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Unformatted text preview: as T goes to inﬁnity. When this happens for a random process, we say that the random process is ergodic in the mean. * This estimate yields a single number, so obviously it only makes sense to consider processes for which m X ( t ) = m , a constant. Ergodicity Theorem . Let X ( t ) be a WSS process with m X ( t ) = m , then lim T →∞ h X ( t ) i T = m in the mean square sense, if and only if lim T →∞ VAR [ h X ( t ) i T ] = lim T →∞ 1 2 T Z 2 T-2 T ± 1-| u | 2 T ² C X ( u ) du = We can also deﬁne ergodicity in the higher moments, for example, mean square or power. Theorem . A WSS random process X ( t ) is ergodic in mean square if lim T →∞ " 1 2 T Z T-T X 2 ( t ) dt # = R X ( ) Problem Let X ( t ) = A for all t , where A is a zero-mean, unit-variance random variable. Is X ( t ) ergodic?...
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- Fall '08
- Statistics, Probability theory, random process