The Monotone Density Theorem for regularly varying functions says, in essence:

Theorem (Monotone Density Theorem). Let $f$ be a differentiable regularly varying real-function of index $\rho$ well-defined in $[a,+\infty)$ for some $a\geqslant 0$. If $f'$ is asymptotically monotone, then $f'$ is of regular variation of index $\rho-1$.

A proof (with a slightly different statement) may be found in p. 39 of Bingham's book Regular Variation. Although, this proof makes strong use of Karamata's characterization theorem, and offers no clue how to extend it to $O$-regularly varying functions.

A positive real-function $f$ defined in $[a,+\infty)$ is said to be of regular variation if for all $\lambda > 0$ holds $$f(\lambda x) \sim g(\lambda)f(x)~\text{ as }~x\to +\infty,~\text{with }~ g(\lambda) \in (0,+\infty).$$

An $O$-regular variation is the weaker condition that for all $\lambda > 0$ holds $$f(\lambda x) \asymp f(x)~\text{ as }~x\to +\infty.$$

The "O-version of Monotone Density Theorem" described in p. 119 of Bingham's book (Proposition 2.10.3) seems to be a far generalization of this fact, but I had a little trouble even to understand how to effectively handle the definition of Matuszewska indices (although $O$-regularity clearly implies $BI\cap PI$). The propostion is as follows:

Proposition 2.10.3. [Bingham's book] Let $f$ be a differentiable real-function well-defined in $[a,+\infty)$ for some $a\geqslant 0$. Assume $f'$ is asymptotically positive and $f'\in BI\cup BD$. If $f \in BI\cap PI$, then $f'(x)\asymp f(x)/x$ as $x\to+\infty$.

The acronyms stands for $BI(BD)$ = bounded increase (decrease), $PI(PD)$ = positive increase (decrease) and are defined in terms of Matuszewska indices (p.68) in p.71 of Bingham's book.

So my question is:

Does Proposition 2.10.3 of Bingham implies the Monotone Density Theorem assuming the same hypotheses but changing regular to $O$-regular? If so, is there an easier way to do this?


Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.