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- W93765222 abstract "Abstract In what follows let W(t) = (W1(t)W2(t)W3(t)) be a threedimensional Wiener process (Brownian motion) starting at the origin. P. Levy has shown that the characteristic function of the random variable X = ∫10[W1(t-tW1(1)]d[W2(t)-tW2(1)], which measures the signed area of a random planar loop, is EeizX = (z/2) csch (z/2). (The corresponding density, of importance in polymer physics, is (π / 2) sech2πx). Levy's derivation uses N. Wiener's simple sine-cosine Fourier expansion of white Gaussian noise. This expansion is used here as the starting point in showing that the six-dimensional vector (W(1),∫10W(t)×dW(t)), needed to characterize polarization dispersion effects for high speed communications over optical fibers, also has an elementary closed form characteristic function. Employing the same Fourier expansion we prove the following result, generalizing the two examples: Let V be any finite dimensional vector whose components are arbitrary linear combinations of random variables of the form Wk(1), Wk(1)Wl(1), Wk(1)∫10Wl(t)dt, ∫10Wk(t)dWl(t) (1≤k≤3, 1≤l≤3). The joint characteristic function of V can be expressed in closed form using elementary function." @default.
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- W93765222 date "1991-01-01" @default.
- W93765222 modified "2023-10-16" @default.
- W93765222 title "Closed Form Characteristic Functions for Certain Random Variables Related to Brownian Motion" @default.
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- W93765222 doi "https://doi.org/10.1016/b978-0-12-481005-1.50016-8" @default.
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