×

Compound Poisson approximation for triangular arrays with application to threshold estimation. (English) Zbl 1259.60027

Let \(\left\{Y_{n,j},\;j=1,2,\dotsc,n,\;n=1,2,\dots\right\}\) be triangular array of random variables, whose rows are asymptotically negligible and satisfy some weak dependence condition. Using the Tikhomirov method, the authors obtain compound Poisson approximation for the sums \(S_{n}=\sum_{j=1}^{n}Y_{n,j}\). Moreover, they derive bounds on the Levy distance between the distributions function of the sums S\(_{n}\) and the compound Poisson distribution. The results are applied to threshold estimation in autoregressive models.

MSC:

60F05 Central limit and other weak theorems
62F12 Asymptotic properties of parametric estimators
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)