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Heavy-traffic approximations for fractionally integrated random walks in the domain of attraction of a non-Gaussian stable distribution. (English) Zbl 1254.60035
Let $$(X_i)_{i\geq 1}$$ be a sequence of i.i.d. mean zero random variables having a distribution function $$F$$. Set $$S_0=0$$ and $$S_n=\sum_{0\leq i <n}g_iX_{n-i}$$, $$n\in\mathbb{N}$$, where the series $$g(x)=\sum_{i\geq 0}g_ix^i$$ has a radius of convergence of at least $$1$$. This $$(g,F)$$-process $$(S_n)_{n\geq 0}$$ comprises, e.g., an autoregressive moving average (ARMA) one. Under specified conditions (among them $$g_n =cn^{\gamma -1}(1+o(n^{-\varepsilon}))$$ as $$n\to \infty$$ for some $$c,\varepsilon >0$$ and $$\gamma >1$$; $$F$$ is in the domain of attraction of a stable law of index $$\alpha$$), one can prove that appropriately normalized $$\sup_{n\geq 1}(S_n - a\sum_{0\leq i<n}g_i)$$ converges weakly as $$a\to 0$$ to a certain random variable defined by means of a fractional Lévy stable process.

MSC:
 60F17 Functional limit theorems; invariance principles 60G52 Stable stochastic processes 60G22 Fractional processes, including fractional Brownian motion 60K25 Queueing theory (aspects of probability theory) 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 60G70 Extreme value theory; extremal stochastic processes 62P20 Applications of statistics to economics
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