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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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