Some Extensions of the Innovations Theorem

Some Extensions of the Innovations Theorem* By THOMAS KAILATH* (Manuscript received December 4, 1970) Consider an observed process that is the sum of a Wiener noise process and the integral of a not necessarily gaussian signal process. The innovations process is defined as the difference between the observed process and the integral of the causal minimum-mean-squareerror estimate of the signal process. Then if the integral of the expected value of the absolute magnitude of the signal process is finite, we show that the innovations process is also a Wiener process. The present conditions are a substantial weakening of those previously used in which the integral of the signal variance had to be finite. The new result is obtained by using some recent results in martingale theory. These results also enable us to obtain similar results when the Wiener process is replaced by a square-integrable martingale.