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Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes. (English) Zbl 1400.62194
Summary: We discuss joint temporal and contemporaneous aggregation of $$N$$ independent copies of AR(1) process with random-coefficient $$a\in [0,1)$$ when $$N$$ and time scale $$n$$ increase at different rate. Assuming that $$a$$ has a density, regularly varying at $$a=1$$ with exponent $$-1<\beta <1$$, different joint limits of normalized aggregated partial sums are shown to exist when $$N^{1/(1+\beta )}/n$$ tends to (i) $$\infty$$, (ii) 0, (iii) $$0<\mu <\infty$$. The limit process arising under (iii) admits a Poisson integral representation on $$(0,\infty )\times C(\mathbb R)$$ and enjoys ‘intermediate’ properties between fractional Brownian motion limit in (i) and sub-Gaussian limit in (ii).

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 60F05 Central limit and other weak theorems
longmemo
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