The central limit theorem for eigenvalues

TL;DR

This paper proves a central limit theorem for the spectral radius of certain random walks on groups, showing that the distribution of eigenvalues' moduli converges to a normal distribution under specific conditions.

Contribution

It establishes a CLT for eigenvalues of random walks on reductive groups, extending previous results to broader classes under minimal moment assumptions.

Findings

Spectral radius follows a normal distribution asymptotically.

Results apply to Zariski-dense random walks on reductive groups.

Minimal moment conditions are sufficient for the CLT.

Abstract

We prove that the spectral radius of a strongly irreducible random walk on GLd(R) (or more generally the vector of moduli of eigenvalues of a Zariski-dense random walk on a reductive group) satisfies a central limit theorem under an order two moment assumption.

Equations

$$\int l(g{)}^{i}d\mu (g)<+\infty .$$

$${\lambda }_1(\mu )=\underset{n\to +\infty }{\lim }\frac{1}{n}\ln ||L_n||.$$

$$\frac{\ln \rho (L_n)-n{\lambda }_1(\mu )}{\sqrt{n}}\underset{n\to +\infty }{\stackrel{\mathcal{L}}{⟶}}\mathcal{N}(0,{\sigma }_{\mu }).$$

$$\limsup_{n\to+\infty}\mathbb{P}\left(\frac{\rho(L_{n})}{||L_{n}||}\leq\epsilon\right)\underset{\epsilon\to 0}{\big{\downarrow}}0.$$

$$\mathbb{P}\left(\frac{\rho (L_n)}{||L_n||}\le {ϵ}_n\right)\underset{n\to +\infty }{⟶}0.$$

$$\left({\rho }_1(L_n)-n{\lambda }_1(\mu ),\cdots ,{\rho }_d(L_n)-n{\lambda }_d(\mu )\right)/\sqrt{n}$$

$$v_g^{+}=k_g[e_1],H_g^{-}=[\ker (u_g^{-1}{e}_1^{\ast })]={\left(\mathbb{R}(u_g^{-1}{e}_1)\right)}^{\perp }.$$

$$\forall x=[v],y=[w]\in \text{P}(V),\delta (x,y):=\frac{||v\wedge w||}{||v||||w||}.$$

$$\limsup_{n\to+\infty}\mathbb{P}\left(\delta(v_{L_{n}}^{+},H_{L_{n}}^{-})\leq\epsilon\right)\underset{\epsilon\to 0}{\big{\downarrow}}0.$$

$$\mathbb{P}\left(\delta (v_{L_n}^{+},H_{L_n}^{-})\le {ϵ}_n\right)\underset{n\to +\infty }{⟶}0.$$

$$\frac{\rho (g)}{||g||}\ge \frac{\delta (v_g^{+},H_g^{-})}{2}.$$

$$g\cdot U_{ϵ}\subseteq \overline{B(v_g^{+},r)}\subset B\left(v_g^{+},\frac{a_2(g)}{a_1(g)}\frac{1}{ϵ}\right).$$

$$\forall v\in V\setminus \{0\},{\left[\delta ([v],H_g^{-})\right]}^2\le {\left(\frac{||gv||}{||g||||v||}\right)}^2\le {\left[\delta ([v],H_g^{-})\right]}^2+{\left(\frac{a_2(g)}{a_1(g)}\right)}^2.$$

$$\forall [v]\in U_{ϵ},\frac{||gv||}{||g||||v||}\ge ϵ.$$

$$\underset{x,y\in U_{ϵ}}{\sup }\frac{\delta \left(g\cdot x,g\cdot y\right)}{\delta (x,y)}\le \frac{a_2(g)}{a_1(g)}\frac{1}{{ϵ}^2}.$$

$$\mathbb{P}\left(\frac{\rho (L_n)}{||L_n||}\le ϵ\right)\le \mathbb{P}\left(\frac{\rho (\pi (L_n))}{||\pi (L_n)||}\le C{ϵ}^p\right).$$

$${\Omega }_n:=\left\{\omega \in \Omega ;{\delta }_n^2(\omega )>4\frac{a_{2,n}(\omega )}{a_{1,n}(\omega )}\right\}.$$

$$\mathbb{P}({\Omega }_n)\underset{n\to +\infty }{⟶}1.$$

$$\frac{1}{n}\ln \frac{a_{2,n}}{a_{1,n}}\underset{n\to +\infty }{\stackrel{\mathbb{P}}{⟶}}{\lambda }_2-{\lambda }_1.$$

$$\mathbb{P}\left(\frac{a_{2,n}}{a_{1,n}}\le \exp (-n\gamma )\right)\underset{n\to +\infty }{⟶}1.$$

$$\mathbb{P}\left(\frac{\rho (L_n)}{||L_n||}\le ϵ\right)\le \mathbb{P}(\Omega \setminus {\Omega }_n)+\mathbb{P}\left(\delta (v_n^{+},H_n^{-})\le 2ϵ\right).$$

$$\underset{n\to +\infty }{\mathrm{limsup}}\mathbb{P}\left(\frac{\rho (L_n)}{||L_n||}\le ϵ\right)\le \underset{n\to +\infty }{\mathrm{limsup}}\mathbb{P}\left(\delta (v_n^{+},H_n^{-})\le 2ϵ\right).$$

$$Y_n:=\frac{1}{\sqrt{n}}\ln \frac{||L_n||}{\rho (L_n)}\underset{n\to +\infty }{\stackrel{\mathbb{P}}{⟶}}0.$$

$$\limsup_{n\to+\infty}\mathbb{P}(\|\kappa(L_{n})-\ell(L_{n})\|>M)\underset{M\to+\infty}{\big{\downarrow}}0.$$

$$\frac{\ell (L_n)-n\overrightarrow{{\lambda }_{\mu }}}{\sqrt{n}}\underset{n\to +\infty }{\stackrel{\mathcal{L}}{⟶}}\mathcal{N}(0,K_{\mu }),$$

$$\underset{\text{H projective hyperplane of P(V)}}{\sup }\nu \{x\in \text{P}(V);\delta (x,H)\le t\}\underset{t\to 0}{⟶}0.$$

$$\underset{\text{H projective hyperplane of P(V)}}{\sup }{\int }_{\text{P}(V)}\varphi \left(\frac{1}{\delta (x,H)}\right)d\nu (x)<+\infty .$$

$$\nu \left\{x\in \text{P}(V);\delta (x,H_n)\le \frac{1}{n}\right\}>{ϵ}_0.$$

$$\Big{|}\delta([x],[\ker f])-\delta([x],[\ker f^{\prime}])\Big{|}=\Big{|}\,\frac{|f(x)|}{||x||}-\frac{|f^{\prime}(x)|}{||x||}\,\Big{|}\leq\min\{||f-f^{\prime}||,||f+f^{\prime}||\}\leq\sqrt{2}\,\delta([f],[f^{\prime}]).$$

$$\forall k\in \mathbb{N},\nu \{x\in \text{P}(V);\delta (x,H)\le a_k\}<e^{-k}.$$

Full text

Keywords: Random matrix products, Lyapunov exponents, Stationary measures, Central limit theorem

AMS classification: 60B15, 60F05, 37H15, 20P05.

