BochnerKhintchineRepresentationTheorem

Let $X(t)$ be a weakly stationary stochastic process with covariance function $R(\tau)$, where $\tau = t - s$. The Bochner-Khintchine representation theorem states that:

$$ R(\tau) = \int_{-\infty}^{\infty} e^{i\lambda\tau} dF(\lambda) $$

where $F(\lambda)$ is a non-negative, finite measure on the real line $\mathbb{R}$, known as the spectral distribution function of the process.

Furthermore, the covariance function $R(\tau)$ is positive definite, which means for any finite set of times $t_1, t_2, \ldots, t_n$ and any set of complex numbers $c_1, c_2, \ldots, c_n$ the following inequality holds:

$$ \sum_{j=1}^{n} \sum_{k=1}^{n} c_j R(t_j - t_k) \overline{c_k} \geq 0 $$

This theorem provides a spectral representation of the covariance function, linking the time domain properties of the process to its frequency domain characteristics.