Precise large deviation results for products of random matrices

Abstract

Abstract. The theorem of Furstenberg and Kesten provides a strong law of large numbers for the norm of a product of random matrices. This can be extended under various assumptions, covering nonnegative as well as invertible matrices, to a law of large numbers for the norm of a vector on which the matrices act. We prove corresponding precise large deviation results, generalizing the Bahadur-Rao theorem to this situation. Therefore, we obtain a third-order Edgeworth expansion for the cumulative distribution function of the vector norm. This result in turn relies on an application of the Nagaev-Guivarch method. Our result is then used to study matrix recursions, arising e.g. in financial time series, and to provide precise large deviation estimates there.

Precise large deviation asymptotics for products of random matrices

“…. The asymptotic (1.2) clearly implies a large deviation result due to Buraczewski and Mentemeier [8] which holds for invertible matrices and positive matrices: for q = Λ ′ (s) and s ∈ I • µ , there exist two constants 0 < c s < C s < +∞ such that c s lim inf n→∞ P(log |G n x| nq) 1 √ n e −nΛ * (q) lim sup n→∞ P(log |G n x| nq) 1 √ n e −nΛ * (q) C s . (1.5) Consider the Markov chain X x n := G n x/|G n x|.…”

“…As a special case of (1.6) with l = 0 and ψ compactly supported we obtain Theorem 3.3 of Guivarc'h [20]. With l = 0, ψ the indicator function of the interval [0, ∞) and ϕ = r s , we get the main result in [8].…”

“…Our proof is different from the standard approach of Dembo and Zeitouni [11] based on the Edgeworth expansion, which has been employed for instance in [8]. In contrast to [8], we start with the identity…”

“…For any s ∈ I µ , for invertible matrices and for positive matrices, the following limit exists (see [21] and [8]):…”

“…A simple sufficient condition established in [28] for the measure µ to be non-arithmetic is that the additive subgroup of R generated by the set {log λ g : g ∈ Γ µ , g ∈ G • + } is dense in R (see [8,Lemma 2.7]). Note that for positive matrices, condition A4 is used to ensure that σ 2 s = Λ ′′ (s) > 0.…”