Homogeneous mappings of regularly varying vectors

TL;DR

This paper extends the theory of regular variation from univariate variables to multivariate vectors, providing conditions for products of such vectors and matrices to maintain regular variation, with applications to stochastic recurrences.

Contribution

It offers sharp sufficient conditions for regular variation of product-type functions of multivariate regularly varying vectors, generalizing univariate results and applying to matrices and stochastic equations.

Findings

Conditions for regular variation of products of vectors

Characterization of regular variation in matrix products

Application to affine stochastic recurrence solutions

Abstract

It is well known that the product of two independent regularly varying random variables with the same tail index is again regularly varying with this index. In this paper, we provide sharp sufficient conditions for the regular variation property of product-type functions of regularly varying random vectors, generalizing and extending the univariate theory in various directions. The main result is then applied to characterize the regular variation property of products of iid regularly varying quadratic random matrices and of solutions to affine stochastic recurrence equations under non-standard conditions.

Equations

$$\mathbb{P}(X>x)=\frac{L(x)}{x^{\alpha }},x>0,$$

$$\mathbb{P}(\pm {\sigma }_t{Z}_t>x)\sim \mathbb{E}[(Z_t^{\pm }{)}^{\alpha }]\mathbb{P}({\sigma }_t>x),x\to \infty .$$

$$\mathbb{P}(Z_t>x)\sim p_{+}\frac{L(x)}{x^{\alpha }}\text{mbox}and\mathbb{P}(Z_t<-x)\sim p_{-}\frac{L(x)}{x^{\alpha }}$$

$$\mathbb{P}(\pm X_t>x)\sim \mathbb{E}[{\sigma }_t^{\alpha }]\mathbb{P}(\pm Z_t>x),x\to \infty ,$$

$$\underset{x\to \infty }{\lim }\frac{\mathbb{P}(XY>x)}{\mathbb{P}(X>x)}=\infty .$$

$$c_0=\underset{t\to \infty }{\lim }\frac{\mathbb{P}(Y>t)}{\mathbb{P}(X>t)}\in [0,\infty )$$

$$\underset{M\to \infty }{\lim }\underset{x\to \infty }{\mathrm{limsup}}\frac{\mathbb{P}(XY>x,M<X\le x/M)}{\mathbb{P}(X>x)}$$

$$\underset{t\to \infty }{\lim }\frac{\mathbb{P}(XY>t)}{\mathbb{P}(X>t)}=\mathbb{E}[X^{\alpha }]+c_0\mathbb{E}[Y^{\alpha }].$$

$$\displaystyle\lim_{x\to\infty}\sup_{y\leq\varepsilon}\Big{|}\dfrac{\mathbb{P}(X>x/y)}{\mathbb{P}(X>x)}-y^{\alpha}\Big{|}=0\,.$$

$$\displaystyle\mathbb{P}\Big{(}\dfrac{{\bf X}}{\|{\bf X}\|}\in\cdot\,,\dfrac{\|{\bf X}\|}{x}\in\cdot\left|\|{\bf X}\|>x\Big{)}\right.\stackrel{{\scriptstyle w}}{{\rightarrow}}\,\mathbb{P}(\mbox{\boldmath$\Theta$}_{\mathbf{X}}\in\cdot)\,\mathbb{P}(Z\in\cdot)\,,\qquad x\to\infty\,,$$

$${\mu }_t^{\mathbf{X}}(\cdot )=\frac{\mathbb{P}(t^{-1}\mathbf{X}\in \cdot )}{\mathbb{P}(\Vert \mathbf{X}\Vert >t)}\stackrel{v}{\to }{\mu }^{\mathbf{X}}(\cdot ),t\to \infty ,$$

$$\int f(\mathbf{x}){\mu }_t^{\mathbf{X}}(d\mathbf{x})\to \int f(\mathbf{x}){\mu }^{\mathbf{X}}(d\mathbf{x}),t\to \infty .$$

$${\mu }^{\mathbf{X}}(tA)=t^{-{\alpha }_{\mathbf{X}}}{\mu }^{\mathbf{X}}(A),t>0.$$

$$\mathbb{P}(\Vert \mathbf{X}\Vert >x)=\frac{L(x)}{x^{{\alpha }_{\mathbf{X}}}},$$

$$\psi :{\mathbb{R}}^{d_{\mathbf{X}}}\times {\mathbb{R}}^{d_{\mathbf{Y}}}\to {\mathbb{R}}^{d_{\mathbf{Z}}}$$

$$\psi (s\mathbf{x},t\mathbf{y})=s^{a_{\mathbf{X}}}{t}^{a_{\mathbf{Y}}}\psi (\mathbf{x},\mathbf{y}),s,t\ge 0.$$

$${\mathbf{X}}\in{\rm RV}(\alpha,\mu^{\mathbf{X}})\quad\mbox{and}\quad\mathbb{E}\big{[}\|{\mathbf{Y}}\|^{\alpha+\delta}\big{]}<\infty\quad\mbox{ for some $\delta>0$,}$$

$$\displaystyle\dfrac{\mathbb{P}(t^{-1}{\mathbf{X}}\cdot{\mathbf{Y}}\in\cdot)}{\mathbb{P}(\|{\mathbf{X}}\|>t)}\stackrel{{\scriptstyle v}}{{\to}}\ \eta(\cdot):=\mathbb{E}\big{[}\mu^{\mathbf{X}}\big{(}\big{\{}{\bf x}\>:\>{\bf x}\cdot{\mathbf{Y}}\in\cdot\big{\}}\big{)}\big{]}.$$

$$\displaystyle\mu^{{\bf Z}}(\cdot)=\dfrac{\eta(\cdot)}{\eta\big{(}\big{\{}{\bf z}\>:\>\|{\bf z}\|>1\big{\}}\big{)}}\,.$$

$$\underset{t\to \infty }{\lim }\frac{\mathbb{P}(\Vert \mathbf{X}{\Vert }^{a_{\mathbf{X}}}>t)}{\mathbb{P}(\Vert \mathbf{X}{\Vert }^{a_{\mathbf{X}}}\cdot \Vert \mathbf{Y}{\Vert }^{a_{\mathbf{Y}}}>t)}=c_{\mathbf{X}},\underset{t\to \infty }{\lim }\frac{\mathbb{P}(\Vert \mathbf{Y}{\Vert }^{a_{\mathbf{Y}}}>t)}{\mathbb{P}(\Vert \mathbf{X}{\Vert }^{a_{\mathbf{X}}}\cdot \Vert \mathbf{Y}{\Vert }^{a_{\mathbf{Y}}}>t)}=c_{\mathbf{Y}}.$$

$$\underset{t\to \infty }{\lim }\frac{\mathbb{P}(\Vert \mathbf{Y}{\Vert }^{a_{\mathbf{Y}}}>t)}{\mathbb{P}(\Vert \mathbf{X}{\Vert }^{a_{\mathbf{X}}}>t)}=c_0\in [0,\infty )$$

$$\underset{M\to \infty }{\lim }\underset{t\to \infty }{\mathrm{limsup}}\frac{\mathbb{P}(\Vert \mathbf{X}{\Vert }^{a_{\mathbf{X}}}\Vert \mathbf{Y}{\Vert }^{a_{\mathbf{Y}}}>t,M<\Vert \mathbf{X}{\Vert }^{a_X}\le t/M)}{\mathbb{P}(\Vert \mathbf{X}{\Vert }^{a_{\mathbf{X}}}>x)}=0$$

$$c_{\mathbf{X}}=\frac{1}{\mathbb{E}[\Vert \mathbf{Y}{\Vert }^{{\alpha }_{\mathbf{Y}}}]+c_0\mathbb{E}[\Vert \mathbf{X}{\Vert }^{{\alpha }_{\mathbf{X}}}]}\text{mbox}and{c}_{\mathbf{Y}}=c_{\mathbf{X}}{c}_0.$$

