# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.