Random matrix-valued multiplicative functions and linear recurrences in Hilbert-Schmidt norms of random matrices

TL;DR

This paper introduces random matrix-valued multiplicative functions, extending classical multiplicative functions to matrices, and analyzes their moments using linear recurrences and spectral radius bounds.

Contribution

It generalizes Rademacher multiplicative functions to matrices and derives asymptotic second moments and bounds for higher moments based on spectral properties.

Findings

Asymptotic second moment formula for matrix-valued functions

Upper bounds for higher even moments

Linear recurrence relations for Hilbert-Schmidt norms

Abstract

We introduce the notion of a random matrix-valued multiplicative function, generalizing Rademacher random multiplicative functions to matrices. We provide an asymptotic for the second moment based on a linear recurrence property for Hilbert-Schmidt norms of sucessive products of random matrices. Moreover, we provide upper bounds for the higher even moments related to the generalized joint spectral radius.

Equations

$$\mathbb{E}\bigg{[}\Big{(}\sum_{n\leq x}f(n)\Big{)}^{2}\bigg{]}=\frac{6}{\pi^{2}}x+O(\sqrt{x})$$

$$\mathbb{E}\bigg{[}\Big{(}\sum_{n\leq x}f(n)\Big{)}^{k}\bigg{]}\sim C_{k}x^{k/2}(\log x)^{\binom{k}{2}-k}.$$

$$\mathbb{E}\Big{[}\big{\|}\sum_{n\leq x}f(n)\big{\|}_{HS}^{2k}\Big{]},$$

$$\mathbb{E}\left[\Vert X{\Vert }_{HS}^2\text{mathbbm}1(\Vert X{\Vert }_{HS}^2>R)\right]\stackrel{R\to \infty }{\to }0,$$

$$T:{\mathbb{C}}^{d\times d}$$

$$A$$

$$\mathbb{E}\Big{[}\big{\|}\sum_{n\leq x}f(n)\big{\|}_{HS}^{2}\Big{]}=x\sum_{m=1}^{N}\sum_{i=1}^{l}C_{i,m}(\log x)^{\lambda_{i}-m}+O\left(x(\log x)^{\lambda_{1}-N-1}\right)$$

$$\mathbb{E}\left[\Vert X{\Vert }_{HS}^{2k}\text{mathbbm}1(\Vert X{\Vert }_{HS}^{2k}>R)\right]\to 0$$

$$a_n:=a_n^{(2k)}:=\mathbb{E}\left[\Vert X_1\cdots X_n{\Vert }_{HS}^{2k}\right].$$

$$l^{\mathbb{C}}:=\left(\genfrac{}{}{0ex}{}{k+d^2-1}{k}\right).$$

$$l^{\mathbb{R}}:=\left(\genfrac{}{}{0ex}{}{k+\left(\genfrac{}{}{0ex}{}{d+1}{2}\right)-1}{k}\right).$$

$$p(x)=x^l+c_1{x}^{l-1}+\cdots +c_l=(x-{\lambda }_1{)}^{m_1}\cdots (x-{\lambda }_t{)}^{m_t}$$

$$a_{n+l}+c_1{a}_{n+l-1}+\cdots +c_l{a}_n=0$$

$$a_n=g_1(n){\lambda }_1^n+\cdots +g_t(n){\lambda }_t^n.$$

$$V^{\mathbb{C}}:={\mathrm{Sym}}^k({\mathbb{C}}^{d\times d})\text{and}{V}^{\mathbb{R}}:={\mathrm{Sym}}^k(S_d).$$

$$T^{\mathbb{C}}:V^{\mathbb{C}}$$

$$v$$

$$p_T(x)=x^l+c_1{x}^{l-1}+\cdots +c_l$$

$$(A_1\otimes A_2)(A_3\otimes A_4)=A_1{A}_3\otimes A_2{A}_4$$

$$a_n=\mathbb{E}\left[\mathrm{Tr}(X_n^{\ast }\cdots X_1^{\ast }{X}_1\cdots X_n{)}^k\right]$$

$$=\mathrm{Tr}\left(\mathbb{E}\left[(X_n^{\ast }{)}^{\otimes k}\cdots (X_1^{\ast }{)}^{\otimes k}{X}_1^{\otimes k}\cdots X_n^{\otimes k}\right]\right).$$