$$\frac{\mathbb{P}\left(t^{-1}\mathbf{Z}\in \cdot \right)}{\mathbb{P}(\Vert \mathbf{X}{\Vert }^{a_{\mathbf{X}}}\cdot \Vert \mathbf{Y}{\Vert }^{a_{\mathbf{Y}}}>t)}\stackrel{v}{\to }\eta (\cdot )$$

$$\mu^{{\bf Z}}(\cdot)=\dfrac{\eta(\cdot)}{\eta\big{(}\big{\{}{\bf z}\>:\>\|{\bf z}\|>1\big{\}}\big{)}}\,.$$

$$\eta (\cdot )=\frac{1}{\mathbb{E}[\Vert \mathbf{Y}{\Vert }^{{\alpha }_{\mathbf{X}}{a}_{\mathbf{Y}}/a_{\mathbf{X}}}]}{\mu }^{\mathbf{X}}(\{\mathbf{x}:\psi (\mathbf{x},\mathbf{Y})\in \cdot \}).$$

$$\eta (\cdot )=\mathbb{E}\left[{\mu }^{\mathbf{X}}(\{\mathbf{x}:\psi (\mathbf{x},{\Theta }_{\mathbf{Y}})\in \cdot \})\right].$$

$$\eta (\cdot )$$

$$\mathbb{E}\left[{\mu }^{\mathbf{X}}(\{\mathbf{x}:\psi (\mathbf{x},{\Theta }_{\mathbf{Y}})\in \cdot \})\right]=$$

$$=$$

Full text

Piotr Dyszewski was partially supported by the National Science Centre, Poland (Sonata Bis, grant number DEC-2014/14/E/ST1/00588). This work was initiated while the first author was visiting the Department of Mathematics, University of Copenhagen in February 2018. He gratefully acknowledges financial support and hospitality.

Thomas Mikosch is partially supported by an Alexander von Humboldt Research Award. He takes pleasure in thanking his colleagues at Mathematische Fakultät of Ruhruniversität Bochum for hosting him December 2018 - May 2019.

1. Introduction

1.1. Closure of regular variation under multiplication – the univariate case

Consider a non-negative random variable $X$ and assume that $X$ is regularly varying with index $\alpha>0$ in the sense that

[TABLE]

where $L$ denotes some slowly varying function; we refer to Bingham et al. [3] for an encyclopedic treatment of univariate regularly varying functions and to Resnick [16, 17] for the case of regularly varying random vectors.

A natural question appears in this context: given $Y$ is a non-negative random variable independent of $X$, under which conditions is the product $X\,Y$ regularly varying with index $\alpha$? This is a natural problem indeed: in numerous contexts of applied probability one studies models which involve products of independent random variables. Among those are classical time series models such as the ARCH-GARCH family and the stochastic volatility model; see Andersen et al. [1] for an extensive treatment of these models in financial time series analysis. In both cases, the real-valued time series $(X_{t})$ is given via the relation $X_{t}=\sigma_{t}\,Z_{t}$, where $(\sigma_{t})$ is a strictly stationary sequence of positive random variables which is either predictable with respect to the natural filtration of the iid sequence $(Z_{t})$ (such as for ARCH-GARCH) or $(\sigma_{t})$ and $(Z_{t})$ are mutually independent (such as for the stochastic volatility model). In both cases, there is strong interest in the tail behavior of the products $X_{t}=\sigma_{t}Z_{t}$ (notice that, under the aforementioned conditions, $\sigma_{t}$ and $Z_{t}$ are independent). In the ARCH-GARCH the condition $\mathbb{E}[|Z|^{\alpha}]<\infty$ and the dynamics of the volatlity sequence $(\sigma_{t})$ ensure that $\mathbb{P}(\sigma_{t}>x)\sim c\,x^{-\alpha}$ for some positive constants $c,\alpha$ (for more details we refer the reader to Section 4). In turn, the condition $\mathbb{E}[|Z|^{\alpha}]<\infty$ and the so-called Breiman lemma imply that

[TABLE]

Breiman’s result [4] is contained in the following useful lemma; for a proof, see Appendix C.3 in [5].

Lemma 1.1**.**

Assume $X,Y$ are independent non-negative random variables, $X$ is regularly varying with index $\alpha>0$ in the sense of (1.1), and $\mathbb{E}[Y^{\alpha+\delta}]<\infty$ for some $\delta>0$ or $\mathbb{P}(X>x)\sim c\,x^{-\alpha}$ for some positive $c>0$ and $\mathbb{E}[Y^{\alpha}]<\infty$. Then $\mathbb{P}(XY>x)\sim\mathbb{E}[Y^{\alpha}]\,\mathbb{P}(X>x)$ as $x\to\infty$.

Thus the regular variation of $X$ is preserved under multiplication with an independent non-negative random variable $Y$ if the corresponding assumptions on $Y$ hold, ensuring that $Y$ has a lighter tail than $X$. We already mentioned the case of an ARCH-GARCH process $(X_{t})$ when $\sigma_{t}$ is regularly varying with index $\alpha>0$ and $X_{t}$ inherits this property if $\mathbb{E}[|Z_{t}|^{\alpha}]<\infty$. In the stochastic volatility model, $X_{t}$ is regularly varying with index $\alpha>0$ if either $\sigma_{t}$ is regularly varying with the same index and $\mathbb{E}[|Z_{t}|^{\alpha+\delta}]<\infty$ for some $\delta>0$ and then (1.2) holds, or $Z_{t}$ is regularly varying with index $\alpha$, satisfying the tail balance condition :

[TABLE]

for constants $p_{\pm}$ such that $p_{+}+p_{-}=1$ and a slowly varying function $L$, and $\mathbb{E}[\sigma_{t}^{\alpha+\delta}]<\infty$ for some $\delta>0$, and then

[TABLE]

holds.

We mention that power-law tail behavior of a stationary sequence $(X_{t})$ is essential for the asymptotic behavior of their extremes and partial sums, and related point process convergence and functionals acting on them. For example, if $(Z_{t})$ is iid and regularly varying with index $\alpha>0$, then the sequence of the maxima $(a_{n}^{-1}M_{n})$, where $M_{n}=\max_{i=1,\ldots,n}Z_{i}$, and $(a_{n})$ satisfies $n\,\mathbb{P}(Z>a_{n})\to 1$, converges in distribution to a Fréchet distribution $\Phi_{\alpha}(x)=\exp(-x^{-\alpha})$, $x>0$; see Embrechts et al. [10], Section 3.3. Moreover, the process of the points $(a_{n}^{-1}X_{i})_{i=1,\ldots,n}$ converges in distribution to an inhomogeneous Poisson process on $(0,\infty)$ with intensity function $\alpha x^{-\alpha-1}dx$; see Resnick [16, 17], Embrechts et al. [10], Chapter 5. Similarly, if $\alpha\in(0,2)$ and $Z$ is regularly varying in the sense of (1.3) then $(a_{n}^{-1}(S_{n}-c_{n}))$ converges in distribution (with suitable centering constants $(c_{n})$) to an infinite variance $\alpha$-stable limit; see Feller [11] or Resnick [17]. Moreover, there is a vast literature that extends these results from the iid to the dependent case.

For the completeness of presentation, we mention some related results for independent non-negative random variables $X,Y$ when both are regularly varying with the same index $\alpha$. This situation is much more subtle than the Breiman case. Still, $XY$ is regularly varying with index $\alpha$:

Lemma 1.2**.**

Assume that $X,Y$ are independent non-negative random variables and $X$ is regularly varying with index $\alpha>0$. Then the following statements hold:

1. (1)

If either $Y$ is regularly varying with index $\alpha$ or \mathbb{P}(Y>x)=o\big{(}\mathbb{P}(X>x)\big{)} as $x\to\infty$ then $X\,Y$ is regularly varying with index $\alpha$. 2. (2)

If $\mathbb{E}[Y^{\alpha}]=\infty$ then

[TABLE] 3. (3)

Assume that $X,Y$ are regularly varying with index $\alpha>0$, $\mathbb{E}[X^{\alpha}+Y^{\alpha}]<\infty$,

[TABLE]

and

[TABLE]

Then

[TABLE]

The proof of this result is given in Appendix A.1.