1. 1 Statement of the results
2. 2 Preliminary reduction
3. 3 Estimates with a moment of order one
4. 4 End of the proof of Theorem

1 Statement of the results

Let $V$ be a real vector space of dimension $d\geq 1$ and $||\cdot||$ a norm on $V$. For simplicity of notation, the operator norm on $\textrm{End}(V)$ is also denoted by $||\cdot||$. For every $g\in\textrm{GL}(V)$, we denote by $\rho(g)$ the spectral radius of $g$ and we let $l(g):=\max\{\ln^{+}||g||,\ln^{+}||g^{-1}||\}$ with $x^{+}=\textrm{max}\{x,0\}$. If $\mu$ is a Borel probability measure on $\textrm{GL}(V)$, we say that $\mu$ has moment of order $i\in\mathbb{N}$ if

[TABLE]

The right (resp.  left) random walk at time $n\in\mathbb{N}$ will be denote by $R_{n}=X_{1}\cdots X_{n}$ (resp.  $L_{n}=X_{n}\cdots X_{1}$) where $(X_{i})_{i\in\mathbb{N}^{*}}$ is a family of independent and identically distributed $\textrm{GL}(V)$-valued random variables of law $\mu$. All our random variables will be defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and the expectation operator is denoted by $\mathbb{E}$.

When $\mu$ has a moment of order one, we denote by $\lambda_{1}(\mu)$ the top Lyapunov exponent of $\mu$, i.e.

[TABLE]

This convergence holds almost surely and is a simple consequence of Kingman’s subadditive ergodic theorem (it was first proved by Furstenberg and Kesten [FK60] prior to Kingman’s theorem). We denote by $\Gamma_{\mu}$ the semi-group generated by the support of $\mu$. We say that $\Gamma_{\mu}$ is strongly irreducible if it does not stabilize a finite union of non-trivial subspaces of $V$.

The convergence (1) can be thought as a law of large numbers for the non-commutative random product $L_{n}$. A corresponding central limit theorem for $\ln||L_{n}||$ has been established long ago under the assumption of strong irreducibility of $\Gamma_{\mu}$ and an exponential moment assumption on $\mu$ ([LP82], [GG96], [BL85]). Recently, Benoist–Quint gave in [BQ16a, Theorem 1.1] another proof of the CLT, which is valid under the optimal moment hypothesis: namely, that of a moment of order two. Our first main result gives the analogous statement for $\rho(L_{n})$, namely

Theorem 1.1**.**

Let $\mu$ be a probability measure on $\textrm{GL}(V)$ such that:

• •

$\mu$* has a moment of order two,*

• •

$\Gamma_{\mu}$* is strongly irreducible and has unbounded image in $\textrm{PGL}(V)$.*

Then there exists $\sigma_{\mu}>0$ such that the following convergence in law holds:

[TABLE]

This theorem is established in [BQ16c, Theorem 13.22] under a more restrictive exponential moment condition. The main contribution of this note is to establish it under the optimal order two moment assumption.

Remark 1.2**.**

It will follow from the proof that the limit distribution $\mathcal{N}(0,\sigma_{\mu})$ is the same as the limiting distribution of $\frac{\ln||L_{n}||-n\lambda_{1}}{\sqrt{n}}$.

Remark 1.3**.**

As we show in Example 4.2, the sequence of random variables $\frac{\ln\rho(L_{n})-n\lambda_{1}(\mu)}{\sqrt{n}}$ may fail to converge in distribution if $\Gamma_{\mu}$ is not assumed to be strongly irreducible. Note that even if it exists, the limit distribution is not necessarily Gaussian as one can see by considering a random walk on diagonal matrices. It is worth mentioning that, on the contrary, $\ln\rho(L_{n})$ satisfies always a law of large numbers (without any algebraic assumption on the support of $\mu$) as is recently shown in [AS19].

In view of the known CLT for $\ln||L_{n}||$, the proof of Theorem 1.1 reduces to proving that $\frac{1}{\sqrt{n}}\ln\frac{\rho(L_{n})}{||L_{n}||}$ converges in probability to zero when $\mu$ has a moment of order two. We will actually give estimates of the ratio $\frac{\rho(L_{n})}{||L_{n}||}$ with only the assumption of a moment of order one for $\mu$. The main technical result of this note is therefore the following:

Theorem 1.4**.**

Let $\mu$ be a probability measure on $\textrm{GL}(V)$ such that:

• •

$\mu$* has a moment of order one,*

• •

$\Gamma_{\mu}$* is strongly irreducible.*

Then

[TABLE]

Equivalently, for every numerical sequence $(\epsilon_{n})_{n\in\mathbb{N}^{*}}$ that tends to zero,

[TABLE]

Remark 1.5**.**

The speed of convergence when $\epsilon\to 0$ in (2) depends on the regularity of the unique stationary probability measure on the projective space of some strongly irreducible and proximal representation. This is formulated in Theorem 3 which is a more precise statement than the one given in Theorem 3.

