ABSTRACT
If the RW probability distribution used in Chapter 2 is Gaussian the convolution is “exact”: the convolution of two Gaussian distributions is again Gaussian (for the obvious reason that the characteristic functions are Gaussian and the product of two Gaussians again is a Gaussian, see detailed discussion in Problem Set 1) and we obtain: https://www.w3.org/1998/Math/MathML"> ∫ ∏ k = 1 n − 1 d x ( k ) P ( x ( k ) - x ( k - 1 ) ) = e − | x ( n ) − x ( 0 ) | 2 2 σ 2 n ( 2 π σ 2 n ) 1 2 , P ( x ) = e − | x | 2 2 σ 2 ( 2 π σ 2 ) 1 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math3_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
