Premium
SPECTRAL RADIUS, KRONECKER PRODUCTS AND STATIONARITY
Author(s) -
Liu Jian
Publication year - 1992
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/j.1467-9892.1992.tb00110.x
Subject(s) - mathematics , bilinear interpolation , kronecker product , spectral radius , eigenvalues and eigenvectors , kronecker delta , spectrum (functional analysis) , combinatorics , statistics , physics , quantum mechanics
. We provide a stochastic proof of the inequality ρ( A ⊗ A + B ⊗ B ) ≥ρ( A ⊗ A ), where ρ( M ) denotes the spectral radius of any square matrix M , i.e. max{|eigenvalues| of M }, and M ⊗ N denotes the Kronecker product of any two matrices M and N. The inequality is then used to show that stationarity of the bilinear modelwill imply stationarity of the linear part, i.e. the linear ARMA modelfor r = 1 and q = 1. Furthermore, it is shown that stationarity of the subdiagonal model, i.e. the bilinear model with b ij =0 for i < j , again implies stationarity of its linear part, provided that the stationarity condition given by Bhaskara Rao and his colleagues is met. Interestingly, the conclusion that stationarity of the subdiagonal models, implies that the linear component models cannot be extended to the general non‐subdiagonal bilinear models. The last observation is demonstrated via a simple example with p = m = 1, r = 0 and q = 2.