the square roots of the $p$ nonnegative eigenvalues of $$\begin{aligned}
\label{at}
A_T:=X_TX_T^T~.\end{aligned}$$ Therefore, the main purpose of this paper is on the limiting behaviors of the eigenvalues of $A_T$. First, we derive exact moment formula for its LSD. Next, this moment sequence is shown to satisfy the Carleman condition so that it determines uniquely the LSD. Then using power expansion, the Stieltjes transform of the LSD $$s(z):=\int \frac{1}{x-z}dF(x)$$ can be found to satisfy the following equation: $$\begin{aligned}
y^2z^2s^3(z)+y^2zs^2(z)-yzs^2(z)-zs(z)-1=0~.\end{aligned}$$ This result is derived in [@li] using a totally different Stieltjes transform method. Second, we show the convergence of the largest eigenvalue of $A_T$ to the right end point of the support of $F$ almost surely (trivially, this implies the convergence of the largest singular value of $X_T$). Both these two results are derived using moment method. A distinctive feature here is that the matrix $A_T$ can be considered as the product of four matrices involving $\varepsilon_t$, new methodology is needed with respect to the existing literature on random matrix theory. In particular, we provide in Section \[de\] some recursion formula related to the number of random walks, which further leads to our moment result, and the method may be of independent 