Remark 1.3**.**

Condition (1.4) is a very technical assumption. To verify it one would need to have very precise information about the tail behavior of $X$. This condition does not follow from the uniform convergence theorem for regularly varying functions; the latter result ensures that for any $\varepsilon>0$,

[TABLE]

However, for the verification of (1.4) we need information about the deviation of $\mathbb{P}(X>x/y)/\mathbb{P}(X>x)$ from $y^{\alpha}$ in the range $y\in[M,x/M]$ for any $M>0$ and large $x$, i.e., for large values of $y$. Part (3) was proved as Proposition 3.1 by Davis and Resnick [8] in the case when $X,Y$ are iid. In this case, (1.4) is necessary for $\mathbb{P}(XY>t)/\mathbb{P}(X>t)\to 2\mathbb{E}[X^{\alpha}]$ to hold.

We mention in passing that regular variation of $XY$ does in general not imply regular variation of $X$ or $Y$; see Jacobsen et al. [12].

1.2. Closure of regular variation under multiplication – the multivariate case

Our main goal in this paper is to extend some of the aforementioned results to the multivariate case. We start by introducing regular variation of random vectors. For this reason we equip $\mathbb{R}^{d_{\bf X}}$ with an arbitrary norm $\|\cdot\|$. A random vector ${\bf X}$ has a multivariate regularly varying distribution if $\|{\bf X}\|$ has a univariate regularly varying distribution and is asymptotically independent of ${\bf X}/\|{\bf X}\|$ given $\|{\bf X}\|>x$. More precisely, we say that a random vector ${\bf X}\in\mathbb{R}^{d_{\bf X}}$ and its distribution are regularly varying if

[TABLE]

where $Z$ is Pareto distributed with $\mathbb{P}(Z>y)=y^{-\alpha}$, $y>1$, and \mbox{\boldmath\Theta}_{\mathbf{X}} assumes values in the unit sphere ${\mathbb{S}}^{d_{\bf X}-1}=\{{\bf x}\in{\mathbb{R}}^{d_{\bf X}}:\|{\bf x}\|=1\}$. The distribution of \mbox{\boldmath\Theta}_{\mathbf{X}} is the spectral distribution of ${\bf X}$.

We will often refer to an equivalent formulation of multivariate regular variation. Namely, a random vector ${\bf X}\in\mathbb{R}^{d_{\bf X}}$ and its distribution are regularly varying if and only if, there exists a non-null Radon measure $\mu^{\mathbf{X}}$ on $\mathbb{R}_{\mathbf{0}}^{d_{\bf X}}=\mathbb{R}^{d_{\bf X}}\setminus\{\mathbf{0}\}$ such that

[TABLE]

where $\stackrel{{\scriptstyle v}}{{\rightarrow}}$ denotes vague convergence in the space of measures on $\mathbb{R}_{\mathbf{0}}^{d_{\bf X}}$. Recall that for measures $\mu^{{\mathbf{X}}}_{t}$, $\mu^{\mathbf{X}}$ on $\mathbb{R}_{\mathbf{0}}^{d_{\bf X}}$, $\mu^{{\mathbf{X}}}_{t}\stackrel{{\scriptstyle v}}{{\to}}\mu^{\mathbf{X}}$ if for any function $f$ from the set $C_{c}^{+}(\mathbb{R}_{\mathbf{0}}^{d_{\bf X}})$ of non-negative continuous functions on $\mathbb{R}_{\mathbf{0}}^{d_{\bf X}}$ with compact support111 In the context of regular variation, the origin is excluded from consideration. Therefore a set $K\subset\mathbb{R}_{\mathbf{0}}^{d_{\bf X}}$ is compact if it is compact in $\mathbb{R}_{\mathbf{0}}^{d_{\bf X}}$ but bounded away from zero. we have

[TABLE]

It turns out that the limiting measure $\mu^{\mathbf{X}}$ has the homogeneity property. More precisely, there exists $\alpha_{\bf X}>0$ such that for any set $A$ in the Borel $\sigma$-algebra of $\mathbb{R}_{\mathbf{0}}^{d_{\bf X}}$ we have

[TABLE]

We call $\alpha_{\bf X}$ the index of regular variation or tail index of ${\bf X}$ and, for short, we write ${\bf X}\in{\rm RV}(\alpha_{\bf X},\mu^{\mathbf{X}})$. Of course, we necessarily have

[TABLE]

for some slowly varying function $L$. We refer to Resnick [16, 17] as general references to multivariate regular variation and its applications.

Now consider two independent vectors ${\bf X}\in{\rm RV}(\alpha_{\bf X},\mu^{\mathbf{X}})$ and ${\bf Y}\in{\rm RV}(\alpha_{\bf Y},\mu^{\mathbf{Y}})$ with values in ${\mathbb{R}}^{d_{\bf X}}$ and ${\mathbb{R}}^{d_{\bf Y}}$, respectively. Our goal is to establish sufficient conditions under which ${\bf Z}=\psi({\bf X},{\bf Y})$ is also regularly varying where

[TABLE]

is continuous, $a_{\bf X}$-homogeneous with respect to the first argument and $a_{\bf Y}$-homogeneous with respect to the second one for positive $a_{\bf X},a_{\bf Y}$, i.e., for any ${\bf x}\in\mathbb{R}^{d_{\bf X}}$ and ${\bf y}\in\mathbb{R}^{d_{\bf Y}}$,

[TABLE]

Example 1.4** **(Products of independent regularly varying matrices).

If $d_{\bf X}=n_{1}\cdot d_{1}$ then one can identify $\mathbb{R}_{\mathbf{0}}^{d_{\bf X}}$ with the set of non-zero $n_{1}\times d_{1}$ matrices $\mathbb{M}_{n_{1}\times d_{1}}$. Similarly, if $d_{\bf Y}=d_{1}\cdot m_{1}$, $\mathbb{R}_{\mathbf{0}}^{d_{\bf Y}}=\mathbb{M}_{d_{1}\times m_{1}}$. We define $\psi({\bf x},{\bf y})={\bf x}\cdot{\bf y}$ where ${\bf x}\cdot{\bf y}$ denotes ordinary matrix multiplication of an $n_{1}\times d_{1}$ matrix ${\bf x}$ with a $d_{1}\times m_{1}$ matrix ${\bf y}$. Then $d_{\bf Z}=n_{1}\cdot m_{1}$, $a_{\bf X}=a_{\bf Y}=1$, and ${\bf Z}$ is a product of two independent regularly varying matrices ${\bf X}$ and ${\bf Y}$.

In this case, regular variation of ${\bf Z}$ was proved in Basrak et al. [2]; it is a multivariate analog of the Breiman Lemma 1.1: if

[TABLE]

then

[TABLE]

In particular, if $\eta$ is non-null then {\bf Z}={\mathbf{X}}\cdot{\mathbf{Y}}\in{\rm RV}\big{(}\alpha_{\bf X},\mu^{{\bf Z}}\big{)} where

[TABLE]

Example 1.5** **(Kronecker products of independent regularly varying matrices).

Suppose that $d_{\bf X}=n_{1}\cdot d_{1}$ and $d_{\bf Y}=d_{2}\cdot n_{2}$, so we can identify $\mathbb{R}_{\mathbf{0}}^{d_{\bf X}}=\mathbb{M}_{n_{1}\times n_{2}}$, $\mathbb{R}_{\mathbf{0}}^{d_{\bf Y}}=\mathbb{M}_{d_{1}\times d_{2}}$. Now define $\psi\colon\mathbb{R}^{d_{\bf X}}\times\mathbb{R}^{d_{\bf Y}}\to\mathbb{R}^{n_{1}d_{1}n_{2}d_{2}}=\mathbb{M}_{n_{1}d_{1}\times n_{2}d_{2}}$ via the Kronecker product $\psi({\bf x},{\bf y})={\bf x}\otimes{\bf y}$. As for ordinary matrix multiplication, we have $a_{\bf X}=a_{\bf Y}=1$.

