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Lecture 5: Gaussian processes & Stationary processes

WebFor the autocovariance function γof a stationary time series {Xt}, 1. γ(0) ≥ 0, 2. γ(h) ≤ γ(0), 3. γ(h) = γ(−h), 4. γis positive semidefinite. Furthermore, any function γ: Z → R that satisfies (3) and (4) is the autocovariance of some stationary time series (in particular, a Gaussian process). e.g.: (1) and (2) follow from ... WebOct 18, 2024 · 1. Let us observe the situation when τ = 0. The expectation of X t 2 should not depend on t hence we should have either σ = 0 or cos 2 ( c t) = 1 for all t. The last … WebExercise 5.2. Show, from the definition above, that the Wiener process has stationary independent incre-ments, i.e. (a)the distribution of W t W s depends only on t s; (b)the variables W tj W sj, 1 j n are independent whenever the intervals (s j;t j] are disjoint. farm sound animals