Probability of constructing prediction model for observable of a dynamical process via time series

The paper proves that, for linear systems, one can almost surely construct an order-n linear prediction model for the scalar output from a length-2n time series by inverting a Hankel matrix H and exploits the factorization H = Q M(A,x0), where Q is the observability matrix and M(A,x0) is a Krylov/controllability-from-x matrix, to identify the coefficients (Theorem 6). The measure-one statement is given as Theorem 19 and is supported by results showing that matrices with distinct eigenvalues have full measure and, for such A, generic x and c are cyclic/observable (Propositions 15–17) . The model’s solution also builds H from yj = cAjx, uses H = O C (observability × Krylov), and proves generic invertibility by noting det(H) is a nonzero polynomial in (c,A,x), hence its zero set has Lebesgue measure zero; from H a = −Y it derives the recurrence and further shows p(A) = 0 and that p is the characteristic polynomial. Thus both establish essentially the same conclusion, but by different genericity arguments (paper: distinct-eigenvalue + cyclicity/observability; model: algebraic variety of det(H)=0). Notably, the paper contains a clear error in Remark 7 (it asserts the set {x: det M(A,x)=0} is open and dense, which is the opposite of what is used later), and Theorem 6 includes an unnecessary assumption |A|≠0; however, these fixable issues do not invalidate the main results (Corollary 3’s recurrence and Theorem 19’s measure-one claim) .