrm{\parallel }{X}_n{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}^2\mathrm{/}{N}_n$ is almost surely eventually bounded. Thus the directional laws are eventually supported on one compact interval; [F3] makes them probability measures and [F8] gives their exact spectral-measure meaning. They are therefore tight. The half-line is Polish: it is closed in the complete real line and has dense subset $\mathbb{Q}\cap [0,\mathrm{\infty })$, as required by [F6]. By [A1] and [F4], each subsequence has a weakly convergent further subsequence. Its transform equals $m_{c,H}$ on the dense set by step 3.1; continuity of Stieltjes transforms extends equality to ${\mathbb{C}}^{+}$, and [F5] identifies the subsequential limit as $F_{c,H}$. The bounded-continuous-test definition [F10] yields the stated weak convergence. Every subsequence has the same further limit, proving the asserted almost-sure limit.