It is shown that a stochastic process obtained by taking random sums of dilations and translations of a given function converges to Gaussian white noise if a dilation parameter grows to infinity and that it converges to Gaussian colored noise if a scaling parameter for the translations grows to infinity. In particular, the question of when one obtains fractional Brownian motion by integrating this colored noise is studied.
"White and colored Gaussian noises as limits of sums of random dilations and translations of a single function." Electron. Commun. Probab. 16 507 - 516, 2011. https://doi.org/10.1214/ECP.v16-1650