Example 1.6** **(Random quadratic form).

If $d_{\bf Y}=d_{\bf X}^{2}$, identifying $\mathbb{R}_{\mathbf{0}}^{d_{\bf Y}}=\mathbb{M}_{d_{\bf X}\times d_{\bf X}}$, we define $\psi\colon\mathbb{R}^{d_{\bf X}}\times\mathbb{R}^{d_{\bf Y}}\to\mathbb{R}$ by $\psi({\bf x},{\bf y})={\bf x}^{\top}{\bf y}{\bf x}$. In this case, $a_{\bf X}=2$ and $a_{\bf Y}=1$.

1.3. Organization of the article

Our main result (Theorem 2.1) yields sharp sufficient conditions for regular variation of the homogeneous function $\psi({\bf X},{\bf Y})$ acting on independent regularly varying random vectors ${\bf X},{\bf Y}$. The proof is given in Section 3. We apply these results in Section 4. In particular, in Section 4.1 we derive the regular variation properties of products of iid regularly varying quadratic matrices while, in Section 4.2, we prove regular variation of solutions to affine stochastic recurrence equations under non-standard conditions.

2. Main result

In what follows, ${\bf X}$ and ${\bf Y}$ are independent random variables with values in ${\mathbb{R}}^{d_{\bf X}}$ and ${\mathbb{R}}^{d_{\bf Y}}$, respectively, and we also assume ${\bf X}\in{\rm RV}(\alpha_{\bf X},\mu^{{\bf X}})$ and ${\bf Y}\in{\rm RV}(\alpha_{\bf Y},\mu^{{\bf Y}})$. We will study the regular variation property of the $a_{\bf X}$-$a_{\bf Y}$-homogeneous function ${\bf Z}=\psi({\bf X},{\bf Y})$; see (1.7). We also need a tail balance condition : the following limits exist and are finite

[TABLE]

We observe that $\|{\bf X}\|^{a_{\bf X}}$ and $\|{\bf Y}\|^{a_{\bf Y}}$ are regularly varying with indices $\alpha_{\bf X}/a_{\bf X}$ and $\alpha_{\bf Y}/a_{\bf Y}$, respectively. Therefore Lemmas 1.1 and 1.2 apply:

• •

if $\alpha_{\bf X}/a_{\bf X}<\alpha_{\bf Y}/a_{\bf Y}$ then $\|{\bf X}\|^{a_{\bf X}}\|{\bf Y}\|^{a_{\bf Y}}$ is regularly varying with index $\alpha_{\bf X}/a_{\bf X}$, $c_{\bf X}=1/\mathbb{E}[\|{\bf Y}\|^{a_{\bf Y}\,\alpha_{\bf X}/a_{\bf X}}]\in(0,\infty)$ and $c_{\bf Y}=0$.

• •

if $\alpha_{\bf X}/a_{\bf X}=\alpha_{\bf Y}/a_{\bf Y}$ then $\|{\bf X}\|^{a_{\bf X}}\|{\bf Y}\|^{a_{\bf Y}}$ is regularly varying with index $\alpha_{\bf X}/a_{\bf X}$.

• •

if $\alpha_{\bf X}/a_{\bf X}=\alpha_{\bf Y}/a_{\bf Y}$ and $\mathbb{E}[\|{\bf Y}\|^{\alpha_{\bf Y}}]=\infty$ then $c_{\bf X}=0$.

• •

if $\alpha_{\bf X}/a_{\bf X}=\alpha_{\bf Y}/a_{\bf Y}$, $\mathbb{E}[\|{\bf X}\|^{\alpha_{\bf X}}+\|{\bf Y}\|^{\alpha_{\bf Y}}]<\infty$, the limit

[TABLE]

exists and

[TABLE]

holds then

[TABLE]

Now we formulate the first result of this paper.

Theorem 2.1**.**

Assume that the ${\mathbb{R}}^{d_{\bf X}}$-valued ${\bf X}\in{\rm RV}(\alpha_{\bf X},\mu^{{\bf X}})$ and the ${\mathbb{R}}^{d_{\bf Y}}$-valued ${\bf Y}\in{\rm RV}(\alpha_{\bf Y},\mu^{{\bf Y}})$ random vectors are independent and the balance condition (2.1) is satisfied for positive $a_{\bf X},a_{\bf Y}$. Then the following relation holds for the $a_{\bf X}$-$a_{\bf Y}$-homogeneous function ${\bf Z}=\psi({\bf X},{\bf Y})$:

[TABLE]

In particular, if $\eta$ is non-null, then ${\bf Z}\in{\rm RV}(\alpha_{\bf Z},\mu^{{\bf Z}})$, where $\alpha_{\bf Z}=\frac{\alpha_{\bf X}}{a_{\bf X}}\wedge\frac{\alpha_{\bf Y}}{a_{\bf Y}}$ and

[TABLE]

Combining the discussion before Theorem 2.1 and the aforementioned results, we obtain the following consequencees.

Corollary 2.2**.**

Assume the conditions of Theorem 2.1.

1. (1)

If $\frac{\alpha_{\bf X}}{a_{\bf X}}<\frac{\alpha_{\bf Y}}{a_{\bf Y}}$ then $c_{\bf Y}=0$, $c_{\bf X}=1/\mathbb{E}[\|{\mathbf{Y}}\|^{\alpha_{\bf X}a_{\bf Y}/a_{\bf X}}]$, and (2.5) holds with

[TABLE] 2. (2)

If $\mathbb{P}(\|{\mathbf{X}}\|^{a_{\bf X}}>t)+\mathbb{P}(\|{\mathbf{Y}}\|^{a_{\bf Y}}>t)=o(\mathbb{P}(\|{\mathbf{X}}\|^{a_{\bf X}}\cdot\|{\mathbf{Y}}\|^{a_{\bf Y}}>t))$ then $c_{\bf X}=c_{\bf Y}=0$, and (2.5) holds with

[TABLE] 3. (3)

If $\alpha_{\bf Y}/a_{\bf Y}=\alpha_{\bf X}/a_{\bf X}$ and \mathbb{E}\big{[}\|{\mathbf{X}}\|^{\alpha_{\bf X}}+\|{\mathbf{Y}}\|^{\alpha_{\bf Y}}\big{]}<\infty, (2.3) holds, and the limit $c_{0}$ in (2.2) exists, then $c_{\bf X}$ is given in (2.4), $c_{\bf Y}=c_{0}\,c_{\bf X}$, and (2.5) holds with

[TABLE]

Remark 2.3**.**

As regards statement (2), one can verify that $\eta$ is symmetric with respect to ${\mathbf{X}}$ and ${\mathbf{Y}}$. In this case, necessarily $\frac{\alpha_{\bf X}}{a_{\bf X}}=\frac{\alpha_{\bf Y}}{a_{\bf Y}}$, and we can write

[TABLE]

3. Proof of Theorem 2.1

Throughout this section we consider an ${\mathbb{R}}^{d_{\bf X}}$-valued ${\mathbf{X}}\in{\rm RV}(\alpha_{\bf X},\mu^{\bf X})$ random vector independent of an ${\mathbb{R}}^{d_{Y}}$-valued ${\mathbf{Y}}\in{\rm RV}(\alpha_{\bf Y},\mu^{\bf Y})$. Recall that ${\bf Z}=\psi({\bf X},{\bf Y})\in{\mathbb{R}}^{d_{\bf Z}}$. Take any function $f$ from the set $C^{+}_{c}(\mathbb{R}_{\mathbf{0}}^{d_{\bf Z}})$ of nonnegative continuous functions with compact support in $\mathbb{R}_{\mathbf{0}}^{d_{\bf Z}}$. Write

[TABLE]

Then (2.5) turns into $\eta_{t}\stackrel{{\scriptstyle v}}{{\rightarrow}}\eta$ as $t\to\infty$ which can be re-formulated as

[TABLE]

Since $\psi$ is continuous

[TABLE]

It is also $a_{\bf X}$-$a_{\bf Y}$-homogeneous and therefore

[TABLE]

Then we also have for any set $A_{r}=\{{\bf z}:\|{\bf z}\|>r\}$, $r>0$, in view of regular variation of $\|{\bf X}\|^{a_{\bf X}}\|{\bf Y}\|^{a_{\bf Y}}$,

[TABLE]

It follows from Resnick [16], Proposition 3.16, that $(\eta_{t})$ is vaguely relatively compact. Hence $(\eta_{t_{k}})$ converges vaguely along sequences $t_{k}\to\infty$ as $k\to\infty$, and it remains to show that these limits coincide with $\eta$.

