I am not familiar with the definition of total positivity. I am not sure about the link between log-concavity and total positivity.
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In a paper On Variation-Diminishing Integral Operators of the Convolution Type of Schoenberg, he defines Pólya frequency functions as a totally positive one.
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In Estimating a Polya Frequency Function, Pal, Woodroofe, and Meyer define Pólya frequency functions as log-concave functions.
Then, my question is the following, what are the links between total positivity and log-concavity ? It seems that total positivity implies log-concavity, do we have the converse statement ?
Best Answer
$\newcommand{\R}{\mathbb R}$For a positive integer $r$, a measurable function $f\colon\R\to\R$ is called a Pólya frequency function of order $r$ (abbreviated as PF$_r$) if the matrix $(f(x_i-y_j))_{i,j,=1}^r$ is totally positive for all real $x_i$ and $y_j$ such that $x_1\le\dots\le x_r$ and $y_1\le\dots\le y_r$, that is, if $\det(f(x_i-y_j))_{i,j,=1}^m\ge0$ for all integers $m=1,\dots,r$ and all real $x_i$ and $y_j$ such that $x_1\le\dots\le x_m$ and $y_1\le\dots\le y_m$. So, clearly PF$_1\supseteq\,$PF$_2\supseteq\cdots$.
Consider now the case $r=2$. Note that the set of all matrices of the form $(x_i-y_j)_{i,j,=1}^2$ for real $x_i$ and $y_j$ such that $x_1\le x_2$ and $y_1\le y_2$ coincides with the set of all matrices of the form $\begin{pmatrix}a&b\\c&d \end{pmatrix}$ for real $a,b,c,d$ such that $a+d=b+c$ and $\{a,d\}\subseteq[b,c]$.
So, if $f$ is log concave, then $f$ is PF$_2$. Vice versa, if $f$ is PF$_2$, then, taking $a=d=(b+c)/2$, we see that $f$ is midpoint log concave, and hence, being measurable, $f$ is log concave (see e.g. Theorem II).
Thus, a function $f\colon\R\to\R$ is PF$_2$ iff it is log concave.
However, it is easy to find examples of log-concave functions that are not in PF$_3$, and hence not in PF$_r$ for any natural $r\ge3$; for instance, take $f=1_{(0,8)}$ and \begin{equation} (x_1,x_2,x_3,y_1,y_2,y_3)= (0, 1, 3, -5, -4, 0); \end{equation} then $\det(f(x_i-y_j))_{i,j,=1}^3=-1\not\ge0$.
So, the definitions of Pólya frequency functions in the two papers referred to in your post are equivalent to each other for $r=2$ -- but not for $r\ge3$.