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Convergence and convergence rate to fractional Brownian motion for weighted random sums. (English) Zbl 1079.60025

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.

MSC:

60F17 Functional limit theorems; invariance principles
60F15 Strong limit theorems
60G18 Self-similar stochastic processes
60G15 Gaussian processes
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