StronglyHarmonizableStochasticProcess

The covariance function $R(t, s)$ of a strongly harmonizable stochastic process $X(t)$ is characterized by the Bochner-Khintchine representation theorem. It states that the covariance function can be represented as a Fourier transform of a finite complex-valued measure $F$ on $\mathbb{R}^2$. Formally, the covariance function is given by:

$$ R(t, s) = \int_{\mathbb{R}^2} e^{i(\lambda t - \mu s)} dF(\lambda, \mu) $$

where $t, s \in \mathbb{R}$, and $F$ is a complex-valued measure satisfying the positive definiteness condition:

$$ \int_{\mathbb{R}^2} \int_{\mathbb{R}^2} e^{i(\lambda t - \mu s)} dF(\lambda, \mu) \overline{dF(\lambda', \mu')} \geq 0 $$

for all finite sets ${t_j}$ and ${s_j}$ in $\mathbb{R}$.

This representation implies that the covariance structure of strongly harmonizable processes is determined by the spectral measure $F$, which captures the frequency content of the process. The process is "strongly harmonizable" if this spectral measure is finite, which is a stronger condition than the mere existence of a spectral density function as in weakly stationary processes. This allows for a broader class of non-stationary processes to be included under the umbrella of strongly harmonizable processes.