By T. T. Soong
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Extra resources for Random Differential Equations in Science and Engineering
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.