What does jointly stationary mean?
Two stochastic processes and are called jointly wide-sense stationary if they are both wide-sense stationary and their cross-covariance function depends only on the time difference .
What is stationary and non stationary random process?
Another characteristic of a random walk is that the variance evolves over time and goes to infinity as time goes to infinity; therefore, a random walk cannot be predicted. A non-stationary process with a deterministic trend has a mean that grows around a fixed trend, which is constant and independent of time.
When is a random process said to be stationary?
A random process at a given time is a random variable and, in general, the characteristics of this random variable depend on the time at which the random process is sampled. A random process X ( t) is said to be stationary or strict-sense stationary if the pdf of any set of samples does not vary with time.
How does the stationary random process ensure orthogonality?
In other words, the values of τ for which R W(τ) = 0 or sin (2πτ) = 0 (i.e., τ = k 2, k = 0, ± 1, ± 2, …) can ensure orthogonality of X ( t) and Y ( t ). With the knowledge of the expectation and variance of a density distribution, we can reexamine the spectrum of random variables of importance to environmental fluid mechanics.
When is a random process called a WSS?
A random process is called weak-sense stationary or wide-sense stationary (WSS) if its mean function and its correlation function do not change by shifts in time. More precisely, X(t) is WSS if, for all t1, t2 ∈ R and all Δ ∈ R, E[X(t1)] = E[X(t2)], E[X(t1)X(t2)] = E[X(t1 + Δ)X(t2 + Δ)].
Can a stationary random process yield a unique autocorrelation function?
It is important to note that a wide-sense stationary random process yields a unique autocorrelation function, but the converse is not true (i.e., two different wide-sense stationary random processes may have the same autocorrelation function). where A and B are uncorrelated random variables with zero-mean and unit-variance.