Proof of a step of a lemma on the asymptotics of maximum likelihood where a Taylor expansion is used

Question

I am trying to understand a proof of quite a long theorem that I report completely for the sake of completeness. This is From Jensen and Rahbek Asymptotic Inference for Nonstationary GARCH (2004). My question is actually basic and only regards the Taylor expansion done in the proof so one can skip to the end if he wishes.

At the start of the proof through the mean value theorem $\lambda^*$ is proven to exist. Then a Taylor expansion is made.

$$\Big| \frac{\partial f(\lambda^*)}{\partial \lambda} \Big| = \Big| \sum_{i,j,l = 1}^k v_{1,i}v_{2,j} (\varphi_l - \varphi_{0,l}) \partial^3 \ell_T (\varphi_0 - \lambda^* (\varphi - \varphi_0))/ \partial \varphi_i \varphi_j \varphi_l \Big|$$

I think he cuts it off after the second term, to be clear I write the Taylor expansion that I think he does in the univariate form:

$$ f(\lambda^*)+\frac {f'(\lambda^*)}{1!} (\varphi-\lambda^*)+ \cdots. $$

My questions that I think are all connected are:

1. Where does the first term go in the expansion? (where is $ f(\lambda^*)$ ?)
2. Why introduce $\lambda^*$ at all? (notice that condition A.3 of Lemma 1 does not involve $\lambda^*$)?
3. Why does equality hold in this Taylor expansion?

Answer 1

It is the Lagrange form of Taylor's theorem with remainder (https://www.math.hmc.edu/calculus/tutorials/taylors_thm/) I believe.

And it is an expansion of f(1) around f(0), ie $f(1)=f(0)+f'(\lambda^*).(1-0)$, for some $\lambda^*\in (0,1)$.

So hopefully it is clear that this explains your 3 questions.