# Asymptotic expansions in central and local limit theorems for products of random matrices

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## Notation and preliminaries

We first fix some notation. We denote by c or C a constant whose value may change from line to line. For a set B, we use the symbols 1B, B, Bo and ∂B = B \ Bo to denote respectively its indicator function, its closure, interior and boundary. For t ∈ R, we write φ(t) = √2π e− y2/ 2, Φ(t) = −∞ φ(u)du, and φσ (t) = σ√2π e− t2/ (2 σ2 ). For a measure 1 t 1 function f weR ν measurable denote ν(f) = f dν. For two functions f and g, we and fa t ) = o g t )) or f t ) = O g t )) ( t R f t /g t ) = 0 or f t /g t ) is write ( ( ( ( ( ( → 0) when limt→0 ( ) ( ( ) ( bounded for |t| small enough, respectively. Denote by L1 the class of complexed valued measurable and Lebesgue integrable functions on R; for f ∈ L1, denote its L1 norm by kfkL1 = RR |f(x)|dx.
Let M(d, R) be equipped with the operator norm kak = supx∈Sd−1 |ax| for a ∈ M(d, R). Denote by µ := [supp µ] the smallest closed semigroup of M(d, R) generated by the support of µ. Let us recall some definitions in matrix theory. A matrix a is said to be proximal if it has an algebraic simple dominant eigenvalue. Denote by M+ the set of matrices with nonnegative entries. A matrix a ∈ M+ is said to be allowable if every row and every column has a positive entry.
For invertible matrices, we will use the strong irreducibility and proximality conditions. M1. (i) (Strong irreducibility) There is no finite union W = Sni=1 Wi of subspaces 0 6= Wi ( Rd which is µ-invariant (in the sense that µW = W). (ii) (Proximality) µ contains at least one proximal matrix. Notice that when d = 1, the strong irreducibility and proximality conditions are always satisfied. Limit theorems for branching random walks and products of random matrices Thi Thuy Bui 2020 57 3.2. Main results For nonnegative matrices, we will need the allowability, positivity and non-arithmeticity conditions. M2. (i) (Allowability) Every a ∈ µ is allowable. (ii) (Positivity) µ contains at least one matrix belonging to Mo+. We say that the measure µ is arithmetic if there are t > 0, θ ∈ [0, 2π) and a function ϑ : Sd+−1 → R such that for all a ∈ µ and all x ∈ V ( µ), exp{it log |ax| − iθ + i(ϑ(a · x) − ϑ(x))} = 1.
where Sd+−1 = {x ≥ 0 : |x| = 1} is the intersection of the unit sphere with the positive quadrant. Notice when d = 1, we have Sd+−1 = {1}, and the above arithmetic condition reduces to the following more usual form: log a is almost surely concentrated on an arithmetic progression a0 + a2N for some a0, a2 ≥ 0. M3. (Non-arithmeticity) The measure µ is non-arithmetic.

### Notation and assumptions on products of random matrices

Note that in our model, along each branch we encounter a product of random matrices. In this section, we introduce some notation and the necessary assumptions on products of random matrices in order to formulate our main results. We shall consider two cases, the case when the matrices are nonnegative and the case when the matrices are invertible.
Let M(d, R) be equipped with the operator norm: for any a ∈ M(d, R) we set kak = supx∈Sd−1 |ax|, where | · | is a given vectorial norm on Rd, and Sd−1 = {x ∈ Rd : |x| = 1} is the unit sphere in Rd. Denote by µ := [supp µ] the smallest closed semigroup of M(d, R) generated by the support of µ. A matrix a ∈ M(d, R) is said to be proximal if it has an algebraic simple dominant eigenvalue. Denote by M+ the set of matrices with nonnegative entries. A nonnegative matrix a ∈ M+ is said to be allowable if every row and every column has a strictly positive entry.
We say that the measure µ is arithmetic if there are t > 0, θ ∈ [0, 2π) and a function Limit theorems for branching random walks and products of random matrices Thi Thuy Bui 2020
83 4.2. Main results ϑ : Sd+−1 → R such that ∀a ∈ µ, ∀x ∈ supp ν : exp[it log |ax| − iθ + i(ϑ(a · x) − ϑ(x))] = 1, where Sd+−1 = {x ≥ 0 : |x| = 1} is the intersection of the unit sphere with the positive quadrant, and ν is the µ-invariant measure (cf. (4.5)). Notice when d = 1, we have Sd+−1 = {1}, and the above arithmetic condition reduces to the following more usual form: log a is a.s. concentrated on an arithmetic progression a0 + a1N for some a0, a1 > 0. We will need the following assumptions on the law µ. B1.
1. For invertible matrices:
(a) (Strong irreducibility)There is no finite union W = Sni=1 Wi of proper subspaces 0 6=Wi ( Rd which is µ-invariant (in the sense that aW = W for each a ∈ µ).
(b) (Proximality) µ contains at least one proximal matrix.
2. For nonnegative matrices:
(a) (Allowability) Every a ∈ µ is allowable.
(b) (Positivity) µ contains at least one matrix belonging to int(M+).
(c) (Non-arithmeticity) The measure µ is non-arithmetic.

