## 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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### References:

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