Joint distribution of ESACF arrays within triangular zero boundary

An obvious of ESACF approach for model identification by Tsay and Tiao(1984) is that the user may be confused by the elements which are in triangular but marginally larger two standard deviation values. To avoid this drawback, the joint limiting distribution of the vector whose elements are in triangular of ESACF arrays is verified and the statistics to test the nulity of the vector suggested. We illustrate this approach with three examples.