What is a weakly stationary process?

What is a weakly stationary process?

Weak-Sense Stationary Processes: Here, we define one of the most common forms of stationarity that is widely used in practice. 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.

What are stationary stochastic processes?

A stochastic process is said to be stationary if its mean and variance are constant over time and the value of the covariance between the two time periods depends only on a distance or gap or lag between the two time periods and not the actual time at which the covariance is computed.

Is an IID process always weakly stationary?

This process is always strictly stationary by definition. But that is trivially the case due to the iid assumption on the ϵt.

What is the difference between stationary and non stationary stochastic process?

A stationary time series has statistical properties or moments (e.g., mean and variance) that do not vary in time. Stationarity, then, is the status of a stationary time series. Conversely, nonstationarity is the status of a time series whose statistical properties are changing through time.

What is the difference between weak and strong stationarity?

Time series is strong stationery given that unconditional joint probability distribution does not change when shifted in time, this means that the distribution is the same through time. In weak stationary, a time series is when the mean and the variance are not constant through time. They keep changing with time.

Does weak stationarity imply strong stationarity?

The reason strong stationarity does not imply weak stationarity is that it does not mean the process necessarily has a finite second moment; e.g. an IID process with standard Cauchy distribution is strictly stationary but has no finite second moment⁴ (see [Myers, 1989]).

Are stochastic processes IID?

Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time.

Is IID stationary process?

The simplest example of a strictly stationary process is an IID sequence, that is, a sequence whose terms are mutually independent and have the same distribution.

What is the difference between strict stationarity and weak stationarity?

Strict stationarity implies weak stationarity when the mean and variance of the time series exist. The two forms of stationarity are equivalent if the time series follows the normal distribution. Overall, weak stationarity is less restrictive than strict stationarity.

Why is white noise weakly stationary?

White noise is stationary in every sense. because the joint distribution of any set of ϵi does not depend on time at all. Observations are independent and identically distributed.

What is Poisson process in stochastic process?

A Poisson process is a simple and widely used stochastic process for modeling the times at which arrivals enter a system. It is in many ways the continuous-time version of the Bernoulli process that was described in Section 1.3. 5.

Is white noise weakly stationary?

White Noise Process: A white noise process is a serially uncorrelated stochastic process with a mean of zero and a constant and finite variance. Note that this implies that every white noise process is a weak stationary process.

What is IID and non IID?

Literally, non iid should be the opposite of iid in either way, independent or identical . So for example, if a coin is flipped, let X is the random variable of event that result is tail, Y is the random variable of event the result is head, then X and Y are definitely dependent. They can be decided by each other.

What is weakly stationary in time series?

A time series is considered as weakly stationary if the associated mean and covariance function do not vary with respect to time. That is to say, the original time series has statistical properties similar to those of the ‘time-shifted’ series.

What is the difference between strictly stationary and weakly stationary?

A time series model which is both mean stationary and covariance stationary is called weakly stationary. A time series model for which all joint distributions are invariant to shifts in time is called strictly stationary.