Talks given at NuHAG events

On almost periodic random signals

  Ioan Goletz
    University of Timisoara

  given at  strobl07 (17.06.07)
  id:  771
  length:  min
  status:  accepted
  type:  poster
\scshape{Ioan Goletz} \\ \it{ "Politehnica " University of
Timi\c{s}oara}\\E-mail: io$_-$ \\

The signal prediction is one of major target that plays an important
role in numerous applications. Time series show either a
combination of periodic phenomena with stochastic components or
chaotic behavior. Usually, the computing of nonlinear
characteristics indicates the real complexity of the system. In many
cases, the separation of frequency bands representing periodic or
almost periodic behaviors, allows comprehension of the hidden
nonlinear or stochastic phenomena involved. In this work a signal
separation method of almost periodic components based on
probabilistic norms is described. This method achieves effective
time series forecasting needed in applications of fluctuations in
time phenomena. At a given moment $t$ the signal value of a random
signal is assimilated to a random variable on a space with a
probability measure. In this paper we adopt an another point of
view, at each moment t the value of a random signal is known by
its probability measure, that is, the value of a random signal at
the moment $t$ is considered an element of a probabilistic normed
space. Let $(\Omega, K, P)$ be a complete probability measure space,
i.e., the set $ \Omega $ is a
nonempty abstract set, $ {\mathcal{K}} $ is a $ \sigma $-algebra of
subsets of $ \Omega $ and $ P $ is a complete probability measure
on $ {\mathcal{K}} $. Let $ (X,{\mathcal{B}}) $ be a measurable
space, where $ (X, \vert \vert \cdot \vert \vert) $ is a separable
Banach space and $ {\mathcal{B}} $ is the $ \sigma
$-algebra of the Borel subsets of $ (X, \vert \vert \cdot \vert \vert) $.\\
A mapping f is said to be a random signal defined on the time
subset A of real line with values in a separable Banach space X if,
for each $t\in A$ the mapping $f(t) : \Omega \mapsto X $ is a
X-valued random variable. But, the space of random variable can
be endowed with a probabilistic normed space structure. So, we
define a random signal as a mapping of a subset of a real line with
values into a probabilistic normed space. This new framework is
appropriate to give methods for approximation of random signals,
for describing periodicity types, for defining the time variation
at a moment$ t$ and the informational energy on a time interval.
\footnote{{AMS (1990) Subject Classification : 54E35, 46A19.\\}
Key words and phrases: probabilistic normed space, random signal\\}

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