Expansion of white noise into infinite series using orthogonal polynomials

Question

Having a white random process $s(t)$, is it possible to write $$s(t)=\sum_{i=0}^\infty\alpha_i\phi_i(t)$$ where the $\alpha_i$ are random variables and the $\phi_i$ orthogonal polynomials (Jacobi Polynomials, Legendre Polynomials...)?

Answer 1

Yes, if $\{f_j\}$ is an orthonormal basis of $L^2[0,1]$ and $Z_j$ are i.i.d. standard normal variables, then $S(t)=\sum_j Z_j f_j(t)$ is a white noise. Note that this sum does not converge pointwise and should be understood as a distribution. Alternatively, if $F_j(t)=\int_0^t f_j(r) \,dr$ then $B(t)=\sum_j Z_j F_j(t)$ is a standard Brownian motion. When $f_j$ is the trigonometric basis, this is Wiener's original construction of Brownian motion, see [1]. When $f_j$ is the Haar basis, this yields Paul Levy's interpolation construction, see [2], [3]. When $L_j$ are the Legendre polynomials and $f_j(t)$ equals $L_j(2t-1)$ normalized in $L^2$, one gets the desired polynomial representation. See [4] for a refined study.

[1] Jean-Pierre Kahane. Some random series of functions. Vol. 5. Cambridge University Press, 1993.

[2] https://sites.stat.washington.edu/jaw/COURSES/491-2/HO-492/HaarFcnConstr-HO-492.pdf

[3] http://math.stanford.edu/~ryzhik/STANFORD/STANF227-10/notes-bm.pdf

[4] An Optimal Polynomial Approximation of Brownian Motion James Foster, Terry Lyons, and Harald Oberhauser SIAM Journal on Numerical Analysis 2020 58:3, 1393-1421 https://arxiv.org/abs/1904.06998