d Major’s inequalities if one is only interested in the ‘in probability’ version of Theorems \[main0\] and \[mainu\].

We now show that the classes of functions appearing in the previous sections are of VC type, and the suprema countable. We will do this in all detail for the class $\cal F$ in (\[entr0\]), and will give indications for the rest of the classes of functions used.

First we observe that the sup inside the probability bound in (\[eq1\]) is actually a supremum over the set $\{t\in\mathbb Q, h\in \mathbb Q\cap[h_{2^k},h_{2^{k-1}})\}$ by the continuity properties of $K$ and the indicator of $|t-X_i|<h B$. This observation applies to all the other classes of functions in the previous two sections.

\[vc1\] Let $\cal F$ be as in (\[entr0\]) with $K$ and $f$ satisfying Assumptions \[ass1\]. Then, there exists a universal constant $R$ such that for every Borel probability measure $Q$ on $\mathbb R$, $$\label{ivc1}
N({\cal F},L_2(Q),\varepsilon)\le\left(\frac{R\|K\|_V\|f\|_\infty^{1/2}}{\varepsilon}\right)^{22}$$ where $\|K\|_V$ is the total variation norm of $K$, that is, $\cal F$ is of VC type with envelope $\|K\|_V\|f\|_\infty^{1/2}$ with $A=R$ independent of $K$ and $f$ and $v=22$.

By adding an arbitrarily small strictly increasing function to the positive and negative variation functions of $K$, we have $K=K_1-K_2$ with $K_i$ strictly increasing, positive and bounded, with $\|K_1\|_\infty$ ($\|K_2\|_\infty$) arbitrarily close to the positive (negative) variation of $K$. Let ${\cal K}_1$ be the class of functions obtained from $\cal F$ by replacing $K$ by $K_1$ and deleting the indicator in each of the functions in the class. Then, if we assume $f(x)>0$ for all $x$, the subgraphs of the functions in the class ${\cal K}_1$ have the form $$\left\{(x,u):K_{1}\left(\frac{t-x}{h}f^{1/2}(x)\right)f^{1/2}(x)
\ge u \right\}
=\left\{(x,u):\frac{t-x}{h}f^{1/2}(x)\ge K_{1}^{-1}(u/f^{1/2}(x))\right\}$$ $$=\left\{(x,u):\frac{tf^{1/2}(x)}{h}-\frac{xf^{1/2}(x)}{h}-K_{1}^{-1}(u/f^{1/2}(x))
\ge0\right\},$$ and so 