By Marc Nerlove

During this variation Nerlove and his co-authors illustrate innovations of spectral research and techniques in response to parametric versions within the research of monetary time sequence. The publication offers a method and a style for incorporating monetary instinct and conception within the formula of time-series versions

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**Example text**

And c(0) φ 0. The question of when the limits involved in the definitions of the sample moments above exist is connected with the question of stationarity. In fact the following "law of large n u m b e r s " can be established (Doob, 1953, p. 465): Theorem. If { x j is stationary and if g is any function of the k r a n d o m variables x l 5 . . ,x f c such that g{xu... ,xk) is a well-defined r a n d o m variable and such that, for any τ, (8) exists, then the finite sample means (9) converge with probability 1 to the r a n d o m variable (10) If the process {xt} is also what is called ergodic (or in this case strongly ergodic), the r a n d o m variable g is a constant with probability 1 such that (11) The notion of ergodicity is a rather deep one.

1) for any finite index set Tk and for any τ such that Tk + τ belongs to T. It is apparent that the process {y,}, defined by (12), is stationary if the processes ( 1 ) n) { x , } , . . , {x\ } are jointly stationary. If processes are jointly stationary they 5 are individually so; the converse, however, is not t r u e . This is evident since 4 λ ι + ( Let {xt},t e T, be a process defined as follows: xt = ^ ρ \ where r ~ η(μ, 1), / is a fixed frequency, a n d φ is uniformly distributed on the interval [ - π , π ] ; the definition of stationarity may be seen t o apply by deriving the joint distribution of the pair (x r, .

Define a new subset of T, Tk + τ as Tk + T = {tl + τ , . . , ί * + τ}. Then { x j is called a stationary and only if time series (sometimes, strictly stationary) if FTk(-) = FTk + T(-) (1) for all finite subsets Tk of the index set Τ such that Tk + τ belongs to T ; that is, FTk(-) depends only on differences between the i's. , when Τ consists of a continuum of real numbers, it suffices to take Tk to be a finite set of k points in Τ and τ to be any real number such that r, + τ is in Τ for i = 1 , .