$$a_{n+l}+c_1{a}_{n+l-1}+\cdots +c_l{a}_n=0.$$

$$T{v}_i={\lambda }_i{v}_i\text{ and }I=\sum _{i=1}^l{\alpha }_i{v}_i.$$

$$a_n$$

$$=\sum _{i=1}^l{\alpha }_i{\lambda }_i\mathrm{Tr}\left(\mathbb{E}\left[(X_n^{\ast }{)}^{\otimes k}\cdots (X_2^{\ast }{)}^{\otimes k}{v}_i{X}_2^{\otimes k}\cdots X_n^{\otimes k}\right]\right).$$

$$a_n=\sum _{i=1}^l{\alpha }_i\mathrm{Tr}(v_i){\lambda }_i^n,$$

$$\mu :=\sum _{i=1}^m{p}_i{\delta }_{B_i},$$

$$M:=\{(B_1,\dots ,B_m)\in ({\mathbb{K}}^{d\times d}{)}^m:T\text{ diagonalizable}\}\subseteq {\mathbb{R}}^{\beta m{d}^2}$$

$$\tau :{\mathbb{R}}^{\beta m{d}^2}$$

$$(B_1,\dots ,B_m)$$

$${\mathbb{R}}^{\beta m{d}^2}\to \mathrm{End}(V)\times {\mathbb{R}}^{l+1}\to {\mathbb{C}}^l\times {\mathbb{R}}^{l+1}\to \mathbb{C}$$

$$(B_1,\dots ,B_m)$$

$$(T,(b_l,\dots ,b_0))$$

Full text

The author was partially supported by DFG-SNF lead agency program grant 200020L_175755

1. Introduction

A Rademacher random multiplicative function is a family $(f(n))_{n\in\mathbb{N}}$ (with the convention $0\not\in\mathbb{N}$) of random variables taking values in $\{\pm 1,0\}$ such that

• •

$n\mapsto f(n)$ is supported on squarefree integers,

• •

$(f(p))_{p\text{ prime}}$ are independent, each taking the values $\pm 1$ with probability $\frac{1}{2}$ and

• •

when $n=p_{1}\cdots p_{r}$ is squarefree then we have $f(n)=f(p_{1})\cdots f(p_{r})$.

Moments of these functions have been studied in a great amount of detail. It is a classical fact that

[TABLE]

and it was proven by Harper, Nikeghbali and Radziwiłł in [HarperNR1, Theorem 4] and independently by Heap and Lindqvist [HeapLindqvist1, Theorem 4] in the even case that for all integers $k\geq 3$ there exists a constant $C_{k}>0$ such that

[TABLE]

In this work we will consider the following matrix-valued generalisation of Rademacher multiplicative functions.

Definition 1**.**

Let $d\geq 1$ be an integer. A random matrix-valued multiplicative function is a family $(f(n))_{n\in\mathbb{N}}$ of random variables taking values in $\mathbb{C}^{d\times d}$ such that

• •

$n\mapsto f(n)$ is supported on squarefree integers,

• •

$(f(p))_{p\text{ prime}}$ are independent identically distributed (i.i.d.) and

• •

when $n=p_{1}\cdots p_{r}$ is squarefree with $p_{1}<\dots<p_{r}$ then we have $f(n)=f(p_{1})\cdots f(p_{r})$.

Our goal is to obtain estimates for the even moments

[TABLE]

where $\|\cdot\|_{HS}$ denotes the Hilbert-Schmidt norm defined by $\|A\|_{HS}^{2}=\mathrm{Tr}(A^{*}A)$ for $A\in\mathbb{C}^{d\times d}$.

In section 3 we will prove the following estimate for the second moment based on a linear recurrence property of the Hilbert-Schmidt norm, which will be the subject of section 2.

