Consider an $N\times K$ random matrix $X$ (defined on a probability space $(Ω,F,μ)$) with i.i.d. entries having zero mean and variance $1/K$.

There are a lot of results regarding the asymptotic behavior of the empirical distribution of eigenvalues of $XX^T$, or more precisely, the asymptotic behavior of the Stieltjes transform of the matrix $XX^T$ (the Marcenko-Pastur law), this result J. W. Silverstein, “Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices”, J. Multivar. Anal. 54, 175–192 (1995), etc.

Now, all the asymptotic random matrix results that I saw (in this context) are derived under the assumption that both dimensions of the matrix $X$ grows to infinity, but with fixed ratio. Namely, it is assumed that as $K,N→∞$ we fix $N/K→c∈(0,∞)$.

My question is whether there are some results, in case that both dimensions goes to infinity but with vanishing ratio. Concretely, can we say something regard the asymptotic behavior of, e.g. $$ \frac{1}{N}\text{trace}(XX^T+I_N)^{−1} $$ where $N/K→0$. Note that if $N$ is fixed, then we can readily use the SLLN to draw conclusions regard the asymptotic behavior of the above transform.