I would like to find the largest eigenvalue distribution of the following tridiagonal symmetric random matrix in an analytic way.

All the ${\lambda}_i$ are distributed the same way with chi-square (but any other real positive distribution will be OK if it helps)

\begin{align} \small %%%%% M := \begin{bmatrix} %% 0 & \lambda_1^{-1/2} & 0 &0 & \dots &0&0 \\ %% \lambda_1^{-1/2} & 0 & \lambda_2^{-1/2} & 0 & \dots &0&0 \\ %% 0& \lambda_2^{-1/2} & 0 & \lambda_3^{-1/2} & \dots &0&0 \\ %%% & &&\vdots \\ %%% 0& 0 & 0 & \dots &\lambda_{n-1}^{-1/2} & 0& \lambda_{n }^{-1/2} \\ %%% 0& 0 & 0 & \dots &0 & \lambda_{n }^{-1/2} & 0 %%% \end{bmatrix}. \end{align}

Although Sturm sequences seems to be a good fit for this issue, I couldn’t solve the integral for any positive distribution when ${n>2}$