The proof of the theorem is given through several auxiliary result which we provide first. The main steps of the proof are given at the end of this section.

Limits of \mathbb{E}[f\big{(}t^{-1}\psi({\mathbf{X}},{\mathbf{Y}})\big{)}\mid{\bf Y}]. By regular variation of ${\bf X}$ we have

[TABLE]

Define

[TABLE]

In view of (3.1) we expect that the right-hand side converges to

[TABLE]

However, the function ${\bf x}\mapsto f(\psi({\bf x},{\bf y}))$ may not have compact support and therefore some additional argument is needed.

Lemma 3.1**.**

Relation (3.3) holds for any $f\in C^{+}_{c}(\mathbb{R}_{\mathbf{0}}^{d_{\bf Z}})$.

Proof of Lemma 3.1.

Fix ${\bf y}\in\mathbb{R}^{d_{\bf X}}$. Since $f$ is compactly supported there are constants $M_{f},c_{f}>0$ such that

[TABLE]

For $r\geq 1$ choose any continuous function $\varphi_{r}\colon\mathbb{R}^{d_{\bf X}}\to[0,1]$ such that

[TABLE]

We have

[TABLE]

The contribution of the second term is negligible since in view of (3.1),

[TABLE]

Thus it suffices to prove $\lim_{r\to\infty}\lim_{t\to\infty}I_{1}=g({\bf y})$. The function ${\bf x}\mapsto f(\psi({\bf x},{\bf y}))\varphi_{r}({\bf x})$ is continuous and non-negative for any choice of ${\bf y}\in\mathbb{R}^{d_{\bf Y}}$ and $r>1$, and its support is contained in $\{{\bf x}\in\mathbb{R}^{d_{\bf X}}\>:\>(M_{\psi}\|{\bf y}\|^{a_{\bf Y}}c_{f})^{-1/a_{{\bf X}}}\leq\|{\bf x}\|\leq 2r\}$ which is a compact subset of $\mathbb{R}_{\mathbf{0}}^{d_{\bf X}}$. Regular variation of ${\mathbf{X}}$ and monotone convergence allow one to take the successive limits

[TABLE]

∎

The next result presents a continuity bound for $g_{t}$.

Lemma 3.2**.**

Let $f\in C^{+}_{c}(\mathbb{R}^{d_{\bf Z}}_{\bf 0})$. For any $\varepsilon>0$ one can choose $\delta>0$ and $t_{0}>0$ such that for any ${\bf s}$, ${\bf r}\in{\mathbb{S}}^{d_{\bf X}-1}$ with $\|{\bf s}-{\bf r}\|\leq\delta$ and any $t>t_{0}$,

[TABLE]

Proof of Lemma 3.2.

Fix $\varepsilon_{1}>0$. Choose $M_{f},c_{f}>0$ from (3.4). By uniform continuity of $f$ we can choose $\eta\in(0,\varepsilon_{1})$ such that $\|{\bf z}_{1}-{\bf z}_{2}\|\leq\eta$ implies $\|f({\bf z}_{1})-f({\bf z}_{2})\|\leq\varepsilon_{1}$. Since $\psi$ is uniformly continuous on ${\mathbb{S}}^{d_{\bf X}-1}\times{\mathbb{S}}^{d_{\bf Y}-1}$ we can find $\delta>0$ such that for ${\bf r},{\bf s}\in{\mathbb{S}}^{d_{\bf Y}-1}$ with $\|{\bf r}-{\bf s}\|<\delta$,

[TABLE]

Then by homogeneity of $\psi$,

[TABLE]

and we can write for $t>0$,

[TABLE]

where $L$ is defined in (1.6). Given $\varepsilon>0$ we can choose $\varepsilon_{1}$ sufficiently small such that

[TABLE]

Choosing $t_{0}$ big enough, one ensures that

[TABLE]

which proves the claim. ∎

Note that by continuity of $f$ and $\psi$, $g$ is also continuous on $\mathbb{R}^{d_{\bf X}}$, hence also uniformly continuous on the unit sphere ${\mathbb{S}}^{d-1}$. We will use this comment in the proof of the next lemma.

Lemma 3.3**.**

Let $f\in C^{+}_{c}(\mathbb{R}^{d_{\bf Z}}_{\bf 0})$. Then $g_{t}\to g$ as $t\to\infty$ uniformly on ${\mathbb{S}}^{d_{\bf Y}-1}$.

Proof of Lemma 3.3..

Fix $\varepsilon>0$ and take $\delta>0$, $t_{0}>0$ that satisfy the claim of Lemma 3.2 and

[TABLE]

Let $\{{\bf r}_{k}\}_{k=1}^{N}$ for $N=N(\delta)$ be a $\delta$-covering of ${\mathbb{S}}^{d_{\bf Y}-1}$. Take $t_{1}>0$ so large that

[TABLE]

Then for any ${\bf s}\in{\mathbb{S}}^{d_{\bf Y}-1}$ we have $\|{\bf s}-{\bf r}_{k}\|\leq\delta$ for some $k$ and for $t>t_{0}\vee t_{1}$ we have

[TABLE]

This finishes the proof of the lemma. ∎

Before we proceed with the final steps in the proof of Theorem 2.1 we observe that homogeneity of $\mu^{\mathbf{X}}$ and $\psi$ implies for any $r>0$ and ${\bf y}\in\mathbb{R}^{d_{\bf Y}}$,

[TABLE]

Now we define functions $h_{t}\colon\mathbb{R}^{d_{\bf X}}\to[0,+\infty)$ by

[TABLE]

By a symmetry argument, interchanging the roles of ${\mathbf{Y}}$ and ${\mathbf{X}}$, we conclude that $h_{t}\to h$ as $t\to\infty$ point-wise in ${\mathbb{R}}^{d_{\bf X}}$ and uniformly on ${\mathbb{S}}^{d_{\bf X}-1}$ where

[TABLE]

The limiting function is also homogeneous, i.e., for $r>0$ and ${\bf x}\in\mathbb{R}^{d_{\bf X}}$,

[TABLE]

Main steps in the proof of Theorem 2.1.

Recalling the notation introduced so far, our goal is to prove (2.5) in disguised form by applying an approach via test functions:

[TABLE]

Choose $M_{f}>0$ from (3.4) and consider the following decomposition, for $\eta\in(0,1)$,

[TABLE]

Since $f$ is bounded and ${\bf X},{\bf Y}$ are independent we have $J_{3}(t)=o(\mathbb{P}(\|{\mathbf{X}}\|^{a_{\bf X}}\cdot\|{\mathbf{Y}}\|^{a_{\bf Y}}>t))$. Thus it remains to investigate $J_{1}$ and $J_{2}$. We begin with the analysis of the first term, since it requires more work.

