# Limit law of eigenvalue of random matrix with mean different to 0

If $$X$$ denotes a $$m \times n$$ random matrix whose entries are independent identically distributed random variables with mean $$\mu$$ and $$\sigma^2 < \infty$$, let

$$Y = X X^T$$

with $$X^T$$ the transpose of $$X$$. Let $$\lambda_1 , \lambda_2 ,\ldots, \lambda_m$$ be the eigenvalues of $$Y$$ (viewed as random variables).

If $$\mu = 0$$, it is known that the law of $$\lambda$$ converges to Marchenko–Pastur distribution: https://en.wikipedia.org/wiki/Marchenko%E2%80%93Pastur_distribution

My question is that in the case $$\mu \neq 0$$ what is the limit distribution of $$\lambda$$?

The addition of the same constant $$\mu$$ to all elements of $$X$$ (so that their mean becomes $$\mu$$) is a rank-one perturbation of the matrix, which has no effect on the distribution of the eigenvalues of $$XX^T$$ in the limit $$n,m\rightarrow\infty$$ at fixed $$n/m$$ --- this limiting distribution remains the Marchenko-Pastur distribution. The rank-one perturbation introduces a single outlier, but the bulk of the spectrum is unchanged. See for example The singular values and vectors of low rank perturbations of large rectangular random matrices (2011).