Probability for Statistics and Machine Learning by Anirban DasGupta

Author:Anirban DasGupta
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

Example 12.1(Some Illustrative Processes).

We take a few stochastic processes, and try to understand some of their basic properties. The processes we consider are the following. (a) X 1(t) ≡ X, where X ∼ N(0, 1).

(b) X 2(t) = tX, where X ∼ N(0, 1).

(c) X 3(t) = Acosθt + Bsinθt, where θ is a fixed positive number, t ≥ 0, and A, B are iid N(0, 1).

(d) X 4(t) =  ∫0 t W(u)du, t ≥ 0, where W(u) is standard Brownian motion on [0, ∞), starting at zero.

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