Minimal eigenvalue of symmetric random matrix generated by a random vector

Question

Suppose we have a random row vector $V_n=(v_1,...,v_n)$, where $v_1,...,v_n$ are iid and real-valued.

We now create the matrix $M_n=\frac{1}{n}V^TV$.

Are there any nontrivial assumptions on the distributions of $\{v_i\}_i$ that leads to some results regarding the distribution of the minimal singular value or minimal eigenvalue of $M_n$?

Answer 1

Assuming that $V=V_n,$ your matrix is symmetric and rank one, so its top eigenvalue (and singular value) is $V_n \centerdot V_n,$ and all the other eigen/singular values are $0.$