In a similar fashion, using the various wedge power representations of $\textrm{SL}_{d}(\mathbb{R})$, we can deduce easily from Theorem 3 and from Benoist–Quint’s central limit for the Cartan projection a CLT for the full vector of eigenvalues. Namely:

Theorem 1.6**.**

Let $\mu$ be a probability measure on $\textrm{SL}_{d}(\mathbb{R})$ with a moment of order $2$. Assume that its support generates a Zariski-dense subgroup. Then there is a positive definite quadratic form $K_{\mu}$ on the hyperplane $\{(x_{1},\cdots,x_{d})\in\mathbb{R}^{d};\sum_{1}^{d}x_{i}=0\}$ such that the random vector

[TABLE]

converges in law to the multidimensional gaussian centered gaussian distribution $\mathcal{N}(0,K_{\mu})$. Here $\rho_{1}(L_{n})\geq\cdots\geq\rho_{d}(L_{n})>0$ denote the moduli of the eigenvalues of $L_{n}$ in decreasing order, and $\lambda_{i}(\mu):=\underset{n\rightarrow+\infty}{\lim}\frac{1}{n}\ln\frac{||\bigwedge^{i}L_{n}||}{||\bigwedge^{i-1}L_{n}||}$ is the $i$-th Lyapunov exponent.

Recall that a subgroup of $\textrm{SL}_{d}(\mathbb{R})$ is called Zariski-dense if it is contained in no proper real algebraic subgroup of $\textrm{SL}_{d}(\mathbb{R})$. Theorem 1.6 will be proved in a more general setting (see Theorem 2.3), that of random walks on Zariski dense sub-semigroups of reductive groups.

Remark 1.7**.**

Under the assumption of Theorem 1.6, we have $\lambda_{1}(\mu)>\cdots>\lambda_{d}(\mu)$ as it follows from the combination of Guivarc’h–Raugi’s theorem [GR85] and Goldsheid–Margulis’s one [GM89] (see also Benoist–Labourie [BL93] and Prasad [PR03]). In particular, by Theorem 1.6, all the eigenvalues of $L_{n}$ are real with a probability tending to one.

Remark 1.8**.**

Theorems 3, 1.1 and 1.6 are also valid for $R_{n}$ verbatim, since $R_{n}$ and $L_{n}$ have the same law for every $n$.

Remark 1.9**.**

When the field $\mathbb{R}$ is replaced by another local field (as $\mathbb{C}$ or a p-adic field for instance), Theorem 3 remains true verbatim with the same proof. The limiting distributions provided in Theorem 1.1 and 1.6 exist, remain Gaussian but can be however degenerate. See [BQ16c, Example 12.21] for more details.

Acknowledgements

The author has the pleasure to thank Emmanuel Breuillard and Ça$\check{\text{g}}$rı Sert for fruitful discussions. Part of this project was sponsored by the Center of Advanced Mathematical Sciences (CAMS).

2 Preliminary reduction

In this section, we reduce the proofs of Theorem 3 and Theorem 1.1 to Theorem 2.1 below which says essentially that the attracting point of $L_{n}$ is fairly far from its repelling hyperplane.

First, we introduce some notation. We set $V=\mathbb{R}^{d}$. Let $\textrm{P}(V)$ be the projective space of $V$. For every non zero vector $v$ (resp.  non zero subspace $E$) of $V$, we denote by $[v]=\mathbb{R}v$ (resp. $[E]$) its projection to $\textrm{P}(V)$. The action of $g\in\textrm{GL}(V)$ on a vector $v$ will be simply denoted by $gv$, while the action of $g$ on a point $x\in\textrm{P}(V)$ will be denoted by $g\cdot x$.

We endow $V$ with the canonical basis $(e_{1},\cdots,e_{d})$ and the usual Euclidean dot product and norm. Let $K=O_{d}(\mathbb{R})$ be the orthogonal group. Denote by $A\subset\textrm{GL}_{d}(\mathbb{R})$ the subgroup of diagonal matrices and $A^{+}\subset A$ the sub-semigroup made of matrices with positive entries and arranged in decreasing order. The KAK decomposition (or the singular value decomposition) states that $\textrm{GL}_{d}(\mathbb{R})=KA^{+}K$. For every $g\in\textrm{GL}(V)$, we denote by $g=k_{g}a(g)u_{g}$ a KAK decomposition of $g$ in the basis $(e_{1},\cdots,e_{d})$ of $V$. We write $a(g)=\left(a_{1}(g),\cdots,a_{d}(g)\right)$ with $a_{1}(g)\geq\cdots\geq a_{d}(g)>0$. Even though $k_{g}$ and $u_{g}$ are not uniquely defined, we can always fix once for all a privileged choice of a KAK decomposition. We call *attracting point * and repelling hyperplane the following respective point in $\textrm{P}(V)$ and projective hyperplane of $\textrm{P}(V)$:

[TABLE]

In the definitions above, $(e_{1}^{*},\cdots,e_{n}^{*})$ denotes the dual basis of $(e_{1},\cdots,e_{n})$ in the dual vector space $V^{*}$ of $V$. Also $\textrm{GL}(V)$ acts on $V^{*}$ by $(gf)(x)=f(g^{-1}x)$, $g\in\textrm{GL}(V)$, $f\in V^{*}$ and $x\in V$.

Endow the vector space $\bigwedge^{2}V$ with the canonical norm associated to the basis $(e_{i}\wedge e_{j})_{1\leq i<j\leq d}$. We endow $\textrm{P}(V)$ with the standard metric $\delta$ defined by:

[TABLE]

This is just the sine of the angle between the lines $x=\mathbb{R}v$ and $y=\mathbb{R}w$. Finally, an endomorphism $g\in\textrm{End}(V)$ is said to be proximal if it has a unique eigenvalue with maximal modulus and a sub-semigroup $\Gamma$ of $\textrm{GL}(V)$ is said to be proximal if it contains a proximal element.

We are now able to state our main technical result:

Theorem 2.1**.**

Assume that $\mu$ has a moment of order one and that $\Gamma_{\mu}$ is strongly irreducible and proximal. Then

[TABLE]

Equivalently, for any sequence of real numbers $(\epsilon_{n})_{n}$ such that $\epsilon_{n}\underset{n\rightarrow+\infty}{\longrightarrow}0$,

[TABLE]

In order to deduce Theorem 3 from Theorem 2.1, we need the following geometric lemma. It is borrowed from Benoist-Quint [BQ16c, Lemma 13.14]. For the convenience of the reader, we include a proof.

Lemma 2.2**.**

Let $g\in\textrm{GL}(V)$. If $\delta(v_{g}^{+},H_{g}^{-})>2\sqrt{\frac{a_{2}(g)}{a_{1}(g)}}$, then

[TABLE]

Moreover, in this case, $g$ is necessarily a proximal element.

Proof.

