*4.2. Proof of Corollary 1*

By assumption *<sup>f</sup>* : <sup>2</sup><sup>Ω</sup> <sup>→</sup> <sup>R</sup> is a rank function, which implies that 0 <sup>≤</sup> *<sup>f</sup>*(<sup>T</sup> ) <sup>≤</sup> *<sup>f</sup>*(Ω) for every T ⊆ <sup>Ω</sup>. Since (by definition) *<sup>f</sup>* is submodular with *<sup>f</sup>*(∅) = 0, and (by assumption) the function *g* is convex and monotonically increasing, then (from (22), while replacing *k* with *kn*)

$$\mathbb{P}\left(\binom{n}{k\_n} \text{ g}\left(\frac{f(\Omega)}{n}\right) \le \sum\_{\substack{\mathcal{T} \subseteq \Omega \colon |\mathcal{T}| = k\_n}} \text{g}\left(\frac{f(\mathcal{T})}{k\_n}\right) \le \binom{n}{k\_n} \text{g}\left(\frac{f(\Omega)}{k\_n}\right), \qquad n \in \mathbb{N}.\tag{75}$$

By the second assumption in Corollary 1, for positive values of *x* that are sufficiently close to zero, we have


In both cases, it follows that

$$\lim\_{\mathfrak{x}\to 0^{+}} \mathfrak{x} \log \mathfrak{g}(\mathfrak{x}) = 0.\tag{76}$$

In light of (75) and (76), and since (by assumption) *kn* −→*n*→<sup>∞</sup> <sup>∞</sup>, it follows that

$$\lim\_{n \to \infty} \frac{1}{n} \left[ \log \left( \sum\_{\substack{\mathcal{T} \subseteq \Omega \colon |\mathcal{T}| = k\_n}} \lg \left( \frac{f(\mathcal{T})}{k\_n} \right) \right) - \log \binom{n}{k\_n} \right] = 0. \tag{77}$$

By the following upper and lower bounds on the binomial coefficient:

$$\frac{1}{n+1} \exp\left(n \operatorname{\mathsf{H}}\_{\mathsf{b}}\left(\frac{k\_n}{n}\right)\right) \le \binom{n}{k\_n} \le \exp\left(n \operatorname{\mathsf{H}}\_{\mathsf{b}}\left(\frac{k\_n}{n}\right)\right),\tag{78}$$

the combination of equalities (77) and (78) gives equality (23). Equality (24) holds as a special case of (23), under the assumption that lim*n*→<sup>∞</sup> *kn <sup>n</sup>* = *β* ∈ [0, 1].
