Articles of stationary processes

About stationary and wide-sense stationary processes

I have just started with stochastical calculus, and I need some help with a pair of problems: $\bullet$If $X(t)$ is a mean square differentiable wide-sense stationary stochastical process then the processes $X(t)$ and $X’ (t)$ are orthogonal. $\bullet$If $X(t)$ is a twice mean square differentiable, stationary and Gaussian stochastical process, such that $E[X(t)] = 0$, […]

Markovian Gaussian stationary process with continuous paths

Could you, please, help me figure out the following problem. We call a stationary Gaussian process $\xi_t$ (with continuous paths) an Ornstein-Uhlenbeck process if its correlation function $\mathbb{E}\xi_{s+t}\bar \xi_s$ is $$ \mathbb{E}\xi_{s+t}\bar \xi_s=R(t)=R(0)e^{-\alpha|t|}, \alpha\ge 0 $$ The question is that Ornstein-Uhlenbeck process is the unique Markovian stationary Gaussian process. By Markovian I mean that $$ […]

Does the power spectral density vanish when the frequency is zero for a zero-mean process?

A wide-sense stationary random time series $\zeta(t)$ is characterized by its mean value and its autocovariance function, which in the Wiener–Khinchin theorem is equivalent to the Fourier transform of the power spectral density $$\langle[\zeta-\langle\zeta\rangle][\zeta’-\langle\zeta\rangle]\rangle = \sigma^2 C(\tau)-\langle\zeta\rangle^2 = \frac 1{2\pi}\int_{-\infty}^\infty S(\omega)exp(-\omega\tau)d\omega$$ where $\sigma^2 C(\tau)$ is the autocovariance function with rms amplitude $\sigma$, $\tau = t […]