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I am reading the paper [1]. At page 18, eq 115, it is claimed the following:

Given a separable process $(X_t)_{t\in T}$, we have $\lim_{n\to\infty}\mathbb E[\sup_{t\in T}(X_t-X_{\pi_n(t)})]=0$.

Here the only hypothesis on the labelling space $T$ is that it is a bounded metric space. $(X_t)$ is actually assumed to be a subgaussian process, but this fact should not matter for the property stated, or at least the inequality is justified only using the separability in the paper. $\pi_n$ defines a sequence of refining nets. This means that $d(\pi_n(t), t)\leq \varepsilon_n$ for all $t$ and all $n$ (where $d$ is the distance defined on $T$ and $\varepsilon_n$ is a vanishing decreasing positive sequence), and $\pi_n(T)$ is a finite set.

The definition used for separability (Definition 5 in the paper) is the following:

A random process $(X_t)_{t\in T}$ is said separable if there is a countable set $T_0\subseteq T$, dense in $T$, such that $$\mathbb P\left(X_t\in\lim_{s\to t, s\in T_0}X_s, \quad\forall t\in T\right) = 1\,.$$

In the definition above $x\in \lim_{s\to t, s\in T_0}x_s$ means that there is a sequence $x_s$ in $T_0$ which tends to $x$.

Does the separability of a process actually guarantee such a strong convergence of $X_{\pi_n(t)}$ to $X_t$? What strikes me the most is that the separability involves only the existence of a dense subset, which might have nothing to do with the support of the nets $\pi_n(T)$.

The justification of the property given in the paper is

(see proof of Theorem 5.24 in [2]).

However, looking at the referenced proof I could not find anything suggesting that $\lim_{n\to\infty}\mathbb E[\sup_{t\in T}(X_t-X_{\pi_n(t)})]=0$.

Am I actually missing something obvious?

[1] Asadi et al., Chaining Mutual Information and Tightening Generalization Bounds, 2018. https://arxiv.org/abs/1806.03803
[2] Van Handel, Probability in high dimensions. https://web.math.princeton.edu/~rvan/APC550.pdf

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1 Answer 1

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The inequality $$\lim_{n\to\infty}\mathbb E[\sup_{t\in T}(X_t-X_{\pi_n(t)})]=0 \quad (*) $$ certainly does not follow from separability alone: The indicator of ${\mathbb Q} \cap [0,1]$ is a deterministic separable process with $T_0=\Bigl({\mathbb Q} \cup ({\mathbb Q}+\sqrt{2})\Bigr) \cap [0,1]$. Even seperability and the subgaussian property do not suffice if the metric space is too large. (E.g. consider a Gaussian process indexed by the unit ball $B$ of an infinite dimensional Hilbert space, with $E[(X_t-X_s)^2]=\|t-s\|^2$ for all $t,s \in B$. This process is unbounded in any ball $B(z,r) \subset B$.)

However, if you also assume summability of the series (107) in [1], then $(*)$ does follow, and the proof is analogous to the proof of Theorem 5.24 in [2], using the series (107) instead of the Dudley series, and invoking Theorem 7 of [1] instead of Lemma 5.1 of [2].

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  • $\begingroup$ Thank you @Yuval for your answer. However I am still confused and cannot see how to easily pass from the proof in [2] to the conclusion. Could you please detail a bit more? From my understanding, in [2] they control the reminder $E[\sup_{t\in T}(X_t-X_{\pi_n(t)})]$ by using that it is $0$ if $T$ is finite and $n$ large enough. But to use this argument we would need something like $\lim_{n\to\infty}\sup_k E[\sup_{t\in T^{(k)}}(X_t-X_{\pi_n(t)})]=\sup_k\lim_{n\to\infty} E[\sup_{t\in T^{(k)}}(X_t-X_{\pi_n(t)})]=0$, where $T^{(k)}$ are finite subset whose union is $T_0$. But how to show it? $\endgroup$
    – ECL
    Jan 25 at 17:11
  • $\begingroup$ Moreover I don't see how the series (107) helps in solving the question. The random variable $W$ does not appear in (*), so we might choose any random variable like that which is independent of the process, and all the mutual information would always be $0$, making the series (107) summable. $\endgroup$
    – ECL
    Jan 25 at 17:15
  • $\begingroup$ The series (107) helps, with this $W$ in view of Theorem 7 of [1]. Take $T^{(k)}$ as the first $k$ elements of $T_0$ in some enumeration. $\endgroup$ Jan 26 at 17:47

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