Entropy describes a class; maximal inequalities convert that description into a numerical bound for its empirical fluctuations. The most useful inequalities are local: an estimator only explores functions close to its target, and their variance can be much smaller than that of the global envelope.
Uniform entropy inequality
Let \(F\) be an envelope and define \[ J(\delta,\mathcal F,F) = \sup_Q\int_0^\delta \sqrt{1+\log N\!\left( \varepsilon\|F\|_{Q,2}, \mathcal F,L_2(Q) \right)} \,\mathrm{d}\varepsilon. \]
Theorem. Suppose \(0\in\mathcal F\), the class is suitably measurable, \(PF^2<\infty\), and \(J(1,\mathcal F,F)<\infty\). Then \[ E^*\|\mathbb G_n\|_{\mathcal F} \leq C J(1,\mathcal F,F)\|F\|_{P,2} \] for a universal constant \(C\).
Proof. Symmetrization gives \[ E^*\|\mathbb G_n\|_{\mathcal F} \leq2E^*E_\varepsilon \left\|\frac1{\sqrt n}\sum_{i=1}^n\varepsilon_if(X_i)\right\|_{\mathcal F}. \] Conditional on the data, the sign process is sub-Gaussian under \(L_2(P_n)\), has diameter at most \(2\|F\|_{P_n,2}\), and its normalized covering numbers are bounded by the uniform entropy. Dudley's theorem therefore gives \[ E_\varepsilon\|\mathbb G_n^\varepsilon\|_{\mathcal F} \leq C J(1,\mathcal F,F)\|F\|_{P_n,2}. \] Jensen's inequality yields \(E\|F\|_{P_n,2}\leq(PF^2)^{1/2}\). Combine the three displays. \(\square\)
Corollary (VC-type class). If \[ \sup_QN\!\left( \varepsilon\|F\|_{Q,2}, \mathcal F,L_2(Q) \right) \leq(A/\varepsilon)^v, \qquad 0<\varepsilon\leq1, \] then \[ E^*\|\mathbb G_n\|_{\mathcal F} \lesssim\sqrt v\,\|F\|_{P,2} \] when \(A\) is regarded as a fixed constant.
Localized VC-type bounds
Suppose now that every \(f\in\mathcal F\) satisfies \(Pf^2\leq\delta^2\), while \(\|F\|_\infty\leq b\). A global bound pays \(\|F\|_{P,2}\), which may be much larger than \(\delta\). Bernstein rather than purely sub-Gaussian chaining captures the smaller variance.
Theorem (localized maximal inequality). Under the VC-type covering bound \[ \sup_QN(\varepsilon\|F\|_{Q,2},\mathcal F,L_2(Q)) \leq(A/\varepsilon)^v, \] with \(|f|\leq b\) and \(\sup_fPf^2\leq\delta^2\), one has \[ E^*\|\mathbb G_n\|_{\mathcal F} \leq C\left[ \delta\sqrt{v\log\!\left(\frac{A b}{\delta}\right)} + \frac{vb}{\sqrt n} \log\!\left(\frac{A b}{\delta}\right) \right], \] whenever \(0<\delta\leq b\), with a universal \(C\).
The first term is the Gaussian fluctuation at variance scale \(\delta^2\). The second is the finite-sample price for bounded but non-Gaussian increments. It becomes negligible when \[ \sqrt n\,\delta \gg\sqrt{v\log(Ab/\delta)}. \]
Bracketing maximal inequality
For distribution-specific classes define \[ J_{[]}(\delta,\mathcal F,L_2(P)) =\int_0^\delta \sqrt{1+\log N_{[]}(\varepsilon,\mathcal F,L_2(P))} \,\mathrm{d}\varepsilon. \]
Theorem. Suppose \(\mathcal F\) is measurable, \(Pf^2\leq\delta^2\) for every \(f\in\mathcal F\), and \(|f|\leq b\). A standard bracketing bound is \[ E^*\|\mathbb G_n\|_{\mathcal F} \leq C J_{[]}(\delta,\mathcal F,L_2(P)) \left[ 1+ \frac{b\,J_{[]}(\delta,\mathcal F,L_2(P))} {\delta^2\sqrt n} \right]. \]
The first factor accumulates bracket increments. The correction appears because a bracket controls \(L_2(P)\)-width but its empirical width is random; truncation and Bernstein's inequality compare the two.
Peeling
Lemma (peeling principle). Let \(r(f)\geq0\) be a size functional and define shells \[ \mathcal F_j =\{f:2^{j-1}r_0<r(f)\leq2^jr_0\}, \qquad j\geq1. \] For any nonnegative threshold \(a(r)\), \[ P^*\!\left\{\sup_{r(f)>r_0}\frac{|Z_n(f)|}{a(r(f))}>1\right\} \leq \sum_{j\geq1} P^*\!\left\{ \sup_{f\in\mathcal F_j}|Z_n(f)| > \inf_{2^{j-1}r_0<r\leq2^jr_0}a(r) \right\}. \]
Peeling turns a family of local bounds into one uniform statement. It is used in estimator-rate proofs because the random estimator may lie in any shell, while deterministic curvature grows with distance from the target.
Balancing fluctuation and curvature
Proposition (rate template). Let \(\hat\theta_n\) nearly maximize \(M_n(\theta)\), and suppose for \(d(\theta,\theta_0)\leq r_0\), \[ M(\theta)-M(\theta_0)\leq-c\,d(\theta,\theta_0)^\alpha, \qquad \alpha>1. \] If \[ E^*\sup_{d(\theta,\theta_0)\leq r} |\mathbb G_n(m_\theta-m_{\theta_0})| \lesssim\phi(r), \] where \(\phi(r)/r^\alpha\) is nonincreasing, then a candidate rate \(r_n\) is determined by \[ \phi(r_n)\asymp\sqrt n\,r_n^\alpha. \] Under consistency and suitable tail bounds, \(d(\hat\theta_n,\theta_0)=O_p(r_n)\).
Proof outline. On the shell \(2^{j-1}r_n<d(\theta,\theta_0)\leq2^jr_n\), deterministic loss is at least order \(2^{j\alpha}r_n^\alpha\). The stochastic oscillation is of order \(n^{-1/2}\phi(2^jr_n)\), which is at most a constant times \(2^{j\alpha}r_n^\alpha\) by the monotonicity assumption and the balancing equation. A maximal inequality followed by Markov or concentration makes the probability that fluctuation defeats curvature summable over sufficiently large shells. \(\square\)
Example. If local entropy produces \(\phi(r)\asymp r^{1-\alpha_0/2}\) with entropy exponent \(0<\alpha_0<2\), and the criterion has quadratic curvature \(\alpha=2\), then \[ r_n^{1-\alpha_0/2}\asymp\sqrt n\,r_n^2, \qquad r_n\asymp n^{-1/(2+\alpha_0)}. \] This is the familiar nonparametric rate generated by a complexity exponent \(\alpha_0\).
The next page adds deviation inequalities. Expectation bounds identify the scale; concentration states how reliably the process remains near that scale.