6.10 To explore the stability of the filter, consider a univariate state-space model. That is, for t

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6.10 To explore the stability of the filter, consider a univariate state-space model. That is, for t = 1, 2, . . ., the observations are yt = xt +vt and the state equation is xt = φxt−1 + wt, where σw = σv = 1 and |φ|

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(b) Use the result of

(a) to verify Pt−1 t approaches a limit (t→∞) P that is the positive solution of P2 − φ2P − 1 = 0.

(c) With K = limt→∞ Kt as given in Property P6.1, show |1−K|

(d) Show, in steady-state, the one-step-ahead predictor, yn n+1 = E(yn+1 **
yn, yn−1, . . .), of a future observation satisfies

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