Analysis of $J_{1}$. We claim that

[TABLE]

Below we will present a detailed argument for

[TABLE]

The lower bound can be established in a similar fashion. Write for ${\bf z}\neq\bf 0$, $\widetilde{\bf z}={\bf z}/\|{\bf z}\|$, and

[TABLE]

where $g_{t}$ is given via (3.2). By virtue of Lemma 3.3, for any $\varepsilon>0$ there is a sufficiently small $\eta>0$ such that

[TABLE]

Thus, since $\varepsilon$ is arbitrary, we only need to investigate the expectation

[TABLE]

If $\mathbb{E}[g(\widetilde{\bf Y})]=0$ then by homogeneity of $g$, $g({\mathbf{Y}})=0$ a.s. which implies $\mathbb{E}[g({\mathbf{Y}})]=0$ and $\mathbb{E}[g(\Theta_{\mathbf{Y}})]=0$, so the claim follows trivially. Now assume $\mathbb{E}[g(\widetilde{\bf Y})]>0$. Let $Y^{\prime}$ be a random variable independent of ${\mathbf{X}}$ and ${\mathbf{Y}}$ with distribution given by

[TABLE]

Then, by regular variation of ${\mathbf{Y}}$, as $t\to\infty$,

[TABLE]

Therefore for any $\delta>0$ there exists $T=T(\delta)$ such that

[TABLE]

Without loss of generality we may assume that $T\uparrow\infty$ when $\delta\downarrow 0$. Consider the following decomposition

[TABLE]

By Breiman’s Lemma 1.1 and definition of $c_{\bf X}$ we have

[TABLE]

For the first term we have by (3.8) ,

[TABLE]

In the last step we used Breiman’s result as $t\to\infty$. Now, recalling the definition of $c_{\bf Y}$, we conclude that

[TABLE]

and the corresponding lower bound can be derived in an analogous way for any small $\delta>0$.

Finally, we deal with the third term. First we observe that, by regular variation,

[TABLE]

Indeed, if $\mathbb{E}[\|{\mathbf{X}}\|^{\alpha_{Y}a_{\bf X}/a_{\bf Y}}]=\infty$ then $c_{\mathbf{Y}}=0$ and therefore the right-hand side is zero; see Lemma 1.2(2). On the other hand, if $\mathbb{E}[\|{\mathbf{X}}\|^{\alpha_{\bf Y}a_{\bf X}/a_{\bf Y}}]<\infty$ then

[TABLE]

and therefore the right-hand side in (3.9) is zero.

With (3.8) and Breiman’s result at hand, we have as $t\to\infty$,

[TABLE]

Now an application of (3.9) and the definition of $c_{\bf Y}$ yield

[TABLE]

This establishes an upper bound; the corresponding lower bound is completely analogous. This proves (3).

**Analysis of $J_{2}$. ** This term is significantly simpler since we have

[TABLE]

Appealing to dominated convergence theorem, we obtain

[TABLE]

Now monotone convergence yields

[TABLE]

∎

4. Applications

4.1. Products of regularly varying random matrices

In what follows, we consider an iid sequence of $d\times d$ random matrices $({\bf A}_{i})$ and we assume that a generic element ${\bf A}\in{\rm RV}(\alpha,\mu^{\bf A})$. We apply Theorem 2.1 to the function $\psi({\bf x},{\bf y})={\bf x}\cdot{\bf y}$.

Next we formulate our findings for a general product ${\mathbf{\Pi}}_{n}={\bf A}_{1}\cdots{\bf A}_{n}$, $n\geq 1$. Here and in what follows, we also use the notation

[TABLE]

where ${\rm Id}_{d}$ is the $d\times d$ identity matrix.

4.1.1. The case of non-equivalent tails

We first state the results in the case $\mathbb{P}(\|\mathbf{\Pi}_{n}\|>t)=o(\mathbb{P}(\|\mathbf{\Pi}_{n+1}\|>t))$ for all $n$. The complementary case is treated in Section 4.1.2.

Corollary 4.1**.**

Consider an iid sequence $({\bf A}_{i})$ of $d\times d$ matrices with ${\bf A}\in{\rm RV}(\alpha,\mu^{\bf A})$. Assume that

[TABLE]

Then for $n\geq 1$

[TABLE]

If \mathbb{P}(\|\mbox{\boldmath\Theta}_{{\bf A}_{1}}\cdots\mbox{\boldmath\Theta}_{{\bf A}_{n}}\|>0)>0 then ${\mathbf{\Pi}}_{n}$ is regularly varying and, as $t\to\infty$,

[TABLE]

In particular, if ${\bf A}$ is orthogonal,

[TABLE]

Remark 4.2**.**

In view of Lemma 1.2(2), (4.2) is satisfied if $\mathbb{E}[\|{\bf A}\|^{\alpha}]=\infty$.

Proof.

We proceed by induction. We will prove that for each $n$, (4.4), (4.3) and

[TABLE]

hold.

We start with $n=2$. In view of (4.2) by Theorem 2.1,

[TABLE]

In particular,

[TABLE]

where $Y$ has a Pareto distribution, $\mathbb{P}(Y>r)=r^{-\alpha}$, $r>1$, independent of the iid random variables \mbox{\boldmath\Theta}_{{\bf A}_{1}}, \mbox{\boldmath\Theta}_{{\bf A}_{2}}. This proves (4.3) for $n=2$. Hence

[TABLE]

We conclude from (4.6) that (4.4) indeed holds for $n=2$ since

[TABLE]

To prove (4.5) for $n=2$ we note that we have already established

[TABLE]

which, in combination with (4.2), constitutes that for any $M>0$ there exists $t_{0}$ sufficiently large such that

[TABLE]

Take $\eta=t_{0}^{-1}$. We observe as $t\to\infty$ that

[TABLE]

The last two lines yield

[TABLE]

and

[TABLE]

respectively. This proves (4.5) for $n=2$ and finishes the proof of the corollary for $n=2$.

Now suppose that it holds $n=k$ for some $k\geq 2$. Since (4.5) holds for $n=k$ the balance conditions

[TABLE]

are satisfied. An application of Theorem 2.1 yields

[TABLE]

An immediate consequence is

[TABLE]

where the Pareto random variable $Y$, \mbox{\boldmath\Theta}_{{\bf A}_{1}} and \mbox{\boldmath\Theta}_{\mathbf{\Pi}_{2,k+1}} are independent. Here we also used the induction assumption on the distribution of $\mathbf{\Pi}_{k}$. Therefore

[TABLE]

This proves (4.4) for $n=k+1$. Finally, we turn to (4.3) for $n=k+1$:

[TABLE]

In the last step we used the induction assumption leading to tail equivalence of $\|{\bf A}_{2}\|\,\|{\bf A}_{3}\cdots{\bf A}_{k+1}\|$ and $\|{\bf A}_{2}\,{\bf A}_{3}\cdots{\bf A}_{k+1}\|$ with factor \mathbb{E}[\|\mbox{\boldmath\Theta}_{{\bf A}_{1}}\cdots\mbox{\boldmath\Theta}_{{\bf A}_{k}}\|^{\alpha}]. To finish the proof we argue in favor of (4.5) for $n=k+1$ in the same fashion as we did that for $n=2$. More precisely, we have shown that

[TABLE]

which, in combination with (4.5) for $n=k$, gives $\mathbb{P}(\|\mathbf{\Pi}_{k}\|>t)=o(\|\mathbf{\Pi}_{k+1}\|>t))$. Consequently for any $M>0$ there exists $t_{0}$ sufficiently large such that

[TABLE]

On the other hand, $\mathbb{P}(\|{\bf A}_{1}\|>t)=o(\mathbb{P}(\|{\bf A}_{1}\|\cdot\|\mathbf{\Pi}_{2,k+1}\|>t))$ and

[TABLE]

Take $\eta=t_{0}^{-1}$. We observe as $t\to\infty$ that

[TABLE]

This proves $\mathbb{P}(\|\mathbf{\Pi}_{k+1}\|>t)=o(\mathbb{P}(\|{\bf A}_{1}\|\cdot\|\mathbf{\Pi}_{2,k+2}\|>t))$ and finishes the proof of the corollary. ∎

4.1.2. The case of tail-equivalent tails

We also assume condition (2.3) which turns into

[TABLE]

which is equivalent to

[TABLE]

An appeal to the following corollary shows that this condition causes tail equivalence of all $\mathbf{\Pi}_{n}$.

