Asymptotics is strange...

$\newcommand{\argmin}{\mathop{\mathrm{argmin}}\limits}$ $\newcommand{\argmax}{\mathop{\mathrm{argmax}}\limits}$

It is a useful trick to “flip” the denominator into numerator when it comes to proving asymptotic properties of errors. Here’s how to do so. Consider a form

\[\frac{1}{b_0 + b_1 h + o(h)},\]

where $h \to 0$ and $b_i$’s are non-zero constants. Then

\[\begin{aligned} &\frac{1}{b_0 + b_1 h + o(h)} \\ &= \frac{1}{b_0\left(1 + \frac{b_1}{b_0} h + o(h)\right)} \\ &= \frac{1}{b_0} \left(1 - \frac{b_1}{b_0} h + o(h)\right). \end{aligned}\]

The trick is not only easy to use but easy to understand as well. Although there isn’t any technically difficult part, staring at the result I could not help but say this. Asymptotics is strange…