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Joint aggregation of random-coefficient AR(1) processes with common innovations. (English) Zbl 1325.62171
Summary: We discuss joint temporal and contemporaneous aggregation of \(N\) copies of stationary random-coefficient AR(1) processes with common i.i.d. standardized innovations, when \(N\) and time scale \(n\) increase at different rate. Assuming that the random coefficient \(a\) has a density, regularly varying at \(a = 1\) with exponent \(- 1 / 2 < \beta < 0\), 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 paper extends the results in [the authors, Stochastic Processes Appl. 124, No. 2, 1011–1035 (2014; Zbl 1400.62194)] from the case of idiosyncratic innovations to the case of common innovations.

MSC:
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60G22 Fractional processes, including fractional Brownian motion
60G15 Gaussian processes
60G18 Self-similar stochastic processes
60H05 Stochastic integrals
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