interest.

The rest of the paper is organized as follows. Preliminary introduction on the related graph theory is provided in Section \[graph\]. Section \[lsd\] derives the exact moment formula for the limiting spectral distribution of $A_T$, which further leads to the expression of its corresponding Stieltjes transform. Section \[largest\] gives details of the convergence of the largest eigenvalue of $A_T$. In Section \[de\], we provide some techniques related to random walks for deriving some recursion formula, which further leads to the limiting moments in Section \[lsd\].

Some graph theory {#graph}
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In order to calculate the moments of the LSD, we need some information from graph theory.

For a pair of vectors of indexes $i=(i_1,\cdots,i_{2k})$ $(1\leq i_l \leq T, ~l\leq 2k)$ and $j=(j_1,\cdots,j_{2k})$ $(1\leq j_l \leq p, ~l\leq 2k)$, construct a graph $Q(i,j)$ in the following way. Draw two parallel lines, referred to as the I-line and J-line. Plot $i_1,\cdots,i_{2k}$ on the I-line and $j_1,\cdots,j_{2k}$ on the J-line, called the I-vertices and J-vertices, respectively. Draw $k$ down edges from $i_{2u-1}$ to $j_{2u-1}$, $k$ down edges from $i_{2u}+s$ to $j_{2u}$, $k$ up edges from $j_{2u-1}$ to $i_{2u}$, $k$ up edges from $j_{2u}$ to $i_{2u+1}+s$ (all these up and down edges are 