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

We study a class of stationary Markov processes with marginal distributionsidentifiable by moments such that every conditional moment of degree say $m$ isa polynomial of degree at most $m\;\text{.}\;$ We show that then under someadditional, natural technical assumption there exists a family of orthogonalpolynomial martingales. More precisely we show that such a family of processesis completely characterized by the sequence $\{(\alpha_{n} ,p_{n})\}_{n \geq0}$ where $\alpha_{n}^{ \prime } s$ are some positive reals while $p_{n}^{\prime } s$ are some monic orthogonal polynomials. Paper of Bakry\&Mazet(2003)assures that under some additional mild technical conditions each such sequencegenerates some stationary Markov process with polynomial regression.  We single out two important subclasses of the considered class of Markovprocesses. The class of harnesses that we characterize completely. The secondone constitutes of the processes that have independent regression property andare stationary. Processes with independent regression property so to saygeneralize ordinary Ornstein-Uhlenbeck (OU) processes or can also be understoodas time scale transformations of L{\'e}vy processes. We list several propertiesof these processes. In particular we show that if these process are time scaletransforms of L{\'e}vy processes then they are not stationary unless we dealwith classical OU- process. Conversely, time scale transformations ofstationary processes with independent regression property are not L{\'e}vyunless we deal with classical OU process.

On stationary Markov processes with polynomial conditional moments