Fix $g\in\textrm{GL}(V)$. To simplify the notation, let $\delta_{g}:=\delta(v_{g}^{+},H_{g}^{-})$. For every $\epsilon>0$, let $U_{\epsilon}\subseteq\textrm{P}(V)$ be the complement of the closed $\epsilon$-neighborhood around $H_{g}^{-}$, i.e.  $U_{\epsilon}:=\{x\in\textrm{P}(V);\delta(x,H_{g}^{-})>\epsilon\}$. The following statements are easy to verify using the definition of the Cartan decomposition and the standard metric $\delta$ (except the statement i.  which directly follows from the triangle inequality)

1. i.

$B(v_{g}^{+},\delta_{g}/2)\subset U_{\delta_{g}/2}$, where $B(x,r)$ refers to the open ball of center $x\in\textrm{P}(V)$ and radius $r$ in the metric space $\left(\textrm{P}(V),\delta\right)$. 2. ii.

For every $\epsilon>0$, and for $r:=\left(1+\frac{a_{1}(g)^{2}}{a_{2}(g)^{2}}\epsilon^{2}\right)^{-1/2}$, one has that

[TABLE] 3. iii.

[TABLE]

In particular,

[TABLE] 4. iv.

For every $\epsilon>0$,

[TABLE]

Since the family $(U_{\epsilon})_{\epsilon>0}$ is decreasing, we deduce from observations i.  and ii.  above that $\overline{g\cdot U_{\epsilon}}\subset U_{\epsilon}$ as soon as $\frac{a_{2}(g)}{a_{1}(g)}\frac{2}{\delta_{g}}\leq\epsilon\leq\frac{\delta_{g}}{2}$. From now, we assume that $\delta_{g}>2\sqrt{\frac{a_{2}(g)}{a_{1}(g)}}$ and we set $\epsilon=\delta_{g}/2$. With these assumptions, inequality iv.  implies that the action of $g$ on the complete metric space $\overline{U_{\epsilon}}$ is contracting. Thus $g$ has a unique fixed point $x_{g}^{+}$ in $\overline{g\cdot U_{\epsilon}}\subset U_{\epsilon}$. This fixed point provides an eigenvalue $\lambda$ of $g$ whose direction is given by $x_{g}^{+}$. By (4), we have $\frac{|\lambda|}{||g||}\geq\epsilon=\delta_{g}/2$. A fortiori, the spectral radius $\rho(g)$ of $g$ satisfies the desired inequality. This proves the desired lower bound. By Tits converse lemma (see e.g. [Bre08, Lemma 4.7]), we have that $g$ is proximal and that the unique fixed point $x_{g}^{+}$ of $g$ in $\textrm{P}(V)$ corresponding to the top eigenvalue belongs to $U_{\epsilon}$. ∎

Proof of Theorem 3 modulo Theorem 2.1:.

First, we show that we can assume without loss of generality that $\Gamma_{\mu}$ is strongly irreducible and proximal (i-p to abbreviate). Indeed, let $p\in\{1,\cdots,d\}$ be the proximality index of $\Gamma_{\mu}$, i.e.  the least integer $k\in\{1,\cdots,d\}$ such that there exists a sequence of scalars $\lambda_{n}\in\mathbb{R}$ and of elements $g_{n}\in\Gamma_{\mu}$ such that $\lambda_{n}g_{n}$ converges in $\textrm{End}(V)$ to a endomorphism of rank $k$. By [BQ16a, Lemma 4.13 ] there exist a $\Gamma_{\mu}$-invariant subspace $W$ of $\bigwedge^{p}V$ such that such that the action of $\Gamma_{\mu}$ on $W$ is i-p and such that $\{\frac{||g||^{p}}{||\pi(g)||};g\in\Gamma_{\mu}\}$ is bounded, where $\pi:\Gamma_{\mu}\to\textrm{GL}(W)$ is the restriction representation222In positive characteristic, by [BQ16a, Lemma 4.13], one has to replace $W$ by $W/U$ for some subspace $U$ of $W$ and $\pi$ by the representation on $W/U$. With these modifications, (5) remains true. . Let $C:=\sup\left\{\frac{||g||^{p}}{||\pi(g)||};g\in\Gamma_{\mu}\right\}\in[1,+\infty)$. Since $\rho\left(\pi(g)\right)\leq\rho(\wedge^{p}g)\leq\rho(g)^{p}$, then for every $\epsilon>0$ and $n\in\mathbb{N}$,

[TABLE]

Thus proving Theorem 3 for $\Gamma_{\pi(\mu)}$ is enough to prove the same estimate for $\Gamma_{\mu}$.

For now on $\Gamma_{\mu}$ is assumed to be i-p. For every $n\in\mathbb{N}$, let $L_{n}=k_{n}a_{n}u_{n}$ be a KAK decomposition of $L_{n}$, $v_{n}^{+}$ the attracting point of $L_{n}$, $H_{n}^{-}$ its repelling hyperplane and $\delta_{n}:=\delta(v_{n}^{+},H_{n}^{-})$. Let $\Omega_{n}\subseteq\Omega$ be the following event

[TABLE]

First, we check that

[TABLE]

Indeed, by definition of the Lyapunov exponents, the following convergence holds in probability:

[TABLE]

Hence for $\gamma:=(\lambda_{1}(\mu)-\lambda_{2}(\mu))/2$,

[TABLE]

Since $\Gamma_{\mu}$ is i-p, Guivarc’h-Raugi’s theorem [GR85] ensures that $\gamma>0$. Applying now Theorem 2.1 for $\epsilon_{n}=3\exp(-n\gamma/2)$, gives that with probability tending to one, $\delta_{n}^{2}>9\exp(-n\gamma)>4\frac{a_{2,n}}{a_{1,n}}$, i.e. $\mathbb{P}(\Omega_{n})\underset{n\rightarrow+\infty}{\longrightarrow}1$.

Let now $\epsilon>0$. By Lemma 2.2, we have for every $n\in\mathbb{N}^{*}$,

[TABLE]

Tending $n\rightarrow+\infty$ and using (6), we deduce that for every $\epsilon>0$,

[TABLE]

Applying Theorem 2.1, we deduce that the quantity above converges to zero as $\epsilon\rightarrow 0$. ∎

We easily deduce the proof of Theorem 1.1. We recall the classical Slutsky’s lemma in probability theory which asserts that if $(X_{n})_{n\in\mathbb{N}}$ and $(Y_{n})_{n\in\mathbb{N}}$ are two sequences of random variables such that $(X_{n})_{n}$ converges in law to a random variable $X$ and $(Y_{n})_{n\in\mathbb{N}}$ converges in probability to a constant $c\in\mathbb{R}$, then the joint vector $(X_{n},Y_{n})$ converges in law to $(X,c)$; a fortiori $X_{n}+Y_{n}$ converges in law towards $X+c$.

