= Canonical identifiability of Gaussian ARMA models
Normalize the autoregressive and moving-average polynomials to have constant coefficient one, cancel common factors, and require all their zeros to lie outside the closed <unit disk>. With positive innovation <variance> these conditions give a causal, invertible, minimal <Gaussian process> representation. To prove identifiability, compare two such models with the same <time-series spectral density>. The ratio of their scaled rational transfer functions and its reciprocal are analytic on a neighbourhood of the closed <unit disk> and have modulus one on its boundary. The <maximum modulus principle> makes the ratio constant; its positive value at zero forces equality of the innovation <variances> and equality of the rational functions. Coprimality and the constant-coefficient normalization then force equality of both polynomials. Without the root restrictions, reciprocal <root reflection of an ARMA representation> and a rescaling of the innovation <variance> can preserve the spectrum, so the unrestricted parameters are not identifiable.
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