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

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

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:

for any interval such that , and

for any point , . A random process , , is harmonizable if and only if its covariance is representable in the form

Examples of harmonizable random processes.

1) A modulated stationary random process. If

is a stationary random process, a process of the form

where , where is a measure on the line, is usually no longer stationary, but will be harmonizable:

where the random measure is defined by the formula

2) A process defined by sliding summation (or moving averages)

where is some random measure on the line and the weight function is of the same type as above:

In this case

where

References

[1]	M. Loève, "Probability theory" , 2 , Springer (1978)