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Q and let Q uk (t). 22) k=1 Assume that the characteristic functions satisfy 2 2 fuk ,1 (ξ1 )1/σk = ful ,1 (ξ1 )1/σl , ∀k, l ∈ {1, 2, . . 23) 2 where σm = Rum (0). Then u(t) is separable. 23). 27) l=1,l=k where we have used that ful ,2 (ξ1 , 0, τ ) = ful ,1 (ξ1 ) in the last equality. 28) fuk ,1 (ξ1 ) = k2 fu1 ,1 (ξ1 )σk /σ1 −1 fu1 ,1 (ξ1 ). 28) we obtain Guk (ξ1 , τ ) = 2 2 Ruk (τ ) fu1 ,1 (ξ1 )σk /σ1 −1 fu1 ,1 (ξ1 ). 26) in the last equality. 1 gives that u(t) is separable. In Nuttall (1958a), two different results for sums of separable processes are presented.

If y(t) = f (u(t)), and the system is said to be nonstatic if y(t) depends on any other u(t − k), k ∈ Z \ {0}. A class of systems that will be discussed later in this thesis is nonlinear finite impulse response (NFIR) systems. An NFIR system can for some M ∈ N be written as y(t) = f (u(t − k))M k=0 . Here, the compact notation f (u(t − k))M k=0 simply means f (u(t), u(t − 1), . . , a function of a finite number of input components. , 1995). Such a system can be written as y(t) = f (ϕ(t)) + e(t), where the vector ϕ(t) has signal components of u and y as elements and where e(t) is a white noise process.

Furthermore, the convergence of the parameter estimates to the true parameter values can be shown in some special cases (Bai and Li, 2004). However, there are no such convergence results available for the general case. Hence, it is important that the initial LTI approximation is as good as possible in the sense that it resembles the true LTI subsystem as well as possible. 2 Stochastic Approaches 31 invariance property holds, since this property implies that the LTI model that minimizes the mean-square error E((y(t) − G(q)u(t))2 ) will be equal to a scaled version of the LTI part of the system.