TITLE: What Guarantees the Legitimacy of Stochastic Taylor Expansion in Maximum Likelihood?
QUESTION [1 upvotes]: Picture of the key step proving normality of MLE in Bhattachary et al. (2016)
The picture is the key step to prove that MLE is asymptotically normal in Bhattachary et al. (2016). The common justification for the legitimacy of Taylor expansion here is that we do expansion point-wise over the underlying probability space and treat the stochastic function as fixed one. However, this justification is not satisfactory, since point-wise Lagrangian remainder over the probability space may not lead to a random variable without guarantees about measurability. Clearly, we have outer measure and outer expectation to study non-measurability, but things would be more convenient if measurable.
Aliprantis and Border (2006) proved univariate stochastic Taylor expansion by using measurable correspondence theory, and following their spirits Yang and Zhou (2021) extended the theorem to multivariate cases. These results are highly relevant, but failed to justify the expansion in MLE directly. The likelihood function we want to expand is a function of both the sample and the estimator, which is substitute for true parameter. So an ideal expansion should be sort of partial expansion only w.r.t. the estimator.
Is there any reference that gives the desired theorem?

REPLY [0 votes]: I have consulted Prof. Van der Vaart. He suggests that the last term in the picture, which includes mean-value remainder, is measurable and bounded by a measurable term converging to zero in the proof. This argument means whether there exists a Taylor's theorem is irrelevant to the proof. He also mentions a direct theoretical gaurantee is not obvious.
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