concentration via covering

The general approach is as follows. Let $C_{ϵ}$ be an $ϵ$-cover of ${\mathbb{B}}^{d-1}$ (see covering and packing). Then

$$\underset{v\in {\mathbb{B}}^{d-1}}{\sup }⟨v,S_n⟩\le \underset{v\in C_{ϵ}}{\sup }⟨v,S_n⟩+\underset{v\in ϵ{\mathbb{B}}^{d-1}}{\sup }⟨v,S_n⟩=\underset{v\in C_{ϵ}}{\sup }⟨v,S_n⟩+ϵ\underset{v\in {\mathbb{B}}^{d-1}}{\sup }⟨v,S_n⟩,$$

so

$$\underset{v\in {\mathbb{B}}^{d-1}}{\sup }⟨v,S_n⟩\le \frac{1}{1-ϵ}\underset{v\in C_{ϵ}}{\sup }⟨v,S_n⟩.$$

From here, $C_{ϵ}$ is a finite set, so we can apply well-known maximal inequalities over finite sets. For instance, if $S_n=X$ is sub-Gaussian, then light-tailed maximal inequalities give

$$\mathbb{E}\underset{v\in C_{ϵ}}{\sup }⟨v,X⟩\le \sqrt{2\mathbb{V}(X)\log |C_{ϵ}|}\le \sqrt{2\mathbb{V}(X)d\log (3/ϵ)},$$

where we use an upper bound on the $ϵ$-covering number of ${\mathbb{B}}^{d-1}$. This gives

$$\mathbb{E}\left|S_n\right|\le \frac{1}{1-ϵ}\sqrt{2\mathbb{V}(X)\log |C_{ϵ}|}.$$

We can then optimize over $ϵ$ to get a final bound. The sub-Gaussian case usually takes $ϵ=1/2$ (e.g., Theorem 1.19 in high-dimensional statistics by Rigollet and Hutter). We can also get a high-probability bound by noticing that

$$\mathbb{P}(\underset{v\in {\mathbb{B}}^{d-1}}{\sup }⟨v,S_n⟩\ge t)\le \mathbb{P}\left(\frac{1}{1-ϵ}\underset{v\in C_{ϵ}}{\sup }⟨v,S_n⟩\ge t\right),$$

which can be bounded with a union bound and a tail bound and then maybe a Chernoff bound (Chernoff method) based on the properties of $S_n$.