can be verified that any characteristic sequence $(d_1~u_1~\cdots~ d_{2k}~u_{2k})$ uniquely defines a pillar $M(Q)$.

![Characteristic sequence associated with the pillar $M(Q)$ in Figure \[sixm\].[]{data-label="cha"}](cha.eps){width="12cm"}

LSD of $A_T$ using moment method {#lsd}
================================

The problem of showing the convergence of the ESD of $A_T$ reduces to showing that the sequence of its moments $m_k(A_T):={\mathop{\text{tr}}}A_T^k/p$, $k\geq 1$, tends to a limit $(m_k)_k$, and this limit determines properly a probability distribution. For example, the later property is guaranteed if the moment sequence $(m_k)_k$ satisfies the Carleman condition: $$\begin{aligned}
\label{cal}
\sum_{k=1}^{\infty}m_{2k}^{-1/2k}=\infty~.\end{aligned}$$

The following theorem gives the exact formula for the limiting moments $(m_k)_k$.

\[th1\] Suppose the following conditions hold:

(a). $(\varepsilon_{t})_t$ is a sequence of independent $p$-dimensional real valued random vectors with independent entries $\varepsilon_{it}$, $i=1,\cdots, p$ satisfying $${\mathbb{E}}(\varepsilon_{ij})=0,~{\mathbb{E}}(\varepsilon_{ij}^2)=1~$$ and for any $\eta>0$, $$\frac{1}{\eta^4 pT}\sum_{i,j}{\mathbb{E}}\left(|\varepsilon_{ij}|^4I_{(|\varepsilon_{ij}|\geq \eta T^{1/4})}\right)\longrightarrow 0~\quad \text{as}~ p, T \rightarrow \infty;$$

(b). $p$ and $T$ tend to infinity proportionally, that is, $$p \rightarrow \infty, ~T \rightarrow \infty,~ y_T:=p/T \rightarrow y \in (0,\infty)~.$$

Then, with probability one, the empirical spectral distribution $F^{A_T}$ of the matrix $A_T$ in tends to a limiting distribution $F$ whose $k$-th moment is given by: $$m_k=\sum_{i=0}^{k-1}\frac 1k \begin{pmatrix}
                               2k \\
                               i \\
                             \end{pmatrix}\begin{pmatrix}
                                            k \\
                                            i+1 \\
                                          \end{pmatrix}y^{2k-1-i}~.$$

Using 