Proof of Theorem 1.1:.

By Benoist–Quint’s central limit theorem [BQ16a, Theorem 1.1] for $\ln||L_{n}||$ and Slutsky’s lemma, all we need to show is the following convergence in probability:

[TABLE]

This convergence is guaranteed by Theorem 3 (using (3)). ∎

We end this section by stating and proving a general version of Theorem 1.6, using the language of reductive groups. Before stating the result, we recall standard notion of reductive groups (we refer for instance to [Kna02]). Let $\mathbf{G}$ be a linear reductive algebraic group assumed to be Zariski connected and denote by $G=\mathbf{G}(\mathbb{R})$ its group of real points. We denote by $K$ a maximal compact subgroup of $G$, $\mathfrak{a}$ the Lie algebra of a maximal $\mathbb{R}$-split torus $A$ with $\mathfrak{a}^{+}$ a positive Weyl chamber, i.e.  the cone in $\mathfrak{a}$ defined by the requirement that all positive roots take non-negative values. Let $A^{+}=\exp\left(\mathfrak{a}^{+}\right)$. One has that $G=KA^{+}K$ called Cartan or KAK decomposition. The $A^{+}$-component of an element of $G$ in this product is unique. This yields the so-called Cartan projection $\kappa:G\longrightarrow\mathfrak{a}^{+}$.

Recall also the Jordan decomposition: any $g\in G$ can be written as a commuting product of a unipotent element, an elliptic element and a hyperbolic element (i.e. an element with a conjugate in $A$). One can then define the Jordan projection $\ell:G\longrightarrow\mathfrak{a}^{+}$ where $\ell(g)$ is the unique element of $\mathfrak{a}^{+}$ such that $\exp(\ell(g))$ is conjugate to the hyperbolic part of $g$ in the Jordan decomposition of $g$. These are projections of the linear group $G$ which encode the information of the moduli of eigenvalues and operator norms on certain linear representations of $G$.

Let now $\mu$ be a probability measure on $G$. We say that $\mu$ has a moment of order $p\geq 1$ if for some, or equivalently any, faithful linear representation $\phi:G\longrightarrow\textrm{GL}_{n}(\mathbb{R})$ of $G$, $\phi(\mu)$ has a moment of order $p$. Let $(L_{n})_{n\geq 1}$ be the left random walk on $G$ associated to $\mu$. The equivalent formulation of (1) reads as follows: when $\mu$ has a moment of order one, the vector $\frac{\kappa(L_{n})}{n}$ converges almost surely to a non random element $\overset{\rightarrow}{\lambda_{\mu}}\in\mathfrak{a}^{+}$, called the Lyapunov vector of $\mu$.

Theorem 2.3**.**

(Generalization of Theorem 1.6) Let $\mathbf{G}$ be a reductive real algebraic group, $G$ its group of real points, $\Gamma$ a Zariski dense sub-semigroup of $G$. Consider a probability measure $\mu$ on $\Gamma$ whose support generates $\Gamma$. Assume that $\mu$ has a moment of order one. Then,

[TABLE]

Moreover, if $\mu$ has a moment of order two, then the following convergence in law holds:

[TABLE]

where $\mathcal{N}(0,K_{\mu})$ is a multidimensional gaussian centered gaussian distribution. Its support is a vector subspace $\mathfrak{a}_{\mu}$ of $\mathfrak{a}$ which contains the intersection of $\mathfrak{a}$ with the Lie subalgebra of the derived group of $\mathbf{G}$. In particular $\mathfrak{a}_{\mu}=\mathfrak{a}$ when $\mathbf{G}$ is semisimple.

Proof.

Let $d$ be the real rank of $\mathbf{G}$ and $d_{S}$ its semisimple rank. There exists a basis $\{\chi_{1},\cdots,\chi_{d}\}$ of the dual $\mathfrak{a}^{*}$ such that each $\chi_{i}$ is a highest weight of some irreducible representation $V_{i}$ of $\mathbf{G}$ (all of these representations are also strongly irreducible by Zariski connectedness of $\mathbf{G}$). Indeed, it is enough to concatenate the $d_{S}$ fundamental weights to $d-d_{S}$ characters of the abelianization $\mathbf{G}/[\mathbf{G},\mathbf{G}]$ of $\mathbf{G}$. But if $(\psi,V)$ is an irreducible representation of $\mathbf{G}$ and $\chi$ is a highest weight, then by using Mostow’s theorem [Mos55], we can find a norm $||\cdot||$ (depending on $\psi$ and $V$) on each $V$ such that for every $g\in\mathbf{G}(\mathbb{R})$, $\chi(\kappa(g))=\ln(||\psi(g)||)$ and $\chi(\ell(g))=\ln\left(\rho(\psi(g))\right)$. Applying now Theorem 3 on each $(V_{i},\rho_{i})$ proves (9).

We deduce (10) from (9) in the same way we deduced Theorem 1.1 from Theorem 3, i.e. using (9), Slutsky’s lemma and Benoist–Quint’s central limit theorem for the Cartan projection [BQ16a, Theorem 4.16]. ∎

Remark 2.4**.**

The Lyapunov vector lies actually in the open Weyl chamber $\mathfrak{a}^{++}$. This is well-known and follows from the combination of Guivarc’h–Raugi’s theorem [GR85] on the simplicity of the Lyapunov spectrum together with Goldsheid–Margulis’s result [GM89] (see also Benoist–Labourie [BL93] and Prasad [PR03]) concerning the existence of proximal elements in Zariski dense subgroups of real algebraic groups with proximal elements.

Remark 2.5**.**

Assume now that $\mu$ has a moment of order two. Breuillard and Sert refined recently the previous result. Indeed, they proved in [BS18, Theorem 1.9] that $\overset{\rightarrow}{\lambda_{\mu}}$ lies in the interior of the Benoist cone of $\Gamma$ introduced by Benoist [Ben97] (it is the closure in $\mathfrak{a}^{+}$ of the positive linear combinations of $\ell(g)$, $g\in\Gamma$).