Corollary 4.3**.**

Consider an iid sequence $({\bf A}_{i})$ of $d\times d$ matrices such that ${\bf A}\in{\rm RV}(\alpha,\mu^{\bf A})$ and (4.7) holds. Then for any $n\geq 2$,

[TABLE]

Additionally, if \mathbb{P}(\|\mathbf{\Pi}_{k-1}\mbox{\boldmath\Theta}_{{\bf A}_{k}}\mathbf{\Pi}_{k+1,n}\|>0)>0 for some $k\leq n$ then ${\mathbf{\Pi}}_{n}$ is regularly varying and as $t\to\infty$,

[TABLE]

where

[TABLE]

Proof.

We proceed by induction. We will prove (4.8) and

[TABLE]

For $n=2$, Theorem 2.1 yields

[TABLE]

In particular, for a Pareto random variable $Y$ independent of ${\bf A}_{1},{\bf A}_{2}$ and \mbox{\boldmath\Theta}_{{\bf A}_{1}},\mbox{\boldmath\Theta}_{{\bf A}_{2}},

[TABLE]

We also have

[TABLE]

Now suppose that our claim holds for some $n\geq 2$. Put \tilde{c}_{n}=\sum_{k=1}^{n}\mathbb{E}\big{[}\|\mathbf{\Pi}_{k-1}\mbox{\boldmath\Theta}_{{\bf A}_{k}}\mathbf{\Pi}_{k+1,n}\|^{\alpha}\big{]}. Since $\|{\bf A}\|$ satisfies (4.7) and $\mathbb{P}(\|\mathbf{\Pi}_{n}\|>t)\sim\tilde{c}_{n}\,\mathbb{P}(\|{\bf A}\|>t)$ we infer that

[TABLE]

Theorem 2.1 yields

[TABLE]

Consequently, by the induction hypothesis,

[TABLE]

With this at hand, the convergence

[TABLE]

follows. ∎

4.2. Stochastic recurrence equations

We turn to the stochastic recurrence equation

[TABLE]

where \big{(}({\bf A}_{t},{\bf B}_{t})\big{)}_{t\in{\mathbb{Z}}} is an iid sequence with generic element $({\bf A},{\bf B})$, ${\bf A}$ is a $d\times d$ random matrix and ${\bf B}$ an ${\mathbb{R}}^{d}$-valued random vector, possibly dependent on each other. A solution $({\bf R}_{t})$ is causal if for every $t$, ${\bf R}_{t}$ is a function only of values \big{(}({\bf A}_{s},{\bf B}_{s})\big{)}_{s\leq t}, and then it constitutes a Markov chain. If a stationary causal solution $({\bf R}_{t})$ with generic element ${\bf R}$ exists its marginal distribution satisfies the fixed point equation in law

[TABLE]

and ${\bf R}$ has the representation in law

[TABLE]

The latter infinite series converges under conditions on the distribution of $({\bf A},{\bf B})$, for example $\mathbb{E}[\log\|{\bf A}\|]<0$ and $\mathbb{E}[\log_{+}\|{\bf B}\|]<\infty$. Under some mild integrability and non-degeneracy assumptions (4.11) is the unique solution to (4.10). Here and in what follows, we refer to the monograph Buraczewski et al. [5] for details concerning the existence, uniqueness and other properties of the solutions to (4.9) and (4.10).

The equations (4.9) and (4.10) have attracted a lot of attention since the seminal paper by Kesten [14] who proved that ${\mathbf{R}}$ has some regular variation property with tail index $\alpha>0$ given by

[TABLE]

If $d=1$, the latter equation reads as $\mathbb{E}[|A|^{\alpha}]=1$. In the Kesten setting, it is typically assumed that $\mathbb{E}[\|{\bf B}\|^{\alpha}]<\infty$ and $\mathbb{E}[\|{\bf A}\|^{\alpha}\log_{+}\|{\bf A}\|]<\infty$, implying the existence and uniqueness of the solution $({\bf R}_{t})$. Under these and further mild conditions on the distribution of $({\bf A},{\bf B})$ one has ${\bf R}\in{\rm RV}(\alpha,\mu^{\bf R})$ and the tail asymptotics

[TABLE]

Since $\mathbb{E}[\|{\bf R}\|^{\alpha}]=\infty$ we have $\mathbb{P}(\|{\bf B}\|>t)=o(\mathbb{P}(\|{\bf R}\|>t))$, and elementary calculations (Lemma C.3.1 in Buraczewski et al. [5]) show that for $\mu^{\bf R}$-continuity sets $C$,

[TABLE]

and the multivariate Breiman result Lemma C.3.1 in [5] yields

[TABLE]

Hence we have the identity

[TABLE]

Using induction on the recursion (4.9) and similar arguments, we find that

[TABLE]

This relation holds, in particular, if ${\bf A}$ is regularly varying with index $\alpha$ but the additional moment condition $\mathbb{E}[\|{\bf A}\|^{\alpha}\log_{+}\|{\bf A}\|]<\infty$ must be satisfied.

Regular variation of $({\bf R}_{t})$ may also arise from regular variation of ${\bf B}$ under the alternative conditions

[TABLE]

Then ${\mathbf{R}}$ is regularly varying with index $\alpha$ and

[TABLE]

where $\nu_{\mathbf{\Pi}}(\cdot)=\sum_{k=0}^{\infty}\mathbb{P}(\mathbf{\Pi}_{k}\in\cdot)$ ia s measure on $\mathbb{M}_{d\times d}$; see Theorem 4.4.24 in [5].

For our purposes we will treat $({\bf A},{\bf B})$ as a random element of ${\mathbb{M}}_{d\times d}\times{\mathbb{R}}^{d}$ equipped with the norm $\|(\bf a,\bf b)\|=\|\bf a\|+\|\bf b\|$, where $\|\bf a\|$ stands for the operator norm of the matrix $\bf a$ (with respect to the Euclidean distance) and $\|\bf b\|$ is the Euclidean norm of the vector $\bf b$. We assume that the following set of conditions $({\bf C})$ on $({\bf A},{\bf B})$ holds:

1. (C1)

A regular variation condition holds for some non-null Radon measure $\mu^{({\bf A},{\bf B})}$ on ${\mathbb{M}}_{d\times d}\times{\mathbb{R}}^{d}$:

[TABLE] 2. (C2)

$X=\|({\bf A}_{1},{\bf B}_{1})\|$ and $Y=\|({\bf A}_{2},{\bf B}_{2})\|$ satisfy (1.4). 3. (C3)

$\mathbb{E}[\|{\bf A}\|^{\alpha}]<1$ and \mu^{({\bf A},{\bf B})}\big{(}\big{\{}\mathbf{a},\mathbf{b})\>:\>\|\mathbf{a}\|>1\big{\}}\big{)}>0.

Some comments

• •

To the best of our knowledge, except for some univariate cases treated in Damek and Dyszewski [6] and Kevei [15], not much is known about regular variation of ${\bf R}$ under regular variation of ${\bf A}$ and (C3). Then (4.12) is violated since $\mathbb{E}[\|{\bf A}\|^{\alpha+\delta}]=\infty$ for any $\delta>0$.

• •

In view of Lemma 1.2 condition (C2) implies

[TABLE]

The following result is a multivariate counterpart of the results obtained in Damek and Dyszewski [6].

