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62) or Eq. v. are easily derivable from the knowledge of its characteristic function. The joint characteristic function of two random variables X and Y , #xy(u, a ) or $(u, u), is defined by J --oo J --m where u and u are two arbitrary real variables. Analogous to the onerandom-variable case, the joint characteristic function $(u, u ) is often called upon for the determination of the joint density function f ( x , y ) of X and Y and their joint moments. 68) The above development can be generalized to the case of more than two random variables in an obvious manner.

P. X ( t ) defined by Eq. ( 3 . ’s A , , A , , . . , A , for all t E T. p. X ( t ) having this property is said to have n degrees of randomness. p. 2, 3n degrees of randomness. Following the definition of a random vector, let us now introduce the notion of a vector stochastic process. In many physical situations, there is a need to consider not a single stochastic process but a sequence of stochastic processes, say, X l ( t ) , X , ( t ) , . . , X,,(t), t E T. Let X ( t ) , t E T, be a vector stochastic process whose components are X l ( t ) , X,(t), .

We have already noted that the power spectral density function S(w) is real and nonnegative. 62) 0 Hence, S(w) is also an even function of cn). When z = 0, the second of Eqs. 63) It is well known that, in nonrandom time series analysis, the Fourier analysis in the frequency domain is a powerful tool from both mathematical and physical points of view. The notion of power spectral density functions plays such a role in the analysis of stochastic processes. Its usefulness in physical applications will be discussed in detail in Chapters 7-9.