The critical point of a sigmoidal curve

Abstract

Let $y(t)$ be a monotone increasing curve
with $\displaystyle \lim_{t\to \pm\infty}y^{(n)}(t)=0$ for all $n$ and let $t_n$ be the location of the global extremum of the $n$th derivative $y^{(n)}(t)$.
Under certain assumptions on the Fourier and Hilbert transforms of $y(t)$, we prove that the sequence $\{t_n\}$ is convergent. This implies in particular a preferred choice of the origin of the time axis and an intrinsic definition of the even and odd components of a sigmoidal function. In the context of phase transitions, the limit point has the interpretation of the critical point of the transition as discussed in previous work \cite{BP2013}.