Theorem 4.4**.**

Assume $({\bf C})$. Then ${\bf R}$ given in (4.11) satisfies

[TABLE]

In particular, if the measure $\nu$ on $\mathbb{R}^{d}_{\bf 0}$ is non-null then {\mathbf{R}}\in{\rm RV}\big{(}\alpha,\mu^{\mathbf{R}}\big{)} with

[TABLE]

The remainder of this section is devoted to the proof of the theorem. A main step in the proof is provided by the following lemma.

Lemma 4.5**.**

Assume that the $\mathbb{R}^{d}$-valued random vector ${\bf X}\in{\rm RV}(\alpha,\mu^{\bf X})$ is independent of $({\bf A},{\bf B})$ which satisfies (C) and there is a positive constant $d_{{\bf X}}$ such that

[TABLE]

Then as $t\to\infty$,

[TABLE]

Proof of Lemma 4.5.

Write ${\mathbf{1}}_{d}=(1,\ldots,1)^{\top}\in\mathbb{R}^{d}$, ${\rm Id}_{d}$ and ${\rm diag}({\bf b})$, ${\bf b}\in\mathbb{R}^{d}$, in $\mathbb{M}_{d\times d}$ for the identity matrix and the diagonal matrix whose consecutive diagonal entries are the consecutive components of ${\bf b}$, respectively. We write

[TABLE]

Then $\widehat{\bf X}$ and $\widehat{\bf A}$ are both regularly varying. Indeed, for $\widehat{\bf X}$ we have

[TABLE]

For $\widehat{\bf A}$, choosing the operator norm $\|\cdot\|$, we have

[TABLE]

We intend to use the fact that

[TABLE]

in combination with Theorem 2.1 to prove the claim. In view of the tail equivalence condition (4.14) we have

[TABLE]

Therefore Theorem 2.1 yields

[TABLE]

which implies both claims.

∎

Consider the Markov chain $({\mathbf{R}}_{n}^{0})_{n\geq 0}$ given by the recursion (4.9) with ${\mathbf{R}}_{0}^{0}=0$. Then

[TABLE]

By Lemma 4.5,

[TABLE]

and the sequence $(\nu_{n})_{n\geq 0}$ of measures on $\mathbb{R}^{d}_{\bf 0}$ satisfies the recursive relation

[TABLE]

We have

[TABLE]

A copy $\widetilde{R}$ of $R$ which is also independent of $({\bf A},{\bf B})$ solves the equation

[TABLE]

and $\|{\mathbf{R}}_{n}^{0}\|\stackrel{{\scriptstyle d}}{{\leq}}R$, $n\geq 0$, where for any non-negative random variables $X,Y$, $X\stackrel{{\scriptstyle d}}{{\leq}}Y$ stands for stochastic domination, i.e., $\mathbb{P}(Y>t)\geq\mathbb{P}(X>t)$ for any $t>0$. From the main result in Damek and Dyszewski [6] (see Lemma A.1) we also have under (C),

[TABLE]

Lemma 4.6**.**

Assume (C). Then

[TABLE]

where $\nu$ is a Radon measure on $\mathbb{R}^{d}_{\bf 0}$.

Proof of Lemma 4.6.

For $k\leq n$ write $\mathbf{\Pi}^{\downarrow}_{n,k}={\bf A}_{n}{\bf A}_{n-1}\cdots{\bf A}_{k}$. We have by (4.2),

[TABLE]

Write

[TABLE]

We intend to show $\nu_{n}\stackrel{{\scriptstyle v}}{{\rightarrow}}\nu$ or, equivalently, $\int fd\nu_{n}\to\int fd\nu$ for any $f\in C_{c}^{+}(\mathbb{R}^{d}_{\bf 0})$. Then there are $c,M>0$ such that $f$ vanishes on $\{{\bf x}:\|{\bf x}\|>c\}$ and $f({\bf x})\leq M<\infty$. Our strategy is to use the following approximations:

[TABLE]

In what follows, we will make these approximations precise.

Approximations (1) and (3). For (1), we will show that

[TABLE]

For any $c>0$ and $k\leq[n/2]$ we have

[TABLE]

where we used (4.18) in the last step. Now (LABEL:eq:a) is immediate in view of condition $\mathbb{E}[\|{\bf A}\|^{\alpha}]<1$ and since $f\leq M$. The proof that $\nu$ is a Radon measure on ${\mathbb{R}}^{d}_{\bf 0}$ follows along the same lines. The proof of

[TABLE]

is an immediate consequence of this fact, proving (3).

Approximation (2). We have

[TABLE]

and we will show that the right-hand side converges to zero as $n\to\infty$. By uniform continuity of $f$,

[TABLE]

Let $(\mathbf{\Pi}_{i}^{\prime})$ be an independent copy of $(\mathbf{\Pi}_{i})$. For $[n/2]<k\leq n$ write

[TABLE]

Since $\mathbf{\Pi}^{\downarrow}_{n,k+1}\stackrel{{\scriptstyle d}}{{=}}\mathbf{\Pi}_{n-k}$ we have

[TABLE]

The following bounds hold

[TABLE]

Using the continuity of $f$, we also have

[TABLE]

These computations yield

[TABLE]

This bound yields that the right-hand side of (4.20) converges to zero by first letting $n\to\infty$ and then $\varepsilon\to 0$. ∎

Final steps in the proof of Theorem 4.4.

Choose $f\in C_{c}^{+}(\mathbb{R}_{\mathbf{0}}^{d})$ and fix constants $c,M>0$ such that (3.4) holds. By uniform continuity of $f$, we can choose $\varepsilon,\delta>0$ such that (4.21) holds. Write

[TABLE]

We have

[TABLE]

Both terms are asymptotically negligible. Indeed, for the first one,

[TABLE]

The right-hand side converges to zero by first letting $t\to\infty$ and then $n\to\infty$, also observing that $\mathbb{E}[\|{\bf A}\|^{\alpha}]<1$. For the second one, using (4.21),

[TABLE]

In view of Lemma 4.6 we may conclude that if we first take $t\to\infty$, then $n\to\infty$ followed by $\varepsilon\to 0$, we may conclude that

[TABLE]

Since $f$ is arbitrary the theorem follows. ∎

Appendix A

A.1. Proof of Lemma 1.2

(1) was proved in Embrechts and Goldie [9], p. 245. We start with (2). Observe that for any $M>0$, by the uniform convergence theorem for regularly varying functions,

[TABLE]

If $\mathbb{E}[Y^{\alpha}]=\infty$ we can make the right-hand side arbitrarily large by letting $M\to\infty$.

We continue with (3). We follow the lines of the proof of Proposition 3.1 in Davis and Resnick [8] who consider the case of iid $X,Y$. Choose any $M>1$. Then

[TABLE]

In view of (1.4), $I_{2}(t)/\mathbb{P}(X>t)$ is asymptotically negligible when first $t\to\infty$ and then $M\to\infty$. In view of Breiman’s Lemma 1.1 we have as $t\to\infty$,

[TABLE]

where $c_{0}=\lim_{t\to\infty}\mathbb{P}(Y>t)/\mathbb{P}(X>t)$ is assumed finite. Now the desired result follows when $M\to\infty$.

A.2. A result from [6]

Lemma A.1**.**

Assume that $\|{\bf A}\|$ is regularly varying with index $\alpha>0$, $\mathbb{E}[\|{\bf A}\|^{\alpha}]<1$, $\mathbb{P}(\|{\bf B}\|>t)=O(\mathbb{P}(\|{\bf A}\|>t))$, and

[TABLE]

Then $R=\sum_{k=0}^{\infty}\|{\bf B}_{k+1}\|\,\prod_{j=1}^{k}\|{\bf A}_{j}\|$ is finite and satisfies $\mathbb{P}(R>t)=O(\mathbb{P}(\|{\bf A}\|>t))$ as $t\to\infty$. In particular, $\mathbb{E}[R^{\alpha}]<\infty$.