**ABSTRACT:**

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\bf{\small{ ON ALMOST PERIODIC RANDOM SIGNALS}}

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\scshape{Ioan Goletz} \\ \it{ "Politehnica " University of

Timi\c{s}oara}\\E-mail: io$_-$goro@yahoo.co.uk \\

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\begin{abstract}

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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