3 Estimates with a moment of order one

In this section, we provide some qualitative estimates concerning the behavior of the random walk with a moment of order one. In particular, item 5 of Proposition 3.3 is the analog of Theorem 2.1, but with $H_{L_{n}}^{-}$ replaced by a deterministic hyperplane $H$. The uniformity of our constants in the hyperplane $H$ will be crucial as showed in the next section, where we fully prove Theorem 2.1. The regularity of the stationary measure on projective space (Lemma 12 below) is a crucial ingredient. We also discuss what is known about the speed of convergence in Remark 3.2 and Remark 3.5, when $\mu$ has higher order moments.

We recall that if $\mu$ is a probability measure on $\textrm{GL}_{d}(\mathbb{R})$, then a probability measure $\nu$ on $\textrm{P}(V)$ is said to be $\mu$-stationary if for every continuous function $f$ on $\textrm{P}(V)$, $\int_{\textrm{P}(V)}{fd\nu}=\iint_{\textrm{GL}(V)\times\textrm{P}(V)}{f(g\cdot x)d\mu(g)d\nu(x)}$.

When $\Gamma_{\mu}$ is strongly irreducible, classical arguments of Furstenberg show that any $\mu$-stationary probability measure on $\textrm{P}(V)$ is non-degenerate, i.e.  $\nu(H)=0$ for every projective hyperplane $H$ (see [BL85, Chapter III, Proposition 2.3]). Now, if $\Gamma_{\mu}$ is strongly irreducible and proximal, Guivarc’h and Raugi proved in [GR85] that there exists a unique $\mu$-stationary probability measure $\nu$ on $\textrm{P}(V)$. The next lemma gives information on the regularity of stationary measures.

Lemma 3.1**.**

Assume that $\mu$ has a moment of order one and that $\Gamma_{\mu}$ is strongly irreducible. Let $\nu$ be any $\mu$-stationary probability measure on $\textrm{P}(V)$. Then,

[TABLE]

Equivalently, there exists a proper map $\phi:[1,+\infty)\longrightarrow[1,+\infty)$ such that

[TABLE]

Proof.

Let $\nu$ be a $\mu$-stationary probability measure on $\textrm{P}(V)$. If (11) was not true, then there would exist some $\epsilon_{0}>$ and a sequence of projective hyperplanes $H_{n}:=[\ker(f_{n})]$ such that for every $n\in\mathbb{N}$,

[TABLE]

Since $\textrm{P}(V^{*})$ is compact, we can extract a convergent subsequence $[f_{n_{k}}]_{k\in\mathbb{N}}$ of $[f_{n}]$, say to $[f]\in\textrm{P}(V^{*})$. Let $H:=[\ker(f)]$ and $a_{n}=\frac{1}{n}+\sqrt{2}\delta([f_{n}],[f])$. Observe that the following inequality is true for every $x\in\textrm{P}(V)$ and every $f,f^{\prime}\in V^{*}$ such that $||f||=||f^{\prime}||=1$,

[TABLE]

We deduce from (13) and (14) that for every $k\in\mathbb{N}$, $\nu(\{x\in\textrm{P}(V);\delta(x,H)<a_{n_{k}}\})>\epsilon_{0}$. But since $H$ is closed in $\textrm{P}(V)$ and $a_{n_{k}}\underset{k\rightarrow+\infty}{\longrightarrow}0$, $\nu(\{x\in\textrm{P}(V);\delta(x,H)<a_{n_{k}}\})\underset{k\rightarrow+\infty}{\longrightarrow}\nu(H)$. Thus $\nu(H)\geq\epsilon_{0}$ contradicting the non degeneracy of $\nu$.

Now we check that (11) is equivalent to (12). Assume first that (11) holds. We can then find a decreasing sequence $(a_{n})_{n\in\mathbb{N}}$ in $(0,1)$ that converges to zero such that for every projective hyperplane $H$,

[TABLE]

For every $k\in\mathbb{N}^{*}$, denote by $U_{k}$ the interval $[\frac{1}{a_{k}},\frac{1}{a_{k+1}})$ with the convention $U_{0}=[1,\frac{1}{a_{1}})$. Let $\phi:[1,+\infty)\longrightarrow[1,+\infty)$ be any proper function such that $\phi_{|_{U_{k}}}\leq e^{k/2}$ for every $k\in\mathbb{N}$ (for instance affine on each $U_{k}$ with $\phi(\frac{1}{a_{k}})=e^{(k-1)/2}$). Let $H$ be a projective hyperplane and $A_{k}=\{x\in\textrm{P}(V)\setminus H;\frac{1}{\delta(x,H)}\in U_{k}\}$, $k\in\mathbb{N}$. Since $(A_{k})_{k\geq 0}$ covers $\textrm{P}(V)\setminus H$ and since $\nu$ is not degenerate on $\textrm{P}(V)$, we deduce that

[TABLE]

The finite sum above being independent of $H$, the forward implication is proved.

Conversely, assume that (12) holds and let $C:=\sup_{H}\int_{\textrm{P}(V)}{\phi\left(\delta^{-1}(x,H)\right)\,d\nu(x)}<+\infty$.

Let $\epsilon>0$. By properness of $\phi$ we can find $\eta>0$ such that $\phi(\frac{1}{t})>\frac{C}{\epsilon}$ for every $0<t<\eta$.

Hence, for every $t\in(0,\eta)$ and for every projective hyperplane $H$,

\nu\big{\{}x\in\textrm{P}(V);\delta(x,H)<t\big{\}}\leq\nu\big{\{}x\in\textrm{P}(V);\phi(\delta(x,H)^{-1})>\frac{C}{\epsilon}\big{\}}. By Markov’s inequality, we deduce that for every $H$ one has that \nu\big{\{}x\in\textrm{P}(V);\delta(x,H)<t\}\leq\epsilon whenever $t\in(0,\eta)$. This proves the backward implication. ∎

Remark 3.2**.**

Assume now that $\Gamma_{\mu}$ is strongly irreducible and proximal, so that the stationary measure $\nu$ is unique.

When $\mu$ has an exponential moment, Guivarc’h showed in **[Gui90]** that $\phi(x)=x^{\alpha}$ works for some $\alpha>0$ small enough. In other terms, $\nu$ has Holder regularity. In particular, $\nu$ has positive Hausdorff dimension. 2. 2.