Theorem 1**.**

Let $d\geq 1$ be an integer, let $X$ be a $\mathbb{C}^{d\times d}$-valued random variable and let $f$ be the associated matrix-valued multiplicative function. Suppose that $\mathbb{E}X=0$ and

[TABLE]

where $\mathbbm{1}(E)$ denotes the characteristic function of an event $E$. Define

[TABLE]

let $l:=d^{2}$ and assume that $T$ is diagonalizable with eigenvalues $\lambda_{1},\dots,\lambda_{l}$ arranged in descending order according to their real parts. Then for any $N\in\mathbb{N}$ there are constants $C_{i,m},\;m=1,\dots,N,\,i=1,\dots,l$ such that

[TABLE]

holds for all $x\geq 2$.

Our argument also extends to the case when $T$ is not diagonalizable, even though our estimate becomes less precise in this case. The exact statement without the assumption of diagonalizability will be given and proven in section 3.

Section 4 will be devoted to proving an upper bound for higher moments that will be related to what is known as the generalized joint spectral radius.

Acknowledgements**.**

The author would like to thank Emmanuel Kowalski and Jori Merikoski for helpful discussions and comments on earlier drafts of this paper.

2. A Linear Recurrence for Hilbert-Schmidt Norms

The goal of this section is to prove the following result which may be of independent interest.

Theorem 2**.**

Let $d,k\geq 1$ be fixed integers. Suppose that $X,X_{1},X_{2},\dots$ is a sequence of i.i.d. $\mathbb{C}^{d\times d}$-valued random variables such that

[TABLE]

as $R\to\infty$, and define

[TABLE]

Then the sequence $(a_{n})_{n}$ satisfies a linear recurrence of length

[TABLE]

If the random variables are in fact $\mathbb{R}^{d\times d}$-valued, then the sequence $(a_{n})_{n}$ satisfies a linear recurrence of length

[TABLE]

Before proving this Theorem, we first recall the following standard fact about linear recurrences.

Lemma 3**.**

Let $(a_{n})_{n}$ be a sequence of complex numbers, and let

[TABLE]

be a polynomial such that

[TABLE]

holds for all $n$, where $\lambda_{1},\dots,\lambda_{t}$ are distinct complex numbers. Then there exist unique polynomials $g_{i}$ of degrees $<m_{i}$ for $i=1,\dots,t$ such that

[TABLE]

Proof of Theorem 2.

Let $\mu$ be the law of $X$. Let $S_{d}$ denote the space of complex-symmetric $d\times d$ matrices and let

[TABLE]

Note that $V^{\mathbb{C}}$ resp. $V^{\mathbb{R}}$ is a complex vector space of dimension $l^{\mathbb{C}}$ resp. $l^{\mathbb{R}}$.

Further, define

[TABLE]

and let $T^{\mathbb{R}}$ be its restriction to $V^{\mathbb{R}}$ whenever $X$ is real-valued.

In the following, we will shorten notation by writing $V,\,T$ and $l$ in place of the corresponding real and complex objects whenever a statement holds in both cases. We will adopt this convention for objects defined later on. Moreover, we will write $\mathbb{K}$ as a placeholder for $\mathbb{R}$ and $\mathbb{C}$.

Finally, we denote by

[TABLE]

the characteristic polynomial of $T$.

Part 1: $\mu$ has finite support.

Inductively applying the mixed-product identity of the Kronecker product

[TABLE]

for $A_{1},A_{2},A_{3},A_{4}\in\mathbb{C}^{d\times d}$ implies that

[TABLE]

We claim that the sequence $(a_{n})_{n}$ satisfies the recurrence defined by the characteristic polynomial of $T$, i.e. for all $n\in\mathbb{N}$ we have

[TABLE]

In order to see this, assume first that $T$ is diagonalizable. Then we can write the identity $I:=I_{d}^{\otimes k}\in V$ as a linear combination of eigenvectors of $T$, i.e. there are $\lambda_{1},\dots,\lambda_{l},\,\alpha_{1},\dots,\alpha_{l}\in\mathbb{C}$ and non-zero $v_{1},\dots,v_{l}\in V$ such that

[TABLE]

This implies that

[TABLE]

Inductively, we obtain

[TABLE]

so that the sequence $(a_{n})_{n}$ indeed satisfies the characteristic polynomial of $T$ under the assumption that this operator is diagonalizable.

