Abstract
For products,A(t)·A(t−1)...A(1), of i.i.d.N×N random matrices, with i.i.d. entries, a triangle law governs theN→∞ distribution of Lyapunov exponents, much like Wigner's quarter-circle law governs the singular values ofA(1). Our proof requires finite fourth moments and a bounded density; the result was previously derived only in the Gaussian case.
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References
[B] Ball, K.: Cube slicing inR n. Proc. Am. Math. Soc.97, 465–473 (1986)
[BL] Bourgerol, P., Lacroix, J.: Products of random matrices with applications to Schrödinger operators. Boston: Birkhäuser 1985
[BYK] Bai, Z.D., Yin, Y.Q., Krishnaiah, P.R.: On the limiting empirical distribution function of the eigenvalues of a multivariateF matrix. Theory Prob. Appl.32, 490–500 (1987)
[CN] Cohen, J.E., Newman, C.M.: The stability of large random matrices and their products. Ann. Prob.12, 283–310 (1984)
[F] Furstenberg, H.: Noncommuting random products. Trans. Am. Math. Soc.108, 377–428 (1963)
[FK] Furstenberg, H., Kifer, Y.: Random matrix products and measures on projective spaces. Ist. J. Math.46, 12–32 (1983)
[GR] Guivarc'h, Y., Raugi, A.: Frontière de Fürstenberg, propriétés de contraction et theorèmes de convergence. Zeit. Wahrscheinlichkeitstheorie Verw. Gebiete69, 187–242 (1985)
[H] Hensley, D.: Slicing the cube inR n and probability (bounds for the measure of a central cube slice inR n by probability methods). Proc. Am. Math. Soc.73, 95–100 (1979)
[MP] Marčenko, V.A., Pastur, L.A.: Distribution of eigenvalues for some sets of random matrices. Math. USSR-Sb.1, 457–483 (1967)
[N1] Newman, C.M.: The distribution of Lyapunov exponents: exact results for random matrices. Commun. Math. Phys.103, 121–126 (1986)
[N2] Newman, C.M.: Lyapunov exponents for some products of random matrices: exact expressions and asymptotic distributions. In: Random matrices and their applications, Cohen, J.E., Kesten, H., Newman, C.M. (eds.). Providence, RI: AMS 1986
[O] Oseledec, V.I.: A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems. Trans. Mosc. Math. Soc.19, 197–231 (1968)
[R] Raghunathan, M.S.: A proof of Oseledec's multiplicative ergodic theorm. Isr. J. Math.32, 356–362 (1979)
[Ro] Rogozin, B.A.: Estimation of the maximum of a convolution of bounded densities. Theory Prob. Appl.32, 48–56 (1987)
[W] Wigner, E.P.: Random matrices in physics. SIAM Rev.9, 1–23 (1967)
[Wi] Wilks, S.S.: Mathematical statistics. New York: Wiley 1962
[Y] Yin, Y.Q.: Limiting spectral distribution for a class of random matrices. J. Multivar. Anal.20, 50–68 (1986)
[YBK] Yin, Y.Q., Bai, Z.D., Krishnaiah, P.R.: On the limit of the largest eigenvalue of the large dimensional sample covariance matrix. Probab. Theory Relat. Fields78, 509–521 (1988)
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Communicated by T. Spencer
Research supported in part by NSF Grants DMS-8902156 and DMS-9196086
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Isopi, M., Newman, C.M. The triangle law for Lyapunov exponents of large random matrices. Commun.Math. Phys. 143, 591–598 (1992). https://doi.org/10.1007/BF02099267
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DOI: https://doi.org/10.1007/BF02099267