Analysis of the fractal sum of pulses

Abstract

Fractal sums of pulses are defined on $\mathop{\mathbb R}^D$ as follows: $F(t)=\sum_{n=1}^\infty n^{-H/D} G\big(n^{1/D}\,(t-X_n)\big)$ where the $X_n$ are independent random variables, $H\in [0,1]$ and $G$ is the elementary “bump” or “pulse”. If the $X_n$ are uniform on a cube and $G$ sufficiently regular we prove the (almost sure) existence of $F$ and show that the box dimension of its graph is $2-H$.