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#### Proof of Theorem 4.2.4

In this section we will prove Theorem 4.2.4 , the precise large deviation asymptotic of Bahadur-Rao type on the counting measure Znx, using a uniform local limit theorem for products of random matrices that we recently established in .

Auxiliary results

In the proof of Theorem 4.2.4 we make use of the following three assertions. The first one is a local limit theorem for products of random matrices under the changed measure Qxs (see Proposition 4.5.1). The second is an exponential bound of the large deviation probability of the products of random matrices under Qxs (see Proposition 4.5.2). The Limit theorems for branching random walks and products of random matrices Thi Thuy Bui 2020 Chapter 4 – Central limit theorem and precise large deviations for branching random walks with products of random matrices 108 third gives a relationship between moment conditions on Ws,x1 and on Ws,x∗ := supn Ws,nx (see Proposition 4.5.3). We start with a uniform local limit theorem for products of random matrices under the changed measure Qxs. Under the initial measure (when s = 0), it has been established in .
Proposition 4.5.1. Under the conditions of Theorem 4.2.4, we have, for any continuous function f on S and any directly Riemann integrable function h on R, n→∞ (x,y) R s EQs [ ( n ) ( + n − Λ ( ))] lim sup σ √ f Xx h y Sx n 0 s n x ∈S× − πs(f) ZR h(z)φ z y dz = 0, (4.57) σs−√ n where φ(x) = √12π e−x2/2 is the density function of the standard normal law. Proof. For λ > 0 suﬃciently small, we introduce the Banach space Bλ = {f ∈ C(S) : kfkλ < +∞}, where kfkλ := kfk∞ with k f k∞ := sup | f(x) , | f |λ := x ∈S | + |f|λ.

1 Introduction
1.1 Context
1.2 Background and main objectives
1.2.1 The classical branching random walk
1.2.2 Branching random walks with products of random matrices
1.3 Main results
1.3.1 Berry-Esseen bound and Cramér moderate deviation expansion for a supercritical branching random walk
1.3.2 Asymptotic expansions in central and local limit theorems for products of random matrices
1.3.3 Central limit theorem and precise large deviations for branching random walks with products of random matrices
1.3.4 Berry-Esseen bound and Cramér moderate deviation expasion for a branching random walk with products of random matrices
2 Berry – Esseen bound and Cramér moderate deviation expansion for a supercritical branching random walk
2.1 Introduction
2.2 Notation and results
2.3 Proof of Theorems 2.2.1 and 2.2.3
2.4 Proof of Theorem 2.2.2
3 Asymptotic Expansions in central and local limit theorems for products of random matrices
3.1 Introduction
3.2 Main results
3.2.1 Notation and preliminaries
3.2.2 Main results
3.3 Spectral gap property
3.4 Proof of Theorem 3.2.1
3.5 Proof of Theorem 3.2.2
4 Central limit theorem and precise large deviations for branching random walks with products of random matrices
4.1 Introduction
4.2 Main results
4.2.1 Notation and assumptions on products of random matrices
4.2.2 Main results
4.3 Proof of Theorem 4.2.1
4.3.1 Basic decomposition
4.3.2 Proof of Theorem 4.2.1
4.4 Proof of Theorem 4.2.2 and Corollary 4.2.3
4.4.1 Proof of Theorem 4.2.2
4.4.2 Proof of Corollary 4.2.3
4.5 Proof of Theorem 4.2.4
4.5.1 Auxiliary results
4.5.2 Proof of Theorem 4.2.4
5 Berry-Esseen bound and precise moderate deviations for branching random walks with products of random matrices
5.1 Introduction
5.2 Main results
5.2.1 Notation and assumptions on products of random matrices
5.2.2 Main results
5.3 Preliminary results on products of random matrices
5.4 Associated martingales
5.5 Proof of Theorems 5.2.1 and 5.2.3
5.5.1 Proof of Theorem 5.5.1
5.5.2 Proof of Theorem 5.5.2
5.6 Proof of Theorem 5.2.2
Bibliography

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