Now fix $m$ and weights $p_{1},\dots,p_{m}>0$ with $\sum p_{i}=1$. Given $B_{1},\dots,B_{m}\in\mathbb{K}^{d\times d}$, define the (finitely supported) probability measure

[TABLE]

where $\delta_{B}$ denotes the Dirac measure at $B\in\mathbb{K}^{d\times d}$. Set $\beta=1$ resp. $2$ when $X$ is real- resp. complex-valued. Endowing $(\mathbb{K}^{d\times d})^{m}=\mathbb{R}^{\beta md^{2}}$ with the Zariski topology, we claim that the set

[TABLE]

is dense. 111We do not claim that this is a dense condition in $\mathbb{C}^{md^{2}}$ in the complex case, but only in $\mathbb{R}^{2md^{2}}$. First, note that this set is non-empty: Choose $B_{1}=\dots=B_{m}=I$ all to be the identity matrix. Then $T$ is the identity on $V$, hence diagonalizable.

The next step is to prove that $M$ is Zariski-open. But this follows from the fact that the map

[TABLE]

is polynomial in the entries of the $B_{i}$ and diagonalizability of $T$ is an open condition on the right-hand side. Hence, $M$ is indeed a dense set. Note that $\tau$ is not polynomial on $\mathbb{C}^{dm^{2}}$ in the complex case.

Lastly, consider for fixed $n$ the composition of maps

[TABLE]

given by

[TABLE]

where in the second map we send an operator to its characteristic polynomial viewed as a vector in its coefficients. It is clear that each of these maps is continuous, and we know that their composition

[TABLE]

vanishes on the dense set $M$, hence everywhere, which settles Part 1.

Part 2: $\mu$ has compact support.

Our goal is to show that the equation

[TABLE]

still holds for all $n$. Let $K=\text{supp }\mu$ and let $(\mu_{m})_{m}$ be a sequence of probability measures with finite support contained in $K$ such that $\mu_{m}\to\mu$ weakly, i.e. for all continuous bounded functions $f:\mathbb{K}^{d\times d}\to\mathbb{R}$ we have

[TABLE]

as $m\to\infty$. Let $(X_{n,m})_{n}$ be i.i.d. sequences of random variables distributed according to $\mu_{m}$, let

[TABLE]

and similarly define $T_{m}$ and $c_{i,m}$ w.r.t. $\mu_{m}$. Since the measures $\mu_{m}$ have finite support, we know that

[TABLE]

holds for all $n,m$. It thus suffices to show that for any fixed $n$ and $i$ we have $a_{n,m}\to a_{n}$ and $c_{i,m}\to c_{i}$ as $m\to\infty$.

It is a standard fact that the weak convergence of $(\mu_{m})_{m}$ implies the weak convergence of the product measures $\mu_{m}^{\otimes n}\to\mu^{\otimes n}$. Moreover, we have

[TABLE]

Letting $f:(\mathbb{K}^{d\times d})^{n}\to\mathbb{R}$ be a bounded continuous function which coincides with $(A_{1},\dots,A_{n})\mapsto\|A_{1}\cdots A_{n}\|_{HS}^{2k}$ on $K^{\times n}$ implies the convergence $a_{n,m}\to a_{n}$ as $m\to\infty$ for any fixed $n$.

To show the convergence of $c_{i,m}$, note that it suffices to show that each entry of $T_{m}$ in some fixed basis converges to the corresponding entry of $T$. But for this, in turn, it suffices in both the real and the complex case to show the same property for the extended operator

[TABLE]

and the corresponding operators $\tilde{T}_{m}$, since $T$ and $T_{m}$ are just restrictions of these operators to a common invariant subspace. Let us take the standard basis given by $e_{i_{1}j_{1}}\otimes\dots\otimes e_{i_{k}j_{k}}$ with $i_{1},j_{1},\dots,i_{k},j_{k}\in\{1,\dots,d\}$, where $e_{ij}\in\mathbb{C}^{d\times d}$ is the matrix with entry $ij$ being $=1$ and the rest $=0$. One verifies that

[TABLE]

where in the complex case the integral is taken over real and imaginary part separately; analogous statements hold for $\tilde{T}_{m}$. Again taking bounded continuous functions $f:\mathbb{K}^{d\times d}\to\mathbb{R}$ which coincide with real resp. imaginary part of $A\mapsto\overline{A_{i_{i}i_{1}^{\prime}}}A_{j_{1}j_{1}^{\prime}}\cdots\overline{A_{i_{k}i_{k}^{\prime}}}A_{j_{k}j_{k}^{\prime}}$ on $K$ implies the claim.

