Harmonizable random process

A complex-valued random function $ X = X( t) $ of a real parameter $ t $ which may be represented as a stochastic integral:

$$ \tag{* } X ( t) = \int\limits _ {- \infty } ^ \infty e ^ {i \lambda t } d \Phi ( \lambda ) = \ \lim\limits \sum _ { k } e ^ {i \lambda t } \Delta _ {k} \Phi ( \lambda ), $$

where $ \Phi ( \lambda ) $, $ - \infty < \lambda < \infty $, is a random process. The increments $ \Delta _ {k} \Phi ( \lambda ) = \Phi ( \lambda _ {k+} 1 ) - \Phi ( \lambda _ {k} ) $ in (*) define random "amplitudes" $ A _ {k} = | \Delta _ {k} \Phi ( \lambda ) | $ and "phases" $ \theta _ {k} = \mathop{\rm arg} \Delta _ {k} \Phi ( \lambda ) $ of elementary vibrations of the form

$$ Ae ^ {i ( \lambda t + \theta ) } = \ e ^ {i \lambda t } \Delta _ {k} \Phi ( \lambda ) $$

of frequencies $ \lambda $, $ \lambda _ {k} \leq \lambda \leq \lambda _ {k+} 1 $, the superposition of which yields, in the limit, $ X = X( t) $. The (mean-square) limit in the representation (*) is taken along a sequence of successively-finer subdivisions of the line $ - \infty < \lambda < \infty $ into intervals $ \Delta _ {k} = ( \lambda _ {k,\ } \lambda _ {k+} 1 ) $ with $ \max _ {k} ( \lambda _ {k + 1 } - \lambda _ {k} ) \rightarrow 0 $. It is usually assumed that

$$ F ( \Delta _ {1} \times \Delta _ {2} ) = \ {\mathsf E} ( \Delta _ {1} \Phi \cdot \Delta _ {2} \overline \Phi \; ) , $$

as a function of the sets $ \Delta _ {1} \times \Delta _ {2} $ in the plane, defines a complex measure of bounded variation; in this case the corresponding process $ \Phi ( \lambda ) $, $ - \infty < \lambda < \infty $( or, more exactly, the corresponding random measure $ d \Phi ( \lambda ) $), is unambiguously defined by the process $ X( t) $, $ - \infty < t < \infty $, itself:

$$ \Delta \Phi ( \lambda ) = \ \lim\limits _ {T \rightarrow \infty } \ { \frac{1}{2T } } \int\limits _ { - } T ^ { T } \frac{e ^ {- i \lambda _ {2} t } - e ^ {- i \lambda _ {1} t } }{- it } X ( t) dt $$

for any interval $ \Delta = ( \lambda _ {1} , \lambda _ {2} ) $ such that $ d \Phi ( \lambda _ {1} ) = d \Phi ( \lambda _ {2} ) = 0 $, and

$$ \Phi ( \lambda ) = \ \lim\limits _ {T \rightarrow \infty } \ \int\limits _ { - } T ^ { T } e ^ {- i \lambda t } X ( t) dt $$

for any point $ \lambda $, $ - \infty < \lambda < \infty $. A random process $ X( t) $, $ - \infty < t < \infty $, is harmonizable if and only if its covariance is representable in the form

$$ B ( s, t) = \ \int\limits _ {- \infty } ^ \infty \int\limits _ {- \infty } ^ \infty e ^ {i ( \lambda s - \mu t) } F ( d \lambda \times d \mu ). $$

Examples of harmonizable random processes.

1) A modulated stationary random process. If

$$ X _ {0} ( t) = \int\limits _ {- \infty } ^ \infty e ^ {i \lambda t } d \Phi _ {0} ( t) $$

is a stationary random process, a process of the form

$$ X ( t) = c ( t) X _ {0} ( t), $$

where $ c( t) = \int _ {- \infty } ^ \infty e ^ {i \lambda t } m ( d \lambda ) $, where $ m ( d \lambda ) $ is a measure on the line, is usually no longer stationary, but will be harmonizable:

$$ X ( t) = \int\limits _ {- \infty } ^ \infty e ^ {i \lambda t } d \Phi ( \lambda ), $$

where the random measure $ d \Phi ( \lambda ) $ is defined by the formula

$$ \Delta \Phi ( \lambda ) = \ \int\limits _ \Delta m ( \Delta - \lambda ) \ d \Phi _ {0} ( \lambda ). $$

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

$$ X ( t) = \int\limits _ {- \infty } ^ \infty c ( t - s) dZ ( s), $$

where $ d Z( t) $ is some random measure on the line and the weight function $ c( t) $ is of the same type as above:

$$ c ( t) = \int\limits _ {- \infty } ^ \infty e ^ {i \lambda t } m ( d \lambda ) . $$

In this case

$$ X ( t) = \int\limits _ {- \infty } ^ \infty e ^ {i \lambda t } d \Phi ( \lambda ), $$

where

$$ \Delta \Phi ( \lambda ) = \ \int\limits _ \Delta \left [ \int\limits _ {- \infty } ^ \infty e ^ {- i \lambda t } \ dZ ( t) \right ] m ( d \lambda ). $$

References

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