Dudley entropy-integral bound for sub-Gaussian processes

First derive a finite-maximum estimate. If $Y_1,\dots ,Y_m$ are centered and each has sub-Gaussian proxy at most $s^2$, then for $\lambda >0$, $\mathbb{E}{e}^{\lambda {\max }_j\mathrm{\mid }{Y}_j\mathrm{\mid }}\le {\sum }_{j=1}^m(\mathbb{E}{e}^{\lambda Y_j}+\mathbb{E}{e}^{-\lambda Y_j})\le 2m{e}^{{\lambda }^2{s}^2\mathrm{/}2}\mathrm{.}$ By [F3], $\mathbb{E}{\max }_j\mathrm{\mid }{Y}_j\mathrm{\mid }\le {\lambda }^{-1}\log (2m)+\lambda s^2\mathrm{/}2$. For $s>0$, choosing $\lambda =\sqrt{2\log (2m)}\mathrm{/}s$ gives $\mathbb{E}{\max }_j\mathrm{\mid }{Y}_j\mathrm{\mid }\le s\sqrt{2\log (2m)}$; for $s=0$, the proxy bound forces every $Y_j=0$ almost surely by [F4] applied to $e^{\pm \lambda Y_j}$ and letting $\lambda \to \mathrm{\infty }$.

If $D=0$, every pair in $T$ has distance zero. The proxy-zero case of step 1.1 applied to each increment, together with separability and [F6] for the countable intersection of probability-one events, makes the supremum zero. Assume $D>0$. Fix $k\ge 1$ and put ${ϵ}_k=2^{-k}D$. Total boundedness gives a finite cover of $T$ by $N({ϵ}_k\mathrm{/}4,T,d)$ closed balls of radius ${ϵ}_k\mathrm{/}4$. Approximate each center by a point of the dense set $T_0$ within ${ϵ}_k\mathrm{/}4$ to obtain a cover of $T$ by the same number of balls of radius ${ϵ}_k\mathrm{/}2$. Since the specified countable dense set is listed, choose the first finite tuple under its induced enumeration that is an ${ϵ}_k$-net and has cardinality at most this cover size; such a tuple exists by the construction. Denote it by $T_k$. For each $t$, choose the least-index point ${\pi }_k(t)\in T_k$ within ${ϵ}_k$. These canonical choices across scales require no choice axiom. Put $T_0^{(0)}=\{t_0\}$ and ${\pi }_0(t)=t_0$.

For a finite subset $A\subset T_0$, telescope $X_t-X_{t_0}$ through ${\pi }_1(t),\dots ,{\pi }_m(t)$ for every $t\in A$. At level $1$ there are at most $\mathrm{\mid }{T}_1\mathrm{\mid }$ increments, each with proxy at most $D^2$; for $k\ge 2$ there are at most $\mathrm{\mid }{T}_k\mathrm{\mid }\mathrm{\mid }{T}_{k-1}\mathrm{\mid }$ increments, each with proxy at most $({ϵ}_k+{ϵ}_{k-1}{)}^2=(3{ϵ}_k{)}^2$. Applying step 1.1 to the finite set of increments at each level, and the triangle inequality for their maxima, yields $\mathbb{E}{\max }_{t\in A}\mathrm{\mid }{X}_{{\pi }_m(t)}-X_{t_0}\mathrm{\mid }\le C_1{\sum }_{k=1}^m{ϵ}_k\sqrt{1+\log N({ϵ}_k\mathrm{/}4,T,d)}$ for a universal $C_1$; the $k=1$ term is absorbed using ${ϵ}_1=D\mathrm{/}2$ and $N({ϵ}_1\mathrm{/}4)\ge 1$.

The residual maximum over $A$ has expectation at most ${ϵ}_m\sqrt{2\log (2\mathrm{\mid }A\mathrm{\mid })}$ by step 1.1, since $d(t,{\pi }_m(t))\le {ϵ}_m$. Letting $m\to \mathrm{\infty }$ for this fixed finite $A$ removes the residual. The dyadic sum in step 2.2 is bounded by a universal multiple of the entropy integral: on each interval $[{ϵ}_k\mathrm{/}8,{ϵ}_k\mathrm{/}4]$, monotonicity of $N$ makes the integral at least $({ϵ}_k\mathrm{/}8)\sqrt{\log N({ϵ}_k\mathrm{/}4,T,d)}$. Also, for $0<ϵ<D\mathrm{/}2$, one ball of radius $ϵ$ cannot cover a set of diameter $D$, so $N(ϵ,T,d)\ge 2$ and the entropy integral is at least $(D\mathrm{/}2)\sqrt{\log 2}$; this absorbs the geometric contribution from the added $1$. Hence $\mathbb{E}{\max }_{t\in A}\mathrm{\mid }{X}_t-X_{t_0}\mathrm{\mid }\le C{\int }_0^D\sqrt{\log N(ϵ,T,d)}dϵ\mathrm{.}$ Taking an increasing sequence of finite subsets whose union is $T_0$, [F5] and the separability assumption give the asserted supremum over $T$.