Definition: Two sequences of random vectors \(\x_n\) and \(\y_n\) are said to be asymptotically equivalent if \(\x_n - \y_n \inP 0\).
Lemma (Asymptotic equivalence): If \(\x_n \inD \x\) and \(\x_n - \y_n \inP 0\), then \(\y_n \inD \x\).
Proof: The basic strategy of this proof is Portmanteau (b \(\to\) a), by which I mean we will show that if \(g\) is any continuous function with compact support, then \(\Ex g(\y_n) \to \Ex g(\x)\). Convergence in distribution then follows from the Portmanteau theorem.
Let \(g: \real^d \to \real\) be a continuous function with compact support, and let \(\eps > 0\). We begin by noting two important properties of \(g\) that are consequences of having compact support.
\[\begin{alignat*}{2} \tag*{$\tcirc{1}$} &\exists \delta > 0: \norm{\x - \y} < \delta \implies \abs{g(\x) - g(\y)} < \eps &\hspace{4em}& g \href{uniformly-continuous.html}{\text{ uniformly continuous}} \\ \tag*{$\tcirc{2}$} &\exists B: \abs{g(\x)} < B \,\forall\,\x && g \textnormal{ continuous} \end{alignat*}\]Now,
\[\begin{alignat*}{2} \abs{\Ex g(\y_n) - \Ex g(\x)} &\le \abs{\Ex g(\y_n) - \Ex g(\x_n)} + \abs{\Ex g(\x_n) - \Ex g(\x)} &\hspace{4em}& \href{norm.html}{\text{Triangle inequality}} \\ &= \abs{\Ex\{ g(\y_n) - g(\x_n) \}} + \abs{\Ex\{ g(\x_n) - g(\x) \}} && \textnormal{Linearity of expectation} \\ &= \abs{\Ex\{ g(\y_n) - g(\x_n) \}} + o(1) && \href{portmanteau.html}{\text{Portmanteau}} \, a \to b; \x_n \to \x \\ &\le \Ex \abs{g(\y_n) - g(\x_n)} + o(1) && \href{jensen.html}{\text{Jensen's inequality}} \\ &= \Ex \abs{g(\y_n) - g(\x_n)}1\{ \norm{\x_n - \y_n} \le \delta \} && \href{law-total-expectation.html}{\text{Law of Total Expectation}} \\ & \quad + \Ex \abs{g(\y_n) - g(\x_n)}1\{ \norm{\x_n - \y_n} > \delta \} \\ & \quad + o(1) \\ &\le \eps + 2B \Pr \{ \norm{\x_n -\y_n} > \delta \} + o(1) && \tcirc{1}, \tcirc{2} \\ &= \eps + o(1) && \href{convergence-in-probability.html}{\x_n - \y_n \inP \zero}; O(1)o(1) = o(1) \end{alignat*}\]Thus, \(\y_n \inD \x\) by the Portmanteau theorem, (b \(\to\) a).
Remark on Taylor series and similar conditions
The following situation often arises: we want to apply a theorem. The theorem has conditions. We can’t really know for sure whether those conditions are met, because they rely on a random quantity. For example, in order to take a Taylor series expansion, we need to know that the function is differentiable at that point; i.e., that \(\x\) lies within a certain neighborhood of \(\x_0\). From the asymptotic equivalence lemma, if our goal is proving convergence in distribution, it is enough to know that \(\x\) lies within the neighborhood with probability going to 1.
To see why, let \(\y_n\) be equal to \(\x_n\) if \(\x_n\) lies within the neighborhood, and let \(\y_n\) be any point in the neighborhood otherwise. Then \(\y_n\) always meets the conditions and we are guaranteed that we can apply a Taylor series expansion to \(f(\y_n)\). Suppose this allows us to prove that \(\y_n \inD \y\). Now, we also know that \(\x_n \inD \y\) because \(\y_n\) and \(\x_n\) are asymptotically equivalent (the probability that they are equal is going to 1).
This kind of argument is sufficiently common that one does not typically go through the details of constructing \(\y_n\), but it is important to know why this argument works.