Part 3: The general case.

Let $R>0$ be sufficiently large so that $\mu(B_{R}(0))>0$. Define the (conditional) probability measure

[TABLE]

Denoting by $(X_{n}^{R})_{n}$ a family of i.i.d. random variables corresponding to $\mu_{c}^{R}$, we can set

[TABLE]

and similarly $T^{R}$ and $c_{i}^{R}$. Since $\mu_{c}^{R}$ has compact support, we know that

[TABLE]

holds for all $n$. It thus suffices to show that for any fixed $n$ and $i$ we have $a_{n}^{R}\to a_{n}$ and $c_{i}^{R}\to c_{i}$ as $R\to\infty$.

We have

[TABLE]

Since $\mu(B_{R}(0))\to 1$ as $R\to\infty$, it suffices to show that this integral converges to $a_{n}$ as $R\to\infty$. But

[TABLE]

by assumption.

For the convergence of $c_{i}^{R}$ to $c_{i}$, it again suffices to show that every entry of $T^{R}$ converges to the corresponding entry of $T$ in some fixed basis. Again, it suffices to show this for the extended operators $\tilde{T}$ and $\tilde{T}^{R}$ defined in the obvious way. But we have

[TABLE]

as $R\to\infty$, hence the claim. ∎

Remark**.**

The idea of reducing matrix dimensions by looking at symmetric algebras in a similar context of Theorem 2 has been considered in [BlondelN1, ParriloJ1] related to Kronecker and semidefinite lifting.

One might be interested in the optimality of $l$. In the real case, we can in fact prove that $l^{\mathbb{R}}$ is optimal in the sense that for all $d,k\geq 1$ there exists $X$ such that the sequence $(a_{n})_{n}$ does not satisfy a linear recurrence of any shorter length.

In this case, it in fact suffices to take $X$ to be a deterministic distribution supported in a single point $A$. Let $\lambda_{1},\dots,\lambda_{d}$ be the eigenvalues of $A$ and assume for simplicity that they are algebraically independent. One verifies that the eigenvalues of

[TABLE]

are given by $\lambda_{i}\lambda_{j}$ for $1\leq i\leq j\leq d$, which we will denote by $\mu_{1},\dots,\mu_{d^{\prime}}$ with $d^{\prime}=\binom{d+1}{2}$. Moreover, it is elementary to see that the eigenvalues of $T$ are then given by $\mu_{i_{1}}\cdots\mu_{i_{k}}$ for $1\leq i_{1}\leq\dots\leq i_{k}\leq d^{\prime}$, and that they are pairwise distinct. A generic choice of $A$ will satisfy $\alpha_{i}\mathrm{Tr}(v_{i})\neq 0$ for all $i$, which then implies the claim.

In the complex case, taking a deterministic $X$ and doing the same construction as in the real case gives an operator $T$ which can have at most

[TABLE]

distinct eigenvalues. The above argument does prove that there are $X$ such that $(a_{n})_{n}$ satisfies a linear recurrence of no shorter length than ${\tilde{l}}^{\mathbb{C}}$, but it does not give optimality of $l^{\mathbb{C}}$. For this, one would need to take a more complicated $X$, for which it is significantly more difficult to explicitly compute the eigenvalues of $T$. Nonetheless, numerical evidence in this case does suggest that $l^{\mathbb{C}}$ might still be optimal.

The eigenvalue of $T$ of largest real part is essentially the generalized joint spectral radius of $X$. More details on this can be found in section 4.

For further reference, we would like to record the following

Corollary 4**.**

Let $X,X_{1},X_{2},\dots$ be i.i.d. $\mathbb{C}^{d\times d}$-valued random variables with

[TABLE]

and define

[TABLE]

Let

[TABLE]

and denote by

[TABLE]

the characteristic polynomial of $T$, where $l:=d^{2}$ is the dimension of $\mathbb{C}^{d\times d}$. Then for any $n\in\mathbb{N}$, we have

[TABLE]

3. Second-Moment Estimate for Random Matrix-Valued Multiplicative Functions

We will fix the following notation: We set

[TABLE]

and

[TABLE]

as well as $P(z):=P(1,z)$ and $F(z):=F(1,z)$.

