Convergence and convergence rate to fractional Brownian motion for weighted random sums

We consider infinite sums of weighted i.i.d. random variables, with finite variance and arbitrary distribution, and derive a necessary and sufficient conditions for the weak convergence (in function space with uniform topology) of normalized sums to fractional Brownian motion (FBM). We consider also convergence rates questions. Using the embedding suggested by the Komlós–Major–Tusnády strong approximations method, we derive (under certain conditions on the weights) estimates for the quality of the functional approximation to FBM.