This is a minor extension of Slutsky’s theorem that I call “Slutsky’s extension”, but be aware that this is not a term that would mean anything to anyone outside of this course.
Theorem: Suppose \(\y_n \inD \y\), where \(\y\) is a \(d \times 1\) random vector, \(\A_n \inP \A\), where \(\A\) is a positive definite matrix, and that \(\y_n = \A_n\x_n\). Then
\[\x_n \inD \A^{-1}\y\]Proof: Given as homework. Note that the result would be trivial if \(\A_n\) were invertible, but we don’t know that it is – only that its limit, \(\A\), is invertible.