Chapter 2

Abstract

¹See Anderson (1971), Sec. 5.7.1, where it is proved that if B(τ) = 0 for |τ| > n, then numbers b₀, b₁, …, b_n can be found which satisfy (2.5). Let us now consider the more general case of the complex-valued function B(τ). Assuming that b_k = 0 for k > n and k < 0, we can write the corresponding generalization of the result (2.5) in the form

$$B(t - s) = \sum\limits_{k = ^{ - \infty } }^\infty {b_{k + t - 8} } \bar b_k = \sum\limits_{j = ^{ - \infty } }^\infty {b_{t - j} \bar b_{8 - j} }$$

(2.1)

. The last relationship implies the possibility of representing the random sequence X(t) in the form (2.4) by virtue of the so-called theorem on generalized spectral representation of random functions (see the closing part of Note 17 below).