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S,) indexed by finite sets of times tl < tz < "'" < t, which are joint probability distributions in $1 . . . 17). In this sense, the finitedimensional distributions completely determine the probabilistic structure of the stochastic process and any special properties desired of the process can be 38 2 UNIVARIATE SPECTRAL M O D E L S imposed on these distributions. For example, a stochastic process is said to be Gaussian if the finite-dimensional distributions are multivariate normal distributions.

3 SUMMARY 17 OF VECTOR S P A C E G E O M E T R Y be shown that ~(xl~ ~) Y ) = ~(xl ~ ' ) + ~ ( x I ,AP). //r177 in Y which are orthogonal to ~ ' , called the orthogonal complement of~g/l in ,IV, is a linear subspace such that ~ = ,//t' O,/d'x • (Halmos, 1951, p. 26). Applying the above property of projections to this expression, it is possible to derive the relation ~ ( ~ ( x l Y ) I -////) = ~ ( x l ~'). 16) Some Specific Hilbert Spaces A number of the properties of time series models that we will need in the book are simply the properties of vector spaces summarized above applied to particular Hilbert spaces.

22) j-- ~n where 20,2• 1, - . , 2+, are fixed frequencies and the Zj's are complex-valued random variables. This expression actually defines a stochastic process through its spectral representation. However, without further conditions the process can be both complex-valued and nonstationary. We will now indicate the properties the Zj's must have in order that X(t) be a real-valued, weakly stationary process. This will provide some insight into the properties of the general spectral representation to be given in the next section.

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Business Statistics by Geis

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