An Ergodic Theorem for the Square of a Wide-sense Stationary Process

Abstract

Let $\{X(t), -\infty < t < \infty\}$ be a stochastic process which is stationary in the wide sense with spectral representation $X(t) = \int^\infty_{-\infty} e^{it\lambda} d\xi(\lambda)$, where the $\xi$ process is centered and has independent increments with $E\xi(\lambda) \equiv 0, E|\xi(\lambda)|^2 < \infty$. It is shown that under weak conditions $$P - \lim_{T\rightarrow\infty} \frac{1}{2T} \int^T_{-T} |X(t)|^2 dt$$ exists and is equal to $\sigma^2 + \sum J_t^2 + \sum \xi_n^2$, where $\sigma^2$ is equal to the variance of the Gaussian component of the continuous part of the $\xi$ process, $\sum J_t^2$ is the sum of the squares of the jumps of the Gaussian component of the $\xi$ process, and $\xi_N = \xi(\lambda_N + 0) - \xi(\lambda_N - 0)$, where $\{\lambda_N\}$ are the fixed discontinuities of the $\xi$ process.