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Conditioned limit theorems for products of random matrices

Abstract

Let \(g_{1},g_{2},\dots \) be i.i.d. random matrices in \(GL\left( d,\mathbb {R}\right) .\) For any \(n\ge 1\) consider the product \(G_{n}=g_{n} \dots g_{1}\) and the random process \(G_{n}v=g_{n}\dots g_{1}v\) in \(\mathbb {R}^{d}\) starting at point \(v\in \mathbb {R}^{d}{\backslash } \left\{ 0\right\} .\) It is well known that under appropriate assumptions, the sequence \(\left( \log \left\| G_{n}v\right\| \right) _{n\ge 1}\) behaves like a sum of i.i.d. r.v.’s and satisfies standard classical properties such as the law of large numbers, the law of iterated logarithm and the central limit theorem. For any vector v with \(\left\| v \right\| >1\) denote by \(\tau _v\) the first time when the random process \(G_{n}v\) enters the closed unit ball in \(\mathbb {R}^{d}.\) We establish the asymptotic as \(n\rightarrow +\infty \) of the probability of the event \(\left\{ \tau _{v}>n\right\} \) and find the limit law for the quantity \(\frac{1}{\sqrt{n}} \log \left\| G_{n}v\right\| \) conditioned that \(\tau _{v}>n\).

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Correspondence to Ion Grama.

Appendix

Appendix

We prove the existence of a measure \(\varvec{\mu }\) satisfying conditions P1P5. Let \(\varvec{\mu }_0\) be a probability measure on \(\mathbb {G}\) satisfying conditions P1P4 which admits \(\varvec{\nu }\) as invariant measure and whose upper Lyapunov exponent is 0. Let \(\lambda >1.\) Define the measure

$$\begin{aligned} \varvec{\mu }_{\lambda }\left( dg^{\prime }\right) =\alpha \varvec{ \delta }_{\lambda I}\left( dg^{\prime }\right) +\left( 1-\alpha \right) \varvec{\mu }_0\left( \frac{1}{\lambda }dg^{\prime }\right) , \end{aligned}$$

where \(\alpha \in \left( 0,1\right) .\) Then \(\varvec{\mu }_{\lambda }\) satisfies conditions P1P3 and \(\varvec{\mu }_{\lambda }*\varvec{\nu }=\varvec{\nu }\) , i.e. \(\varvec{\nu }\) is \( \varvec{\mu }_{\lambda }\)-invariant measure. Moreover, the upper Lyapunov exponent of \(\varvec{\mu }_{\lambda }\) is 0,  i.e.

$$\begin{aligned} \int _{\mathbb {G}}\int _{\mathbb {P}\left( \mathbb {V}\right) }\rho \left( g, \overline{v}\right) \varvec{\mu }_{\lambda }\left( dg\right) \varvec{ \nu }\left( dv\right) =0 \end{aligned}$$

and

$$\begin{aligned} \inf _{v\in \mathbb {S}^{d-1}}\varvec{\mu }_{\lambda }\left( g:\log \left\| gv\right\|>\ln \lambda \right) \ge \alpha >0 \end{aligned}$$

which means that conditions P4 and P5 are satisfied.

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Grama, I., Le Page, É. & Peigné, M. Conditioned limit theorems for products of random matrices. Probab. Theory Relat. Fields 168, 601–639 (2017). https://doi.org/10.1007/s00440-016-0719-z

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  • DOI: https://doi.org/10.1007/s00440-016-0719-z

Keywords

  • Exit time
  • Markov chains
  • Random matrices
  • Spectral gap

Mathematics Subject Classification

  • 60B20
  • 60J05
  • 60J45