A Martingale Characterization of Diffusion Random Fields on the Plane

1991 ◽  
Vol 35 (1) ◽  
pp. 152-157
Author(s):  
Yu. S. Mishura
Keyword(s):  
Random Fields ◽  
Martingale Characterization

scholarly journals Identification and isotropy characterization of deformed random fields through excursion sets

2018 ◽  
Vol 50 (3) ◽  
pp. 706-725
Author(s):  
Julie Fournier
Keyword(s):  
Random Field ◽  
Euler Characteristic ◽  
Random Fields ◽  
Weak Form ◽  
Invariance Property ◽  
Basic Domain ◽  
Excursion Sets ◽  
The Mean ◽  
Isotropic Random

Abstract A deterministic application θ:ℝ2→ℝ2 deforms bijectively and regularly the plane and allows the construction of a deformed random field X∘θ:ℝ2→ℝ from a regular, stationary, and isotropic random field X:ℝ2→ℝ. The deformed field X∘θ is, in general, not isotropic (and not even stationary), however, we provide an explicit characterization of the deformations θ that preserve the isotropy. Further assuming that X is Gaussian, we introduce a weak form of isotropy of the field X∘θ, defined by an invariance property of the mean Euler characteristic of some of its excursion sets. We prove that deformed fields satisfying this property are strictly isotropic. In addition, we are able to identify θ, assuming that the mean Euler characteristic of excursion sets of X∘θ over some basic domain is known.

A martingale characterization of mixed Poisson processes

1987 ◽  
Vol 24 (1) ◽  
pp. 246-251 ◽  
Author(s):  
Dietmar Pfeifer ◽  
Ursula Heller
Keyword(s):  
Poisson Process ◽  
Random Variable ◽  
Poisson Processes ◽  
Birth Process ◽  
Mixed Poisson Process ◽  
Martingale Characterization

It is shown that an elementary pure birth process is a mixed Poisson process iff the sequence of post-jump intensities forms a martingale with respect to the σ -fields generated by the jump times of the process. In this case, the post-jump intensities converge almost surely to the mixing random variable of the process.

scholarly journals A martingale characterization of Pólya-Lundberg processes

2006 ◽  
Vol 43 (03) ◽  
pp. 741-754 ◽  
Author(s):  
Birgit Niese
Keyword(s):  
Poisson Process ◽  
Exponential Family ◽  
Exponential Families ◽  
Counting Processes ◽  
Mixed Poisson Process ◽  
Martingale Characterization

We study exponential families within the class of counting processes and show that a mixed Poisson process belongs to an exponential family if and only if it is either a Poisson process or has a gamma structure distribution. This property can be expressed via exponential martingales.

Sign in / Sign up

Export Citation Format

Share Document