Let $A$ be $(n,n)$ central Wishart matrix with $k$ degrees of freedom. my question is there is a way to estimate the expectation of: \begin{align} E[det(I+(I+A)^{1})] \end{align}
2 Answers
The joint distribution of the eigenvalues $\lambda_i$, $i=1,2,\ldots n$ of $A$ is known, $$P(\lambda_1,\lambda_2,\ldots\lambda_n)=c_{k,n}\prod_{i<j}\lambda_i\lambda_j\prod_m e^{\lambda_m/2}\lambda_m^{(kn1)/2},$$ with $c_{k,n}$ a normalization constant.
The desired expectation value is given by $$U_{n,k}=\mathbb E[\det(I+(I+A)^{1})]=\int_0^\infty d\lambda_1\int_0^\infty d\lambda_2\cdots \int_0^\infty d\lambda_n\,\prod_{i<j}\lambda_i\lambda_j\prod_m e^{\lambda_m/2}\lambda_m^{(kn1)/2}\left(1+(1+\lambda_m)^{1}\right).$$ For small $n$ the integrals can be done by quadrature, but the integrals quickly become unwieldy.
For example, for $n=1$, $k>0$ I find $$U_{1,k}=2^{\frac{k}{2}} \left[2^{k/2}+\sqrt{e} \,\Gamma \left(1\tfrac{1}{2}k,\tfrac{1}{2}\right)\right],$$ with $\Gamma$ the incomplete Gamma function.
If you are satisfied with the expectation value of the logarithm of the determinant, then you can use the MarchenkoPastur distribution to obtain an accurate result for large $n$.
You did not specify whether you have a relation between $n$ and $k$ and whether you care about asymptotics. I will assume that $k/n\to 1$ and that $n\to\infty$, and that your scaling is such that the eigenvalues are of order $1$. (The case $k/n\to\alpha$ can be handled similarly; also, if you meant that the typical eigenvalues are of order $k$, then the situation is much simpler). Let $\lambda_i$ denote the eigenvalues of $A$. Let $L_k=k^{1} \sum \delta_{\lambda_i}$ denote the empirical measure. Then $\log det(I+(I+A)^{1})=\sum \log (1+(1+\lambda_i))^{1}= k \langle L_k,g\rangle$ where $g(x)=\log(1+(1+x)^{1})$ is continuous and bounded on $R_+$. Now, $L_k$ satisfies a LDP at speed $k^2$, which means that the probability that $L_k$ is not in a small neighborhood of the PasturMarchenko law $\mu$ is exponentially small at scale $k^2$. You then have that $$k^{1}\log E \det(I+(I+A)^{1}) =k^{1} \log E(e^{k \langle L_k,g\rangle})\to \langle \mu,g\rangle $$

$\begingroup$ Do you happen to know (or have a reference) if this LDP is known for empirical covariance matrices of iid Subgaussian vectors? (i.e. not necessarily independent entries). I know this is very specific, so no worries if nothing comes to mind. Thank you! $\endgroup$– DJAMar 3, 2022 at 18:33

$\begingroup$ Update: posted my question in longer form mathoverflow.net/questions/417364 $\endgroup$– DJAMar 3, 2022 at 20:46