3.1. The Diagonalizable Case

Using Theorem 2, we are now in a position to prove Theorem 1.

Proof of Theorem 1.

We have

[TABLE]

By definition of $f$, the contribution of squarefree $n$ to this sum depends only on $\omega(n)$ and is given by

[TABLE]

for i.i.d. random variables $X,X_{1},\dots,X_{\omega(n)}$, where $a_{\omega(n)}$ is defined as in Corollary 4. But this implies

[TABLE]

It thus remains to prove the following

Proposition 5**.**

For any $N\in\mathbb{N}$ and $z\in\mathbb{C}$ there are explicit constants $C_{1},\dots,C_{N}$ (depending on $z$) such that

[TABLE]

For example, we have

[TABLE]

where $P_{s}(z)$ denotes the derivative of $P(s,z)$ w.r.t. $s$ evaluated at $s=1$, and where $\gamma$ is the Euler-Mascheroni constant. In fact, the error terms are uniform over $|z|<A$.

This follows from [Dixon1, Theorem, p. 188] by setting

[TABLE]

in the notation there, so that

[TABLE]

which gives

[TABLE]

Taylor expansion of $(s-1)\zeta(s)$ around $s=1$ and application of the Binomial Theorem quickly yields, for example,

[TABLE]

This quickly gives the second part of the assertion.

Since we can compute an arbitrary number of terms in this Taylor series and also the one for $P$ around $s=1$, this gives the first claim by [Dixon1, Theorem, p. 188]. This concludes the proof of Theorem 1.

∎

Remark**.**

The proof shows that the constants $C_{i,m}$ in Theorem 1 are explicit. Let $v_{1},\dots,v_{l}\in V$ be the eigenvectors of $T$ associated to $\lambda_{1},\dots,\lambda_{l}$, and let $\alpha_{1},\dots,\alpha_{l}\in\mathbb{C}$ be such that $I=\sum\alpha_{i}v_{i}$. Then, for example, we have

[TABLE]

and

[TABLE]

By the methods outlined in the proof of Proposition 5 one can compute arbitrarily many such constants.

We also remark that if $X$ is real-valued then by Theorem 2 we can restrict $T$ to $S_{d}$ and set $l=\binom{d+1}{2}$.

Example 6**.**

Let $X$ be the uniform distribution on the set

[TABLE]

and let $f$ be the associated matrix-valued multiplicative function. Then we have $\mathrm{Im}f=\mathrm{SL}_{2}(\mathbb{Z})\cup\{0\}$ almost surely. If

[TABLE]

then it is verified by evaluating at $\begin{pmatrix}1&0\\ 0&0\end{pmatrix},\,\begin{pmatrix}0&1\\ 1&0\end{pmatrix}$ and $\begin{pmatrix}0&0\\ 0&1\end{pmatrix}$ that $T$ can be represented by the matrix

[TABLE]

with eigenvalues

[TABLE]

and eigenvectors

[TABLE]

Moreover, we can write the identity matrix as $I_{2}=\sum_{i=1}^{3}\alpha_{i}v_{i}$ with

[TABLE]

We obtain

[TABLE]

as well as

[TABLE]

We infer

[TABLE]

3.2. The Non-Diagonalizable Case

If $T$ is not diagonalizable, it turns out that we need to find an estimate for a more difficult quantity, and we are only able to prove an ineffective asymptotic. More precisely, we need the following

Proposition 7**.**

For fixed $z\in\mathbb{C}\setminus\mathbb{Z}^{-}$ and $r\in\mathbb{N}_{0}$, we have

[TABLE]

Proof.

The idea is to find an asymptotic as $s\to 1$ for the associated Dirichlet series and then to apply Delange’s Theorem [Delange1, Théorème IV], compare also [Tenenbaum1, Theorem 7.28].