When $\mu$ has a moment of order $p>1$, Benoist and Quint proved in **[BQ16a]** that $\phi(x)=\left(\ln(x)\right)^{p-1}$ works. In particular, when $p\geq 2$, $\nu$ is log-regular (i.e.  $\phi(x)=\ln x$ works). We note that proving the log-regularity of $\nu$ when $p=2$ was crucial for Benoist and Quint to prove the CLT for $\ln||L_{n}||$. 3. 3.

Lemma 12 shows that such a function $\phi$ still exists when $\mu$ has a moment of order one. However, it does not give an explicit rate of growth for $\phi$. It would be interesting to determine such a rate. 4. 4.

If $\Gamma_{\mu}$ is a non-elementary subgroup of $\textrm{SL}_{2}(\mathbb{R})$, more can be said about the regularity of $\nu$ when $\mu$ has a moment of order one. Indeed, using the work of Benoist and Quint **[BQ16b, Section 5]** on central limit theorems on hyperbolic groups, one can deduce that the unique $\mu$-stationary probability measure on the projective line is $\log$-regular, even when $\mu$ has a moment of order one.

We state now the estimates we will use.

Proposition 3.3**.**

Assume that $\mu$ has a moment of order one and that $\Gamma_{\mu}$ is strongly irreducible and proximal. Denote by $\mu^{t}$ the pushforward probability measure on $\textrm{GL}(V^{*})$ of $\mu$ by the map $g\mapsto g^{t}$. Then

For every $\epsilon>0$,

[TABLE] 2. 2.

There exists $C>0$ such that

[TABLE] 3. 3.

There exists $C>0$ such that

[TABLE] 4. 4.

There exists $C>0$, a random variable $Z$ with values in $\textrm{P}(V)$ of law the unique $\mu$-stationary probability measure on $\textrm{P}(V)$ such that

[TABLE] 5. 5.

Similarly, there exists $C>0$, a random variable $Z^{*}\in\textrm{P}(V^{*})$ of law the unique $\mu^{t}$-stationary probability measure on $\textrm{P}(V^{*})$ such that if $H_{L_{n}}^{-}:=[\ker(f_{L_{n}})]$, then

[TABLE]

where $\delta$ denotes again, by abuse of notation, the standard metric on $\textrm{P}(V^{*})$.

Proof.

We will use in all the proof that if $(A_{n})_{n}$ and $(B_{n})_{n}$ are two sequences of subsets of $\Omega$ such that $\mathbb{P}(A_{n})=1-o(1)$ and $\mathbb{P}(B_{n})=1-o(1)$, then $\mathbb{P}(A_{n}\cap B_{n})=1-o(1)$.

By [BL85, Corollary 3.4 item (iii)], we know that for any sequence $(v_{n})_{n}$ in $V$ of norm one, $\frac{1}{n}\mathbb{E}\left(\ln||L_{n}v_{n}||\right)\underset{n\rightarrow+\infty}{\longrightarrow}\lambda_{1}$. Hence $\frac{1}{n}\mathbb{E}\left(\ln\frac{||L_{n}v_{n}||}{||L_{n}||}\right)\underset{n\rightarrow+\infty}{\longrightarrow}0$ for every such sequence $(v_{n})_{n}$. Thus,

[TABLE]

It is enough now to apply Markov’s inequality in order to have the estimate of 1. 2. 2.

Let $x=[v],y=[w]\in\textrm{P}(V)$. Without loss of generality $||v||=||w||=1$. We have by the definition of the metric $\delta$:

[TABLE]

On the one hand, we know by the Guivarc’h-Raugi theorem [GR85] that with our assumptions on the semi-group generated by the support of $\mu$, the first Lyapunov exponent is simple. Hence the following almost sure convergence holds:

[TABLE]

We deduce that for $C:=(\lambda_{1}-\lambda_{2})/2>0$, we have that

[TABLE]

On the other hand, applying estimate 1. for $\epsilon=C/4$ to get that

[TABLE]

Moreover the previous estimate is uniform in $v$ and $w$. Combining (15), (16) and (17) and the remark at the beginning of the proof, we get the desired estimate. 3. 3.

Let $x=[v]\in\textrm{P}(V)$ and $g\in\textrm{GL}(V)$. Observe that

[TABLE]

It is enough now to apply estimates 1. and 2. 4. 4.

Let $x\in\textrm{P}(V)$. We know from [BL85, Theorem 4.3] that there exists a random variable $Z$ on $\textrm{P}(V)$ independent of $x$ of law the unique $\mu$-stationary probability measure on $\textrm{P}(V)$ such that the sequence of random variables $(R_{n}\cdot x)_{n\in\mathbb{N}}$ converges in probability to $Z$. Hence, there exists a non random subsequence $(n_{k})_{k\in\mathbb{N}}$ such that $(R_{n_{k}}\cdot x)_{k\in\mathbb{N}}$ converges almost surely to $Z$. Fix now $n\in\mathbb{N}$ and denote by $C$ the positive constant given in estimate 2. On the one hand, we have by Fatou’s lemma that:

[TABLE]

On the other hand, writing $R_{n_{k}}\cdot x=R_{n}\cdot(X_{n+1}\cdots X_{n_{k}})\cdot x$ for all $n_{k}>n$ and using the independence of the $X_{i}$’s, we get that for all $n_{k}>n$,

[TABLE]

Combining (18) and (19), we deduce that for every $n\in\mathbb{N}$,

[TABLE]

By estimate 2.  and the fact that $R_{n}$ and $L_{n}$ have the same law for every $n\in\mathbb{N}$, we deduce that the quantity above goes to zero as $n$ tends to infinity. This proves the first inequality. The second estimate then follows item 3. 5. 5.

Apply the previous estimate for the probability measure $\mu^{t}$ which satisfies the same assumptions as $\mu$ (see for instance [BL85, Chapter III, Lemma 3.3]).

∎

Remark 3.4**.**

In parts 1,2,3,4 the speed of convergence is

• •

exponential when $\mu$ has an exponential moment **[BL85]**, **[Gui90]**, **[Aou11]**.

• •

of order $C_{n}$ for some sequence $(C_{n})_{n\in\mathbb{N}}$ that satisfies $\sum_{n}{n^{p-2}C_{n}}<+\infty$, when $\mu$ has a moment of order $p>1$ **[BQ16a]**.

Remark 3.5**.**

The role of $R_{n}$ and $L_{n}$ is interchangeable in the statements of Proposition 3.3 except for estimates 4 and 5 where the result fails if we interchange $R_{n}$ and $L_{n}$.

