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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
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