English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

According to the Wiener-Hopf factorization, the characteristic function$\varphi$ of any probability distribution $\mu$ on $\mathbb{R}$ can bedecomposed in a unique way as\[1-s\varphi(t)=[1-\chi_-(s,it)][1-\chi_+(s,it)]\,,\;\;\;|s|\le1,\,t\in\mathbb{R}\,,\]where $\chi_-(e^{iu},it)$ and $\chi_+(e^{iu},it)$ are the characteristicfunctions of possibly defective distributions in$\mathbb{Z}_+\times(-\infty,0)$ and $\mathbb{Z}_+\times[0,\infty)$,respectively.  We prove that $\mu$ can be characterized by the sole data of the upwardfactor $\chi_+(s,it)$, $s\in[0,1)$, $t\in\mathbb{R}$ in many cases includingthe cases where:  1) $\mu$ has some exponential moments;  2) the function $t\mapsto\mu(t,\infty)$ is completely monotone on$(0,\infty)$;  3) the density of $\mu$ on $[0,\infty)$ admits an analytic continuation on$\mathbb{R}$.  We conjecture that any probability distribution is actually characterized byits upward factor. This conjecture is equivalent to the following: {\it Anyprobability measure $\mu$ on $\mathbb{R}$ whose support is not included in$(-\infty,0)$ is determined by its convolution powers $\mu^{*n}$, $n\ge1$restricted to $[0,\infty)$}. We show that in many instances, the sole knowledgeof $\mu$ and $\mu^{*2}$ restricted to $[0,\infty)$ is actually sufficient todetermine $\mu$. Then we investigate the analogous problem in the framework ofinfinitely divisible distributions.

On distributions determined by their upward, space-time Wiener-Hopf factor