I am considering a setting where I am given an iid sample of symmetric positive matrices $\{S_i\}$, $i=1,\dots, n$, of a matrix valued random variable $S$ with distribution $F$. The support of $F$ is the set of positive definite matrices.
I am interested in what happens with the following fixed point equation (and its solution) as $n$ goes to infinity $$ S_{0,n}= \frac{1}{n} \sum_{i=1}^{n}(S_{0,n}^{1/2} S_i S_{0,n}^{1/2})^{1/2}, $$ where $S_{0,n}$ is the solution of the fixed point equation. I know that, for $n$ fixed, there exists a unique solution. This is proven in Agueh, Carlier - Barycenters in Wasserstein space Proof of Theorem 6.1, Step 1 (existence), Step 3 (Uniqueness).
I thought at first, that I could apply the Law of Large Numbers to the random variable $Y_{i,n} := (S_{0,n}^{1/2} S S_{0,n}^{1/2})^{1/2}$. But since $Y_{i,n}$ also depends on $n$ it seems that this is not possible.
Intuitively, I am expecting the sequence of solutions $(S_{0,n})$ to converge to $S_{0,*}$, where $S_{0,*}$ is the solution of the fixed point equation $$ S_{0,*}= \int (S_{0,*}^{1/2} S S_{0,*}^{1/2})^{1/2} dP $$
The context of the problem is Wasserstein barycenters. The Wasserstein barycenter of a set of $n$ Gaussian probability measures $\mu_i \stackrel{def}{=}\mathcal{N}(0, S_i)$ is the measure $\overline{\mu}_n \stackrel{def}{=}\mathcal{N}(0, S_{0,n})$ (Theorem 6.1 in above paper).
I am working on an extension of this to a probabilistic setting where the measures $\mu_i =\mathcal{N}(0, S_i)$ are themselves random obejcts. I assume that they follow a probability distribution $\mathbb{P}_\mathcal{N}$ that is induced by a distribution on the space of covariance matrices. I would like to recover the form of the population barycenter of $\mathbb{P}_\mathcal{N}$.
I know that, if the sequence $(\overline{\mu}_n)_{n \geq 1}$ weakly converges to a measure $\overline{\mu}$, then $\overline{\mu}$ is a barycenter of $\mathbb{P}_{\mathcal{N}}$.
Using the continuity of characteristic functions, I obtain
$$ \lim_{i\rightarrow \infty} \phi_{\overline{\mu}_n}(t)= \exp(- \frac{1}{2} t^T \lim_{n \rightarrow \infty}S_{0,n} t) $$ Hence, if I can show, that the limit $\lim_{n \rightarrow \infty}S_{0,n}$ exists, and, even better, that $\lim_{n \rightarrow \infty}S_{0,n} = S_{0,*}$, then the sequence $(\overline{\mu}_{n})_{n \geq 1}$ weakly converges and the limit has the characteristic function $\lim_{i\rightarrow \infty} \phi_{\overline{\mu}_n}(t)$.