A single \(\varepsilon\)-net approximates the process only at one resolution. Chaining uses a nested sequence of nets and represents every coordinate as a sum of increments from coarse to fine scales. Each increment is small; the number of possible increments grows with resolution. The entropy integral is the accumulated price of this size–multiplicity tradeoff.
Sub-Gaussian processes
Definition. A centered process \(Z=\{Z_t:t\in T\}\) is sub-Gaussian with respect to a semimetric \(d\) if \[ E\exp\{\lambda(Z_t-Z_s)\} \leq\exp\{\lambda^2d(s,t)^2/2\} \] for every \(s,t\in T\) and \(\lambda\in\mathbb R\).
A centered Gaussian process is sub-Gaussian for its canonical metric \[ d(s,t)=\{E(Z_s-Z_t)^2\}^{1/2}. \] Conditional Rademacher processes are sub-Gaussian for \(d_n(f,g)=\|f-g\|_{P_n,2}\). Thus one geometric argument treats both Gaussian limits and symmetrized empirical processes.
Dudley's entropy bound
Theorem (Dudley). Let \(Z\) be a separable, centered, sub-Gaussian process on \((T,d)\). For every \(t_0\in T\), \[ E\sup_{t\in T}|Z_t-Z_{t_0}| \leq C\int_0^{\operatorname{diam}(T)} \sqrt{\log\{2N(\varepsilon,T,d)\}}\,\mathrm{d}\varepsilon \] for a universal constant \(C\), whenever the integral is finite.
Proof. Put \(D=\operatorname{diam}(T)\), \(\delta_j=2^{-j}D\), and choose a minimal \(\delta_j\)-net \(T_j\), with \(T_0=\{t_0\}\). Select \(\pi_j(t)\in T_j\) with \(d(t,\pi_jt)\leq\delta_j\). Separability permits passage from finite levels to \(T\), and \[ Z_t-Z_{t_0} =\sum_{j\geq1}\{Z_{\pi_jt}-Z_{\pi_{j-1}t}\}. \] At level \(j\), every increment has sub-Gaussian scale at most \[ d(\pi_jt,\pi_{j-1}t) \leq\delta_j+\delta_{j-1}\leq3\delta_{j-1}, \] and there are at most \(|T_j||T_{j-1}|\) possible pairs. The finite-class maximal inequality therefore yields \[ E\sup_t|Z_{\pi_jt}-Z_{\pi_{j-1}t}| \lesssim \delta_{j-1}\sqrt{\log\{2|T_j||T_{j-1}|\}}. \] Summing over \(j\) and using monotonicity of covering numbers gives \[ E\sup_t|Z_t-Z_{t_0}| \lesssim\sum_{j\geq1} \delta_j\sqrt{\log\{2N(\delta_j,T,d)\}}. \] This dyadic sum is bounded by a constant multiple of the stated integral. \(\square\)
Corollary (continuity). If \[ \lim_{\delta\downarrow0} \int_0^\delta\sqrt{\log N(\varepsilon,T,d)}\,\mathrm{d}\varepsilon=0, \] then \(Z\) has a version with bounded, uniformly \(d\)-continuous sample paths.
Apply Dudley's argument to the restricted pairs \(d(s,t)\leq\delta\). The expected modulus tends to zero; a subsequence and Borel–Cantelli yield uniform continuity of a separable version.
Truncated chains
Proposition. For every \(a>0\), \[ E\sup_{t\in T}|Z_t-Z_{t_0}| \lesssim E\sup_{t\in T}|Z_t-Z_{\pi_a t}| +\int_a^D\sqrt{\log N(\varepsilon,T,d)}\,\mathrm{d}\varepsilon, \] where \(\pi_a t\) is an \(a\)-net projection.
Truncation is useful when entropy diverges at zero or increments cease to be sub-Gaussian below a sample-dependent scale. One handles the unresolved remainder by an envelope or Bernstein bound and optimizes \(a\).
Sudakov minoration
Theorem (Sudakov). Let \(Z\) be a centered Gaussian process with canonical metric \(d\). Then \[ E\sup_{t\in T}Z_t \geq c\sup_{\varepsilon>0} \varepsilon\sqrt{\log M(\varepsilon,T,d)} \] for a universal constant \(c>0\).
Dudley gives an upper bound by adding all scales; Sudakov says that any single scale containing many well-separated points already forces a large supremum. Together they show why metric entropy is not merely a proof artifact.
Gaussian comparison
Theorem (Sudakov–Fernique). Let \(\{X_t:t\in T\}\) and \(\{Y_t:t\in T\}\) be separable centered Gaussian processes. If \[ E(X_s-X_t)^2\leq E(Y_s-Y_t)^2 \qquad\text{for all }s,t\in T, \] then \[ E\sup_{t\in T}X_t\leq E\sup_{t\in T}Y_t. \]
Proof outline. First take finite \(T=\{1,\ldots,m\}\) and replace the maximum by \[ F_\beta(x)=\beta^{-1}\log\sum_{j=1}^me^{\beta x_j}. \] Interpolate \(Z(u)=\sqrt u\,X+\sqrt{1-u}\,Y\). Gaussian integration by parts expresses \(\frac{\mathrm{d}}{\mathrm{d}u}EF_\beta(Z(u))\) as a weighted sum of differences between the increment variances of \(X\) and \(Y\). The weights have the sign that makes this derivative nonpositive under the assumed ordering. Integrate from \(0\) to \(1\), then let \(\beta\to\infty\). Finite approximations and separability extend the conclusion to general \(T\). \(\square\)
Example (Gaussian width of a sphere). For \(g\sim N_d(0,I_d)\) and \(Z_t=g^\top t\) on \(T=S^{d-1}\), \[ \sup_{t\in S^{d-1}}Z_t=\|g\|_2, \qquad E\|g\|_2\asymp\sqrt d. \] The sphere has \(\log N(\varepsilon)\asymp d\log(1/\varepsilon)\), so Dudley's integral also gives order \(\sqrt d\). Here the entropy upper bound is sharp in order.
Generic chaining
Definition. An admissible sequence \((T_k)_{k\geq0}\) satisfies \[ |T_0|=1,\qquad |T_k|\leq2^{2^k}. \] The generic chaining functional is \[ \gamma_2(T,d) = \inf_{(T_k)} \sup_{t\in T}\sum_{k\geq0}2^{k/2}d(t,T_k). \]
Theorem (majorizing measures). For every centered Gaussian process with canonical metric \(d\), \[ E\sup_{t\in T}(Z_t-Z_{t_0})\asymp\gamma_2(T,d), \] with universal comparison constants.
Dudley's integral chooses nets according only to their worst covering radius. Generic chaining allocates approximating points unevenly across the geometry and is sharp for every Gaussian process. In most statistical applications Dudley is simpler and adequate; \(\gamma_2\) becomes essential for irregular, anisotropic, or high-dimensional index sets.