A crucial point in proving this is that

[TABLE]

Expanding the falling factorials using the Stirling numbers of the second kind, denoted by curly brackets, we obtain

[TABLE]

But we have

[TABLE]

Here, $F_{z}$, denotes the derivative of $F$ in the second component. Inductively we obtain expansions of the form

[TABLE]

where the other terms involve lower powers of $\log\zeta(s)$ as well as derivatives of $F$. We thus obtain an expansion of the form

[TABLE]

where again the other terms involve lower (non-negative, integral) powers of $\log\zeta(s)$ multiplied by functions of $s$ and $z$ which are holomorphic around $s=1$ for any $z$. We are thus in a position to apply Delange’s Theorem, which indeed implies that

[TABLE]

when $z\in\mathbb{C}\setminus\mathbb{Z}^{-}$, as claimed. ∎

This Proposition allows us to prove the following

Theorem 8**.**

Let $d\geq 1$ be an integer, let $X$ be a $\mathbb{C}^{d\times d}$-valued random variable and let $f$ be the associated matrix-valued multiplicative function. Suppose that $\mathbb{E}X=0$ and

[TABLE]

Define

[TABLE]

and let $\lambda_{1},\dots,\lambda_{t}$ be the (distinct) eigenvalues of $T$ arranged in descending order according to their real parts. Let $p_{T}$ be the characteristic polynomial of $T$ and let $c_{1},\dots,c_{l}$ and $m_{1},\dots,m_{t}$ be such that

[TABLE]

Further, define

[TABLE]

where $X_{1},X_{2},\dots$ are i.i.d. copies of $X$, and let $g_{1},\dots,g_{t}$ be the polynomials satisfying $d_{i}:=\deg g_{i}<m_{i}$ and

[TABLE]

(see Theorem 2 and Lemma 3). Let $R$ be the maximal real part among those $\lambda_{i}$ with $d_{i}>0$. Define $L_{1},L_{2}$ and $L_{3}$ to be the collection of $i$ such that $\Re\lambda_{i}>R,\Re\lambda_{i}=R$ and $\Re\lambda_{i}<R$, respectively. Lastly, let $d_{max}=\max_{i\in L_{2}}d_{i}$ and $L_{2}^{\prime}=\{i\in L_{2}:d_{i}=d_{max}\}$. Then for any $N\in\mathbb{N}$ there are explicit constants $C_{i,m}$ for $i\in L_{1},\,m=1,\dots,N$ and $C_{j}^{\prime}$ for $j\in L_{2}^{\prime}$ such that

[TABLE]

Proof.

Writing $b_{i}$ for the leading coefficient of $g_{i}$, the same argument as in Theorem 1 implies

[TABLE]

Regarding the first summand, Proposition 5 directly tells us that for any $N\in\mathbb{N}$ there are explicit constants $C_{i,m}$ such that

[TABLE]

Using Proposition 7 on the second summand directly implies

[TABLE]

for some explicit constants $C_{j}^{\prime}$.

Proposition 7 furthermore implies

[TABLE]

For the last error term, fix $\varepsilon>0$ such that $\lambda_{i}+\varepsilon<R$ for all $i\in L_{3}$. Then

[TABLE]

and the claim follows. ∎

4. An Upper Bound for Higher Even Moments

Let $s\geq 1$, and let $X,X_{1},X_{2},\dots$ be i.i.d. $\mathbb{C}^{d\times d}$-valued random variables with

[TABLE]

Then

[TABLE]

will be called the spectral $s$-radius of $X$. If $S\subset\mathbb{C}^{d\times d}$ is bounded then

[TABLE]

is called the joint spectral radius of $S$. Note that all these quantities are in fact independent of the chosen norm, since all norms on $\mathbb{C}^{d\times d}$ are equivalent.

The joint spectral radius has been studied in great detail in contexts such as dynamical systems, wavelets, optimization and control. We refer the interested reader to [Jungers1]. For the generalized joint spectral radius, its geometric interpretation and relation to Kronecker products, see e.g. [Protasov1, Protasov2].

We note at this point that by Hölder’s inequality, $\rho_{s}$ is monotonically increasing and if $X$ is the uniform distribution on a bounded set $S$ then we have $\rho_{s}\uparrow\rho_{\infty}$ as $s\to\infty$. Also, note that $\rho_{2k}=\lambda_{1}^{1/2k}$, where $\lambda_{1}\geq 0$ is as in Theorem 8.

