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On the rate of approximation in limit theorems for sums of moving averages. (English) Zbl 1152.60041
Theory Probab. Appl. 52, No. 2, 361-370 (2008) and Teor. Veroyatn. Primen. 52, No. 2, 405-414 (2007).
The authors consider a linear process \(X_t=\sum_{j=0}^\infty a_j \varepsilon_{t-j},\;t\geq 1,\) where \((\varepsilon_i,i\in \mathbb{Z})\) are independent identically distributed random variables in the domain of attraction of a stable law with index \(\alpha\in (0,1)\cup (1,2].\) Under some conditions on random variables \(\varepsilon_i\) and coefficients \(a_j,\) they study approximation of sums \(S_n=B_n^{-1}\sum_{t=1}^n X_t\) by sums \(Z_n=B_n^{-1}\sum_{t=1}^n Y_t,\) where \(Y_t=\sum_{j=0}^\infty a_j \eta_{t-j},\) and \((\eta_i,i\in \mathbb{Z})\) are i.i.d copies of \(\alpha\)-stable random variable \(\eta.\) It is shown that in many cases the rate of convergence of \(\triangle_n=\sup_{x\in R} | P(S_n\leq x)-P(Z_n\leq x)| \) is the same as in the case of i.i.d. summands. Moreover, the results apply to situations when \(P(S_n\leq x)\) and \(P(Z_n\leq x)\) do not converge. The authors obtained easily verifiable conditions on the coefficients \((a_j)\) in various dependence situations under which the rate of convergence is achieved.

MSC:
60G52 Stable stochastic processes
60F99 Limit theorems in probability theory
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