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Convergence of functionals of sums of r.v.s to local times of fractional stable motions. (English) Zbl 1049.60019

The author studies the asymptotical properties of the process \((\beta_n/n)\sum_{k=1}^{[nt]}f (\beta_n(S_k/\gamma_n+ x)),\) where \(S_k=\sum_{j=1}^k\sum_{m=0}^\infty c_m\xi_{j-m},\) \((c_m)\) is a sequence of constants, \((\xi_j)\) is a sequence of i.i.d. random variables belonging to the domain of attraction of a strictly stable law with index \(0<\alpha\leq 2,\) \((\beta_n)\) is a sequence of constants such that \(\beta_n\to\infty,\) \(\beta_n/n\to 0.\) The limiting distribution is written in terms of the local time of the linear fractional stable motion. The only conditions on the distribution of \(\xi_1\) are Cramer’s condition and the existence of nonzero absolutely continuous components.

MSC:

60F05 Central limit and other weak theorems
60G18 Self-similar stochastic processes
60J55 Local time and additive functionals
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62J02 General nonlinear regression
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