The goal of this section is to prove the following

Theorem 9**.**

Let $k\geq 2$ be an integer, and let $X$ be a symmetric $\mathbb{C}^{d\times d}$-valued random variable satisfying $\mathbb{E}X=0$ and

[TABLE]

as $R\to\infty$. Let $f$ be the random matrix-valued multiplicative function associated to $X$. Then we have

[TABLE]

where $[\,\cdot\,]$ denotes the integral part.

Proof.

Denoting by $\Box$ a generic square, we have

[TABLE]

But from the definition of $\rho_{2k}$, we see that

[TABLE]

and in particular

[TABLE]

We thus obtain

[TABLE]

It thus remains to prove that

[TABLE]

for all $m\in\mathbb{N}$. We proceed similar to the proof of [HarperNR1, Theorem 4]. To this end, let $g$ be the multiplicative function supported on squarefree integers such that $g(n_{1},\dots,n_{2k})=m^{(\omega(n_{1})+\dots+\omega(n_{2k}))/2}$ when $n_{1}\cdots n_{2k}$ is a square, and [math] otherwise. Then the associated multiple Dirichlet series

[TABLE]

has the Euler product representation

[TABLE]

This factors as

[TABLE]

with $H$ being holomorphic strictly to the left of $s=(\frac{1}{2},\dots,\frac{1}{2})$. The claim follows from [DeLaBreteche1, Theorem 2], choosing each of the linear forms $s_{i}+s_{j}$ for $1\leq i<j\leq 2k$ precisely $m$ times, so that they are $m\binom{2k}{2}$ and have rank $2k$ (this is where we are using that $k\geq 2$). ∎

Remark**.**

Note that our argument in fact implies a stronger statement than (4), namely that for fixed $m\in\mathbb{N}$ we have

[TABLE]

It seems rather natural, also in light of Proposition 5, to conjecture that this asymptotic holds for all fixed $z>0$ (say) in place of $m\in\mathbb{N}$. However, this is not possible when $z$ is small: Looking only at the contribution of tuples $(n_{1},\dots,n_{2k})=(p_{1},\dots,p_{k},p_{1},\dots,p_{k})$, we see that for any fixed $z>0$ we have

[TABLE]

When $z$ is sufficiently small (depending only on $k$) then this is clearly a contradiction. It would be very interesting to know what the correct asymptotic for this expression is, or more generally for any multiple Dirichlet series of this type, i.e. to have a generalisation of [DeLaBreteche1, Theorem 2] to poles of non-integral order. Our remark here suggests that this is not as straightforward as one might expect.

Note also that Theorem 2 implies that we can improve (3) to

[TABLE]

where $0\leq r<l^{\mathbb{C}}$ is the degree of $g_{1}$. This leads in a natural way to the even more general question of obtaining an asymptotic (or upper bound) for multiple Dirichlet series with a pole of non-integral order times a logarithmic pole.

In particular, ($*$) implies that if $T$ is diagonalizable (or more generally if $\deg g_{1}=0$) then we get

[TABLE]

If in addition $\rho_{2k}^{2}$ is an integer, our argument thus gives

[TABLE]

in place of (2). In particular, if $f$ is a Rademacher multiplicative function then $\rho_{2k}=1$ for all $k$ and up to constant we obtain the optimal upper bound. Noting that all our inequalities in the proof are in fact equalities in this case and that (4) can be improved to an asymptotic, we can recover [HarperNR1, Theorem 4], but this leads to the identical argument as it is carried out there.

It would be interesting to know if one can obtain a lower bound for the higher moments, for example in terms of the joint spectral subradius.

Example 10**.**

We continue with example 6. We were not able to find explicit expressions for $\rho_{2k}$ when $k\geq 2$; it seems quite plausible that such expressions don’t exist. However, we can bound $\rho_{2k}$ from above by the joint spectral radius $\rho_{\infty}(S)$. Moreover, using the JSR toolbox for Matlab (see [VHJ] for its documentation and instructions for installation), we could compute that $\rho_{\infty}^{2}=1.8173540\dots<2$. In particular, we see that

[TABLE]

holds for $k\geq 2$ and $x\geq 2$.