Harmonizable random process
A complex-valued random function of a real parameter
which may be represented as a stochastic integral:
![]() | (*) |
where ,
, is a random process. The increments
in (*) define random "amplitudes"
and "phases"
of elementary vibrations of the form
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of frequencies ,
, the superposition of which yields, in the limit,
. The (mean-square) limit in the representation (*) is taken along a sequence of successively-finer subdivisions of the line
into intervals
with
. It is usually assumed that
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as a function of the sets in the plane, defines a complex measure of bounded variation; in this case the corresponding process
,
(or, more exactly, the corresponding random measure
), is unambiguously defined by the process
,
, itself:
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for any interval such that
, and
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for any point ,
. A random process
,
, is harmonizable if and only if its covariance is representable in the form
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Examples of harmonizable random processes.
1) A modulated stationary random process. If
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is a stationary random process, a process of the form
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where , where
is a measure on the line, is usually no longer stationary, but will be harmonizable:
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where the random measure is defined by the formula
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2) A process defined by sliding summation (or moving averages)
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where is some random measure on the line and the weight function
is of the same type as above:
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In this case
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where
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References
[1] | M. Loève, "Probability theory" , 2 , Springer (1978) |
Harmonizable random process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonizable_random_process&oldid=47186