Total positivity of copulas from a Markov kernel perspective

Total positivity of a Cauchy kernel

We study the total positivity of the kernel 1/(x+2 cos(πα)xy+y). The case of infinite order is characterized by an application of Schoenberg’s theorem. We then give necessary conditions for the cases of any given finite order with the help of Chebyshev polynomials of the second kind. Sufficient conditions for the finite order cases are also obtained, thanks to Propp’s formula for the Izergin-Ko...

Total positivity of Riordan arrays

An infinite matrix is called totally positive if its minors of all orders are nonnegative. A nonnegative sequence (an)n≥0 is called log-convex (logconcave, resp.) if aiaj+1 ≥ ai+1aj ( aiaj+1 ≤ ai+1aj , resp.) for 0 ≤ i < j . The object of this talk is to study various positivity properties of Riordan arrays, including the total positivity of such a matrix, the log-convexity of the 0th column an...

Regulators and Total Positivity

The classical regulator R of a number field K is given [F1] by a rapidly convergent series of the form R w = ∞

Total positivity of a shuffle matrix

Copulas selection in pairwise Markov chain

The Hidden Markov Chain (HMC) model considers that the process of unobservable states is a Markov chain. The Pairwise Markov Chain (PMC) model however considers the couple of processes of observations and states as a Markov chain. It has been shown that the PMC model is strictly more general than the HMC one, but retains the ease of processings that made the success of HMC in a number of applic...