The name “Slutsky’s theorem” is widely used in an inconsistent manner to mean a number of similar results. Here, we use Slutsky’s theorem to mean the result below; its various corollaries come from combining this result with the continuous mapping theorem.
Theorem (Slutsky): If \(\x_n \inD \x\) and \(\y_n \inP \a\), where \(\a\) is a constant, then \(\begin{aligned}\left[ \begin{array}{c} \x_n \\ \y_n \end{array} \right] \inD \left[ \begin{array}{c} \x \\ \a \end{array} \right]. \end{aligned}\)
Proof: The proof is rather simple if the asymptotic equivalence lemma has already been proven. The idea is to show that \((\x_n \quad \y_n)\) is asymptotically equivalent to \((\x_n \quad \a)\).
First, note that \((\x_n \quad \a) \inD (\x \quad \a)\) since \(\x_n \inD \x\). This can be shown a number of ways, perhaps the simplest is by using the CMT and noting that \(f(\x) = (\x \quad \a)\) is a continuous function.
Second, letting \(\eps > 0\),
\[\begin{alignat*}{2} \Pr\{ \norm{(\x_n \quad \y_n) - (\x_n \quad \a)} > \eps \} &= \Pr\{ \norm{\y_n - \a} > \eps \} &\hspace{4em}& \sqrt{0^2 + 0^2 + \ldots + x_1^2 + x_2^2 + \ldots} = \norm{\x} \\ &\to 0 && \y_n \inP \a \end{alignat*}\]Thus, by the AEL, \((\x_n \quad \y_n) \inD (\x_n \quad \a)\).
Univariate example
A very common application of Slutsky’s theorem is that if \(A_n \inP a\), \(B_n \inP b\), and \(X_n \inD X\), then
\[A_n X_n + B_n \inD aX + b.\]This comes from using the above result to conclude that \((X_n \quad A_n \quad B_n) \inD (X \quad a \quad b)\), then applying the continuous mapping theorem, recognizing that addition and multiplication are continuous.
Multivariate example
It is often helpful to disentangle Slutsky’s theorem and the continuous mapping theorem, as they can be combined to show results far beyond addition and multiplication. For example, if \(\x_1, \x_2, \ldots\) is an iid sample with mean \(\bm\) and nonsingular variance, then
\[n(\bar{\x}-\bm) \Tr \S_n^{-1} (\bar{\x} - \bm) \inD \chi_d^2.\]Proof:
\[\begin{alignat*}{2} & \sqrt{n}(\bar{\x} - \bm) \inD \Norm(\zero, \bS) &\hspace{4em}& \href{central-limit-theorem.html}{\text{CLT}} \\ & \S_n \inP \bS && \textnormal{Good exercise!} \\ & [\sqrt{n}(\bar{\x} - \bm) \quad \S_n] \inD [\Norm(\zero, \bS) \quad \bS] && \textnormal{Slutsky; just stack $S_n$} \\ & \sqrt{n}\S_n^{-1/2}(\bar{\x} - \bm) \inD \bS^{-1/2} \Norm(\zero, \bS) && \href{continuous-mapping-theorem.html}{\text{CMT}}; \textnormal{Matrix inversion is continuous} \\ & \phantom{\sqrt{n}\S_n^{-1/2}(\bar{\x} - \bm)} =_d \Norm(\zero, \I) \\ & n(\bar{\x}-\bm) \Tr \S_n^{-1} (\bar{\x} - \bm) \inD \chi_d^2 && \href{continuous-mapping-theorem.html}{\text{CMT}}; \textnormal{Inner product is continuous} \end{alignat*}\]