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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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