4 End of the proof of Theorem 2.1

The end of the proof is based on a usual trick about the asymptotic independence of the right and the left random walk. We refer for instance to [Tut68, Section 2.3], [Vir70, Section 6], [GG96, Theorem 1.5] and [Aou13, Lemme 4.3] for a general statement.

Proof of Theorem 2.1:.

Let $\mathcal{H}$ be the set of all projective hyperplanes of $\textrm{P}(V)$. For every $H=\ker(f)$ and $H^{\prime}=\ker(f^{\prime})$ in $\mathcal{H}$, we denote for simplicity $\delta(H,H^{\prime}):=\delta([f],[f^{\prime}])$. By estimates 3. , 4.  and 5.  of Proposition 3.3, there exist a random variable $Z\in\textrm{P}(V)$, $C>0$, $n_{0}\in\mathbb{N}$ such that for every $n\geq n_{0}$:

1. i.

[TABLE]

In particular,

[TABLE] 2. ii.

[TABLE]

The fact that the $X_{i}$’s are i.i.d  is used in the left equality above, while the right one follows from Estimate 5. of Proposition 3.3. Fix now $\epsilon>0$. We deduce that, for $n\geq n_{0}$,

[TABLE]

Estimates (20) and (23) follow immediately from estimate i.  at the beginning of the proof. In line (21), we used estimate ii.  above and identity (14). Identity (22) is due to the independence of the $(X_{i})_{i}$’s.

 Observe now that, by compactness of $\mathcal{H}$, the following convergence holds for $\epsilon>0$ fixed:

[TABLE]

Since the law of the random variable $Z$ is the unique $\mu$-stationary probability measure $\nu$, we deduce from (23) that for every $\epsilon>0$,

[TABLE]

Applying Lemma 12 ends the proof. ∎

The proof of Theorem 3 (combine (5), (7) and (24)) shows the following result which makes clear the link between the regularity of stationary measures on projective space (see Remark 3.2) and the speed of convergence as $\epsilon\rightarrow 0$ of the function $\epsilon\mapsto\limsup_{n\rightarrow+\infty}{\mathbb{P}\left(\frac{\rho(L_{n})}{||L_{n}||}\leq\epsilon\right)}$.

Theorem 4.1**.**

Let $\mu$ be a probability measure on $\textrm{GL}(V)$ such that $\mu$ has a moment of order one and such that $\Gamma_{\mu}$ is strongly irreducible. Let $p$ be the proximality index of $\Gamma_{\mu}$. Then there exists $C=C(\Gamma_{\mu})\geq 1$ such that for every $\epsilon>0$,

[TABLE]

where $\nu$ is the unique $\mu$-stationary measure on the projective space of some irreducible and proximal representation333Assume we are working in characteristic zero. Then $\nu$ is the unique $\mu$-stationary probability measure on the projective space $\textrm{P}(\bigwedge^{p}V)$ whose cocycle average is maximal (i.e. $\iint{\ln\frac{||\bigwedge^{p}g\,v||}{||v||}\,d\mu(g)d\nu([v])}=p\lambda_{1}(\mu)$). of $\Gamma_{\mu}$. ∎

We end by justifying Remark 1.3. The setting is borrowed from [BS18, Example 3.5].

Example 4.2**.**

Let $\lambda>1$, $\sigma=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}$ and $a=\begin{pmatrix}\lambda&0\\ 0&\lambda^{-1}\end{pmatrix}$. Let $\mu$ be a probability measure on $S:=\{\sigma,\sigma a\}$ with full support, $\theta:=\mu(\sigma a)\in(0,1)$, $(X_{i})_{i\in\mathbb{N}}$ a sequence of i.i.d random variables with law $\mu$ and consider the random walk $L_{n}=X_{n}\cdots X_{1}$ on $\textrm{SL}_{2}(\mathbb{R})$ with respect to $\mu$. We claim that $\lambda_{1}(\mu)=0$, $\frac{\ln\rho(L_{n})}{n}\to 0$ almost surely but that $\frac{\ln\rho(L_{n})}{\sqrt{n}}$ does not converge in distribution. On the contrary, $\frac{\ln||L_{n}||}{\sqrt{n}}$ does converge in distribution but to the maximum of two (dependent) Gaussian distributions. The semi-group $\Gamma_{\mu}$ is actually a subgroup of $\textrm{SL}_{2}(\mathbb{R})$ and it is irreducible but not strongly irreducible (as the union of the coordinate axis of $\mathbb{R}^{2}$ is $\Gamma_{\mu}$-invariant). Let us check our claims. Denote by $H\subset\Gamma_{\mu}$ the subgroup of diagonal matrices. Note that $H$ is a subgroup of $\Gamma_{\mu}$ of index two and that the matrices in the coset $\sigma H$ have spectral radius equal to one. On the one hand, since $\mu$ is supported in the coset $\sigma H$, we deduce that $L_{2n+1}\in\sigma H$ for every $n$ and therefore that $\rho(L_{2n+1})=1$ for every $n$. In particular, the only possible limiting distribution of $\frac{\ln\rho(L_{n})}{\sqrt{n}}$ is the Dirac mass at [math]. On the other hand, writing $X_{i}=\sigma a^{\epsilon_{i}}$ with $(\epsilon_{i})_{i\in\mathbb{N}}$ a sequence of independent Bernoulli random variables $\mathcal{B}(\theta)$, and using the relation $a^{k}\sigma=\sigma a^{-k}$ for $k\in\mathbb{Z}$, we get for $Y_{i}:=\epsilon_{2i-1}-\epsilon_{2i}$,

[TABLE]

The sequence $(Y_{i})_{i\in\mathbb{N}}$ is a sequence of i.i.d. random variables on $\{-1,0,1\}$ with $\mathbb{P}(Y_{i}=-1)=\mathbb{P}(Y_{i}=1)=\theta(1-\theta)$, so that $S_{n}$ is a centered random walk on $\mathbb{Z}$. Thus, by the classical central limit theorem, $\frac{\ln\rho(L_{2n})}{\sqrt{n}}=\ln\lambda\,\frac{|S_{n}|}{\sqrt{n}}$ converges in distribution to $|X|$ with $X$ being a non degenerate Gaussian distribution. Thus $\frac{\ln\rho(L_{n})}{\sqrt{n}}$ does not converge in distribution. The other claims follow readily from the discussions above.