Konstantopoulos, Takis; Sakhanenko, A. I. Convergence and convergence rate to fractional Brownian motion for weighted random sums. (English) Zbl 1079.60025 Sib. Èlektron. Mat. Izv. 1, 47-63 (2004). Let \(\xi_k\) be i.i.d. random variables with infinite variance and arbitrary distribution. Consider a stationary sequence of random variables \(X_j=\sum_{k=-\infty}^\infty a_{j-k}\xi_k\), where the coefficients \(a_k\) are deterministic. Depending on the behavior of \(a_k\), the sequence \(X_j\) may be long-range dependent, in the sense that its correlation decays polynomially. Introduce the “random walk” \[ S_0 = 0,\quad S_n = X_1+\dots+ X_n,\quad n=1,2, \dots\,. \] The authors derive necessary and sufficient conditions for the weak convergence (in a function space with uniform topology) of the normalized sums \(Z_{n,H}(t) =S_{[nt]}/n^H\), \(t\in[0,\infty)\), to a fractional Brownian motion; here \(H\) is the Hurst parameter. They also study the rate of convergence. Using the embedding suggested by the Komlós-Major-Tusnády strong approximations method, the authors derive estimates for the quality of the functional approximation to a fractional Brownian motion. Reviewer: D. A. Korshunov (Novosibirsk) Cited in 11 Documents MSC: 60F17 Functional limit theorems; invariance principles 60F15 Strong limit theorems 60G18 Self-similar stochastic processes 60G15 Gaussian processes Keywords:sums of weighted i.i.d. random variables; Komlos-Major-Tusnady strong approximations method PDFBibTeX XMLCite \textit{T. Konstantopoulos} and \textit{A. I. Sakhanenko}, Sib. Èlektron. Mat. Izv. 1, 47--63 (2004; Zbl 1079.60025) Full Text: EuDML