Question

Explain Second order stationarity and weak  Second order stationarity schemes briefly.

Answer 1

The class of linear time series models, which includes the class of autoregressive moving-average (ARMA) models, provides a general framework for studying stationary processes. In fact, SECOND ORDER STATIONARY PROCESS is either a linear process by subtracting a deterministic component. This result is known as World's Decomposition.

A second-order random process {Xt:t?T}{Xt:t?T} is one for which E[X2t]E[Xt2]is finite (indeed bounded) for all t?Tt?T. For us electrical engineers who apply (or mis-apply!) random process models in studying electrical signals, E[X2t]E[Xt2] is a measure of the average power delivered at time tt by a stochastic signal, and so all physically observable signals are modeled as second-order processes. Note that stationarity has not been mentioned at all and these second-order processes might or might not be stationary.

A random process that is stationary to order 22, which we can (but perhaps should not) call a second-order stationary random process provided we agree that second-order modifiesstationary and not random process, is one for which TT is a set of real numbers that is closed under addition, and the joint distribution of the random variables XtXt and Xt+?Xt+?(where t,??T)t,??T) depends on ?? but not on tt. As the link provided by AO shows, a random process stationary to order 22 need not be strictly stationary. Nor is such a process necessarily wide-sense-stationarybecause there is no guarantee that E[X2t]E[Xt2] is finite: consider for example a strictly stationary process in which the the XtXt's are independent Cauchy random variables.