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Suppose $X_n$ is a random matrix, which converges in probability to a matrix of constants, $Y$. It seems intuitive that therefore $X_n^{-1} \xrightarrow{p} Y^{-1}$ - so the limit in probability of an inverse matrix is equal to the inverse of the limit in probability of the matrix.

Indeed, this property is used in many proofs for basic properties of OLS. For instance, in deriving the asymptotic distribution of the OLS estimator $\hat{\beta}$ under standard Gauss-Markov assumptions, it is often assumed that $\text{plim} \frac{X'X}{n} = Q$ where $Q$ is full rank, and then we use the property that $X_n \xrightarrow{p} Y$ implies $X_n^{-1} \xrightarrow{p} Y^{-1}$ (and the fact that $Q$ is full rank, i.e. invertible) to show that $\text{plim} \Big[\frac{X'X}{n}\Big]^{-1} = Q^{-1}$.

However, despite how often this property is used and how intuitive it seems, I am struggling to find a proof of why it holds. Some texts say that this property holds due to Slutsky's theorem, but I'm not sure how Slutsky's theorem applies in the context of taking the inverse of a matrix.

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I would not say that Slutsky's theorem in its oft presented form is a worthwhile theorem to use here.

Slutsky's theorem, as far as I have ever seen, deals with convergence in distribution involving combinations of a sequence of random variables that converges in distribution to some random variable $X$, and a sequence of random variables that converges to a constant $a$. And then only deals with convergence in distribution of combinations of those two sequences, not convergence in probability. It's a nice result for establishing central limit theorem type results to estimators.

I believe that it should be the Continuous mapping theorem that applies here, since it deals directly with convergence in probability; not only that, but also convergence in distribution and almost sure convergence.

The gist of the theorem and how it applies to your case is that if you can show that a getting the inverse of a matrix (supposing it exists) is a continuous mapping, then the convergence in probability holds. This continuity of calculating the inverse is shown in the answer to Continuity of the inverse matrix function (as mentioned in your comments). So if $X_n \overset{p}{\to} Y$ then $X_n^{-1} \overset{p}{\to} Y^{-1}$ by the continuous mapping theorem.

In any situation like this, look for continuity. If you have continuity then you have your convergence. Just make sure you are paying attention to the right type of convergence.

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