# Bootstrap consistency for maximum likelihood

I'm looking for references (textbook if possible) that treat the strong consistency of bootstrapped maximum likelihood (MLH) or more generally M-estimators. By strong consistency I mean that the difference between the cdf of the original statistic and the bootstrap counterpart goes a.s. to zero uniformly (see below).

So far I have been following the textbook:

Shao, Tu: The Jacknife and Bootstrap. 1995.

that provides a really nice account of the bootstrap and theorems of this type. For example, see Theorem 3.1 (page 80) in Shao & Tu, denote by $$\bar{X}_n$$ the sample mean of $$X_1,\ldots,X_n$$ (i.i.d.), and by $$h\colon \mathbb{R}\to\mathbb{R}$$ a function such that $$h'(\mu) \neq 0$$ and $$h'$$ is continuous at $$\mu$$, where $$\mu = \mathbb{E}[X]$$ and $$\mathbb{E}[X^2] < \infty$$. For a statistic $$T_n = h(\bar{X}_n)$$ with bootstrap counterpart $$T^*_n$$ we then have $$\Vert F_{T_n^*} - F_{T_n} \Vert_{\infty} = \sup_{x\in\mathbb{R}} \vert F_{T_n^*}(x) - F_{T_n}(x) \vert \xrightarrow[n\to\infty]{a.s.} 0.$$ Here $$F_{T_n^*}$$ and $$F_{T_n}$$ are the respective distribution functions and the convergence a.s. is necessary since $$F_{T_n^*}$$ is random.

This settles method of moment type estimators (in fact in Shao & Tu the general multivariate version is considered), but I'm wondering about implicitly defined estimators like MLH (or M-estimators).

My question is therefore, if $$\mathcal{M} = \{f_{\theta};\theta\in\Theta\}$$ is a parametric statistical model and $$\widehat{\theta}_n$$ is the MLH estimator based on $$X_1,\ldots,X_n$$ (i.i.d. with density $$f_{\theta_0}$$) what can be said about the limit $$\Vert F_{\widehat{\theta}_n^*} - F_{\widehat{\theta}_n} \Vert_{\infty}.$$

I'm looking for references that either give conditions and a proof, or, preferably, a textbook that outlines the results in a digestible way.