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UPDATE: The result below can be understood as an almost sure stochastic equicontinuity condition. I don't know of any result establishing primitives of almost sure stochastic equicontinuity. If you know, this would very likely help me answer the question.

I need the following almost sure convergence result:

Let $||\cdot||$ denote a norm on a functional space (could be $||\cdot||_{\infty}$ or $||\cdot ||_2$ in $L^2$, for example). $\mathcal{G}=\{g:\mathcal{Z}\rightarrow \mathbb{R}\}$ of measurable functions. Let $Z_1,Z_2,\dots$ be $i.i.d$ random variables with $E[g(Z_i)]=0$ for any $g\in \mathcal{G}$. Let $\{\delta_n\}$ be a sequence of positive numbers such that $\delta_n\rightarrow 0$. Then, under ADDITIONAL ASSUMPTIONS, \begin{equation*} \sup_{g\in\mathcal{G} \text{ s.t. }||g||\leq \delta_n}\left| \frac{1}{\sqrt{n}}\sum_{i=1}^n g(Z_i)\right|\rightarrow_{a.s.}0. \end{equation*}

Note that all random variables are defined in the same probability space and are independent draws of the same distribution. The number $n$ here denotes the size of the sample, and the almost sure convergence is over the distribution of the $Z_i$ as $n\rightarrow \infty$. The space $\mathcal{G}$ does not depend on $n$, and $g$ is not random.

I need to find which reasonable assumptions could be made to establish this result. $\mathcal{G}$ is $P$-Donsker? Lipschitz functions? Bounds on moments? etc.


Attending requests for clarification of the background: this result is needed to establish a stochastic equicontinuity condition on a linear regression estimator with a generated regressor (it means that one of the regressors is estimated). This is a high level result, in the sense that it is a result for a generic estimator for the regressor, we want to give conditions that this estimator must satisfy, but not specify the estimator. I am using results in a paper by Chen, Linton and Keilegom (2003) which establish the consistency of the bootstrap for estimators which are based on the optimization of a function of the data, the parameter of interest and a nuisance infinite dimension parameter. So, suppose the model is $E[Y|X,W]=\beta X+\gamma h_0(W)$, but $h_0$ is not known. The term is thus estimated and the regression is done on $\hat{h}(W)$ instead. If it helps, you can think of $h_0(W)=E[V|V\leq 0, W]$, for another variable $V$ which is observed in the data, for example.(FIY: this is a huge simplification of the true situation, so don't give me suggestions about how to estimate the model I just described. In the actual model $V$ is not observed and must be predicted out of sample with machine learning and optimization methods.)

In order to apply one of the results in the paper I mentioned, I need to prove that some quantities such as the one above are $o_{a.s.}(1)$. For example, I give you a simplified version of one of them: \begin{equation*} \sup_{h,h_0\in\mathcal{H} \text{ s.t. }||h-h_0||\leq \delta_n}\left| \frac{1}{\sqrt{n}}\sum_{i=1}^n \left[X_i(h(W_i)-h_0(W_i))-E[X_i(h(W_i)-h_0(W_i))\right]\right| \end{equation*} I need to show that several objects that look more or less like the one above are $o_{a.s.}(1)$ as $n\rightarrow \infty$ for any sequence of positive $\delta_n\rightarrow 0$. Note that I can make assumptions about $\mathcal{H}$, $X$ and $W$. For example, I can say that $\mathcal{H}$ is the space of measurable negative Lipschitz functions, and I can say that $E[|X|^4]$ and $E[|W|^4]$ are finite. Basically I need conditions that would allow me to establish this result.

At the essence this is an empirical process. Define $\mathbb{G}_nf=\frac{1}{\sqrt{n}}\sum_{i=1}^n f(Z_i)-E[f(Z_i)]$ and let $\mathbb{G}$ be the Brownian Bridge, then if $\mathcal{G}$ is $P$-Donsker, $\mathbb{G}_ng\rightsquigarrow\mathbb{G}g$ for every fixed $g$. This is a convergence in distribution in a functional space. I need to show that $\sup_{||g||_{\mathcal{G}}\leq \delta_n} |\mathbb{G}_n g|\rightarrow_{a.s.} 0$.

I hope this is clearer and someone can help me. I ran out of ideas of things to try and need a fresh take.

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  • $\begingroup$ I think you should wait a little bit more than 9 hours on Math Stack Overflow. $\endgroup$ – Dieter Kadelka Jul 4 '20 at 23:01
  • $\begingroup$ To me this seems to some sort of "Rundumschlag" (german). Can you make your question more concrete, specifying the function space (what is the $\sup$-norm) and clarify your problem. For example is your $\sup$ measurable as a supremum over uncountable many functions? $\endgroup$ – Dieter Kadelka Jul 4 '20 at 23:47
  • $\begingroup$ As stated the claim does not hold, as $Z_i$ could be uniform on $[-1,1]$ and for each $n$, the function $g$ could be an odd function from $[-1,1]$ to $[-\delta_n,\delta_n]$, chosen to satisfy $g(Z_i)=\delta_n$ for all $i \le n$. More care is needed in the supremum. $\endgroup$ – Yuval Peres Jul 5 '20 at 2:49
  • $\begingroup$ @DieterKadelka Define any normed space, suppose that the norm exists. I even took out the $\sup$ from the description (I simply meant $||g||=\sup_{z\in\mathcal{Z}}|g(z)|$ and you can suppose that the functions $g$ are defined in $\mathcal{Z}$ with image in $\mathbb{R}$ and are bounded.) I think it is clear in the question that I have plenty of freedom to decide the functional space. If you know of a theorem that may achieve this, I can probably assume its requirements. I added explanations to the question. See new paragraph. $\endgroup$ – Caetano Jul 5 '20 at 5:27
  • $\begingroup$ @YuvalPeres Yes, it should be obvious that as stated it does not hold, as I mentioned that even to prove this convergence in probability one needs bounded second moments. Your example forgets the requirement in the statement that $E[g(Z_i)]=0$. Please see updated comments. $\endgroup$ – Caetano Jul 5 '20 at 5:50
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A sufficient condition to establish the result above for the convergence in probability is if $\mathcal{G}$ has finite entropy with an envelope function $M$ which satisfies $E[M(Z)^2]<\infty$. This allows us to apply Theorem 1 in https://cowles.yale.edu/sites/default/files/files/pub/d10/d1059.pdf.

Unfortunately I could not find any result for almost sure stochastic equicontinuity, which is probably the answer to this problem. I will look it up further. If you know anything about this, please let me know.

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