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Asymptotic properties of powers of nonnegative matrices, with applications. (English) Zbl 0668.15012
The main result of this paper gives first order approximations to \(f_ n(A)\) as \(n\to \infty\), for certain sequences \(f_ n\) of analytic functions, where A is a finite square reducible matrix in Frobenius form. In particular it holds when \(f_ n(A)=A^ n\), and hence can be used to study local behaviour of entries as well as blocks of \(A^ n\). An application to the behaviour of the vector \(pA^ n\) (“multiplicative processes”) generalizes a theorem of P. Mandl [Casopis Pĕst. Mat. 84, 140-149 (1959; Zbl 0087.136)] who considered only non-degenerate and aperiodic diagonal blocks. The approach is via spectral properties of finite dimensional linear operators.

MSC:
15B48 Positive matrices and their generalizations; cones of matrices
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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[1] Brook, D.; Evans, D.A., An approach to the probability distribution of cusum run length, Biometrika, 59, 539-549, (1972) · Zbl 0265.62038
[2] Campbell, S.L.; Meyer, C.D., Generalized inverses of linear transformations, (1979), Pitman London · Zbl 0417.15002
[3] Cooper, C.D.H., On the maximum eigenvalue of a reducible non-negative real matrix, Math. Z., 131, 213-217, (1973) · Zbl 0261.15006
[4] Darroch, J.N.; Seneta, E., On quasi-stationary distributions in absorbing discrete time finite Markov chains, J. appl. probab., 2, 88-100, (1965) · Zbl 0134.34704
[5] Dunford, N.; Schwarz, J.T., Linear operators, part 1, (1958), Interscience New York
[6] Friedland, S.; Schneider, H., The growth of powers of a nonnegative matrix, SIAM J. algebraic discrete methods, 1, 185-200, (1980) · Zbl 0498.65018
[7] Gantmacher, F.R., The theory of matrices, (1959), Chelsea New York · Zbl 0085.01001
[8] Keilson, J., Markov chain models—rarity and exponentiality, () · Zbl 0411.60068
[9] Lindqvist, B.H., Asymptotic properties of powers of nonnegative matrices, ()
[10] Lindqvist, B.H., Nonnegative multiplicative processes reaching stationarity in finite time, Linear algebra appl., 86, 75-90, (1987) · Zbl 0617.60067
[11] Mandl, P., Sur le comportement asymptotique des probabilités dans LES ensembles des états d’une chaine de Markov homogène, Časopis Pěst. mat., 84, 140-149, (1959), (in Russian) · Zbl 0087.13601
[12] Rothblum, U.G., Bounds on the indices of the spectral circle eigenvalues of a nonnegative matrix, Linear algebra appl., 55, 155-167, (1980)
[13] Rothblum, U.G., Sensitive growth analysis of multiplicative systems I: the dynamic approach, SIAM J. algebraic discrete methods, 2, 25-34, (1981) · Zbl 0498.60047
[14] Rothblum, U.G., Expansions of sums of matrix powers, SIAM rev., 23, 143-164, (1981) · Zbl 0466.15005
[15] Rothblum, U.G.; Tan, C.P., Multiplicative processes reaching stationarity in finite time, Linear algebra appl., 55, 155-167, (1983) · Zbl 0534.60063
[16] Seneta, E., Non-negative matrices, (1973), Allen & Unwin London · Zbl 0278.15011
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