TAILIEUCHUNG - Independent And Stationary Sequences Of Random Variables - Chapter 16

Chapter 16 SOME RESULTS FROM THE THEORY OF STATIONARY PROCESSES In this chapter an account is given of those results from the theory of stationary processes which will be required in the sequel . This chapter has much in common with Chapter 1, but here the proofs will, as a rule | Chapter 16 SOME RESULTS FROM THE THEORY OF STATIONARY PROCESSES In this chapter an account is given of those results from the theory of stationary processes which will be required in the sequel. This chapter has much in common with Chapter 1 but here the proofs will as a rule be given in full although the discussion will be rather condensed. For a more complete and detailed account we refer to chapters X and XI of 31 as well as 163 . 1. Definition and general properties A random process Xt t e T is called stationary in the strict sense if the distribution of the random vector does not depend on h so long as the values tz h belong to T a subset of the real line . The random process is called stationary in the wide sense if E X2 oo for all t and if E XS and E XsXs t do not depend on s. Without loss of generality we can and will take E XS O otherwise we can replace Xt by Xt E Xt . If no confusion can be caused the qualifying parentheses in the strict or wide sense will be omitted. The parameter set T will be taken to be either the whole line or the set of integers positive or negative except when it is specifically stated that only non-negative values of t are considered. We distinguish the two cases as those of continuous time and discrete time stationary processes with a discrete time parameter are often called stationary sequences. It is notationally convenient to write continuous time processes as X t or X s and discrete time processes as Xn or Xj when both are considered together we use the notation Xt or Xs. . DEFINITION AND GENERAL PROPERTIES 285 In the case of continuous time we assume that the process is stochastically continuous in the sense that for all g 0 limP X t s -X t 0. s- 0 When dealing with wide-sense stationary processes however we shall assume the stronger condition lim X t s -X t 2 0 . s- 0 These conditions are very weak they are fulfilled in all cases of interest. A random process Xt is a function Xt œ of two variables t g T .

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