Lower bounds for hypothesis testing based on information theory

C Convex functions of Radon-Nikodym derivatives

Let \(f : \mathbb {R} \to \mathbb {R}\) be convex on \([0, +\infty )\). Remark that \(f(0)\) is finite.

Definition 50
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We define \(f'(\infty ) := \limsup _{x \to + \infty } f(x)/x\). This can be equal to \(+\infty \) (but not \(-\infty \)).

Lemma C.0.1

If \(\mu \) and \(\nu \) are two finite measures and \(f'(\infty ) {\lt} \infty \), then \(x \mapsto f\left(\frac{d\mu }{d\nu }(x)\right)\) is \(\nu \)-integrable.

Proof

By convexity and \(f(0) {\lt} \infty \), \(f'(\infty ){\lt}\infty \), we can sandwich \(f\) between two affine functions of \(\frac{d\mu }{d\nu }\). Those functions are integrable since the measures are finite.

Theorem C.0.2
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For \(f\) convex, \(f\left(\mu [g \mid m]\right) \le \mu [f \circ g \mid m]\).

Proof