S. Viguier-Pla (2017)
Proximity between stationary random functions.
Mathematical Analysis with Applications in Mechanics,
Perpignan, September 6-8th, 2017.
- Classification
- 60G57, 60G10, 60B15, 60H05
- Keywords
- Projector, Unitary operator, Stationary
process, spectral measure
Abstract :
We will expose mathematical tools for the study of stationary random
functions.
In particular, we will speak about projector valued spectral measures,
product of convolution
of spectral measures, and groups of unitary operators.
Thanks to these tools, we will analyze how the proximity between
random measures is linked
with the proximity between the associated random functions.
We also analyze the case where random functions are
cyclostationary. We see how these
functions can be associated with a unique stationary series. So the
methods applied on
stationary series can be extended to cyclostationary functions.
The product of convolution is de ned for spectral measures which
commute. We will analyze
how this commutativity expresses in a practical point of view. Then,
when there is no
commutativity, we introduce the notion of commutator, which will be
able to retreive the
parts which commute from two random functions which do not commute.
This notion of commutator opens a large eld of applications for the
study of the proximity
between any two random functions.
This work has been conducted in collaboration with Alain Boudou, at
University of Toulouse.