# Determining the asymptotic behavior of some scalar function of random matrices

Consider a series of random matrices $X_n\in\mathbb{R}^{n\times m}$ consisting of i.i.d. entries, each with zero mean and variance $1/m$, and let $y_n\in\mathbb{R}^{n\times1}$ be a random vector with i.i.d. entries, zero mean, with each entry having unit variance. It is given that $y_n$ is independent on $X_n$.

I want to find the structure of some "nice"/"simple" "limit" function, $f_n$, of the following term $$e_1^T\left(X_nX_n^T+I_n\right)^{-1}y_n-f_n\to0$$ almost surely (or weakly), as $n,m\to\infty$ with fixed ratio, where $e_1$ is the unit vector.

My problem is with the scaling. If there was a $1/n$ normalization before the left term, then obviously the limit is zero, and actually it is not difficult to show that $$\frac{1}{n}z_n^T\left(X_nX_n^T+I_n\right)^{-1}y_n\to0$$ for any random vector $z_n$ that is independent of the other variables.

Without the scaling, can I say something regarding the limit in the a.s. sense, or maybe in the weak sense?

Second question: Can we claim something regard the convergence of (which is of course an example of the previous question) $$\frac{1}{n}e_i^T\left(X_nX_n^T+I_n\right)^{-1}e_j$$ where $e_i,e_j$ are again the unit vectors with '1' at the $i$ and $j$ indexes, respectively. When $i=j$ it is not very complicated to find the limit, but I'm curious what happens if $i\neq j$.

EDIT: This question is related to this post.

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