3e Some integral geometry............... 36
نویسنده
چکیده
3a An instructive toy model: two paradoxes We start with a very simple random trigonometric polynomial (even simpler than 1c12): (3a1) X(t) = ζ cos t+ η sin t where ζ, η are independent N(0, 1) random variables. Its distribution is the image of γ under the map (x, y) 7→ (t 7→ x cos t+ y sin t). Time shifts of the trigonometric polynomial correspond to rotations of R, (3a2) X(t− α) = ζ(cos t cosα + sin t sinα) + η(sin t cosα− cos t sinα) = = (ζ cosα− η sinα) cos t+ (ζ sinα + η cosα) sin t ; the process (3a1) is stationary, that is, invariant under time shifts; (3a3) EX(t) = 0 , EX(s)X(t) = cos(s− t) . The random variable (3a4) M = max t∈R |X(t)| = √
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