×

Random difference equations and renewal theory for products of random matrices. (English) Zbl 0291.60029


MSC:

60H99 Stochastic analysis
60K05 Renewal theory
15B51 Stochastic matrices
60J35 Transition functions, generators and resolvents
62E20 Asymptotic distribution theory in statistics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bharucha-Reid, A. T., On the theory of random equations.Matscience report 31, Inst. of Math. Sciences, Madras, 1964. · Zbl 0142.13603
[2] Breiman, L.,Probability. Addison-Wesley Publ. Co., 1968.
[3] Cavalli-Sforza, L. & Feldman, M. W., Models for cultural inheritance, I. Group mean and within group variation.Theor. Population Biol. 4 (1973) plus a sequel, to appear later. · Zbl 0281.92022
[4] Dunford, N. & Schwartz, J. T.,Linear operators, Vol. I. Interscience Publishers, 1958. · Zbl 0084.10402
[5] Feller, W.,An introduction to probability theory and its applications, Vol. II,2nd ed. John Wiley & Sons, 1971. · Zbl 0219.60003
[6] Furstenberg, H. &Kesten, H., Products of random matrices.Ann. Math. Statist. 31 (1960), 457–469. · Zbl 0137.35501
[7] Gantmacher, F. R.,The theory of matrices. Chelsea Publ. Co., 1959. · Zbl 0085.01001
[8] Grenander, U.,Probabilities on algebraic structures, 2nd ed. Almqvist & Wiksell, 1968. · Zbl 0131.34804
[9] Halmos, P. R.,Lectures on ergodic theory. Chelsea Publ. Co., 1956. · Zbl 0073.09302
[10] Kaijser, T., Some limit theorems for Markov chains with applications to learning models and products of random matrices. Report Institut Mittag Leffler, 1972. · Zbl 0234.43001
[11] Kalman, R. E., Control of randomly varying linear dynamical systems.Proc. Symp. Appl. Math. 13 (1962), 287–298, Amer. Math. Soc., 1962. · Zbl 0107.35102
[12] Karlin, S., Positive operators.J. Math. Mech. 8 (1959), 907–937. · Zbl 0087.11002
[13] Kesten, H., Renewal theory for functionals of a Markov chain with general state space. To appear inAnn. Prob. · Zbl 0303.60090
[14] Kesten, H. &Stigum, B. P., Additional limit theorems for indecomposable multidimensional Galton-Watson processes.Ann. Math. Statist. 37 (1966), 1463–1481. · Zbl 0203.17402
[15] Konstantinov, V. M. &Nevelson, M. B., Stability of a linear difference system with random parameters.Mat. Zametki 8 (no. 6) (1970), 753–760; translated inMath. Notes 8 (1970), 895–899.
[16] Krein, M. G. &Rutman, M. A., Linear operators leaving invariant a cone in Banach space.Uspehi Mat. Nauk 3 (1948), no. 1 (23), 3–95,Amer. Math. Soc. Translations Series 1, Vol. 10, 199–325. · Zbl 0030.12902
[17] Meyer, P. A.,Probability and potentials. Blaisdell Publ. Co., 1966. · Zbl 0138.10401
[18] Parthasarathy, K. R.,Probability measures on metric spaces. Academic Press, 1967. · Zbl 0153.19101
[19] Paulson, A. S. &Uppuluri, V. R. R., Limit laws of a sequence determined by a random difference equation governing a one-compartment system.Math. Biosci. 13 (1972), 325–333. · Zbl 0232.60020
[20] Rosenblatt, M.,Markov processes, Structure and asymptotic behavior. Springer Verlag, 1971. · Zbl 0236.60002
[21] Rvačeva, E. L., On domains of attraction of multi-dimensional distributions.Lvov. Gos. Univ. Uc. Zap. 29,Ser. Meh. Mat. no. 6 (1954), 5–44;Selected Translations in Math. Statist. and Prob., 2 (1962), 183–205, Amer. Math. Soc.
[22] Solomon, F., Random walks in a random environment. To appear inAnn. Prob. · Zbl 0305.60029
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.