On Contraction Properties for Products of Markov Driven Random Matrices
We describe contraction properties on pro jective spaces for products of matrices governed by Markov chains which satisfy strong mixing conditions. Assuming that the subgroup generated by the corresponding matrices is "large" we show in particular that the top Lyapunov exponent of their pr...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2008
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irk-123456789-1065192016-09-30T03:03:03Z On Contraction Properties for Products of Markov Driven Random Matrices Guivarc'h, Y. We describe contraction properties on pro jective spaces for products of matrices governed by Markov chains which satisfy strong mixing conditions. Assuming that the subgroup generated by the corresponding matrices is "large" we show in particular that the top Lyapunov exponent of their product has multiplicity one and we give an exposition of the related results. 2008 Article On Contraction Properties for Products of Markov Driven Random Matrices / Y. Guivarc'h // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 4. — С. 457-489. — Бібліогр.: 39 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106519 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We describe contraction properties on pro jective spaces for products of matrices governed by Markov chains which satisfy strong mixing conditions. Assuming that the subgroup generated by the corresponding matrices is "large" we show in particular that the top Lyapunov exponent of their product has multiplicity one and we give an exposition of the related results. |
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Article |
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Guivarc'h, Y. |
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Guivarc'h, Y. On Contraction Properties for Products of Markov Driven Random Matrices Журнал математической физики, анализа, геометрии |
| author_facet |
Guivarc'h, Y. |
| author_sort |
Guivarc'h, Y. |
| title |
On Contraction Properties for Products of Markov Driven Random Matrices |
| title_short |
On Contraction Properties for Products of Markov Driven Random Matrices |
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On Contraction Properties for Products of Markov Driven Random Matrices |
| title_fullStr |
On Contraction Properties for Products of Markov Driven Random Matrices |
| title_full_unstemmed |
On Contraction Properties for Products of Markov Driven Random Matrices |
| title_sort |
on contraction properties for products of markov driven random matrices |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/106519 |
| citation_txt |
On Contraction Properties for Products of Markov Driven Random Matrices / Y. Guivarc'h // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 4. — С. 457-489. — Бібліогр.: 39 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT guivarchy oncontractionpropertiesforproductsofmarkovdrivenrandommatrices |
| first_indexed |
2025-07-07T18:35:59Z |
| last_indexed |
2025-07-07T18:35:59Z |
| _version_ |
1837014303960465408 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2008, vol. 4, No. 4, pp. 457�489
On Contraction Properties for Products of Markov
Driven Random Matrices
Y. Guivarc'h
IRMAR CNRS Rennes I, Universit�e de Rennes I, Campus de Beaulieu
35042 Rennes Cedex, France
E-mail:yves.guivarch@univ-rennes1.fr
Received March 28, 2008
We describe contraction properties on projective spaces for products of
matrices governed by Markov chains which satisfy strong mixing conditions.
Assuming that the subgroup generated by the corresponding matrices is
"large" we show in particular that the top Lyapunov exponent of their pro-
duct has multiplicity one and we give an exposition of the related results.
Key words: Lyapunov exponent, Markov chain, martingale, spectral gap,
proximal.
Mathematics Subject Classi�cation 2000: 37XX, 22D40, 60BXX, 82B44.
1. Introduction. Notations
Let V be a d-dimensional Euclidean vector space, i:e, V = R
d with its natural
scalar product. Let G = GL(V ) be the linear group of V and gk(k 2 Z) a sequence
of elements of G. We consider the recurrence relation in V
vn+1 = gn+1vn ; n 2 Z.
Then, given v0 2 V , we can express vn, n 2 N, by vn = Snv0, where Sn =
gn : : : g1 2 G is the product of the elements gk, 1 � k � n. In analogy with the
constant case gk = g, A. Lyapunov was able to describe the asymptotic behaviour
of Snv, v 2 V , n 2 N, in terms of a �nite number of exponents �1; �2; : : : ; �p;
p � d, under a mild growth condition on the sequence gk (Lyapunov regularity).
The numbers �i, 1 � i � p, are called the Lyapunov exponents and the set
f�1; : : : ; �pg is called the Lyapunov spectrum of the sequence (gk)k2Z.
Let (
; �; �) be a measured dynamic system where � is a �nite �-invariant
and ergodic probability measure, and gk(!) a �-stationary sequence, i:e, gk(!) =
g0(�
k!), k 2 Z, such that Logkgk(!)k and Logkg�1k (!)k are �-integrable. Using
the methods of ergodic theory, V.I. Oseledets showed ([33]) the Lyapunov regu-
larity of the sequence (gk(!))k2Z, ��a:e. In particular the product Sn(!) can be
c
Y. Guivarc'h, 2008
Y. Guivarc'h
reduced to a block-diagonal form where each block has a de�nite growth exponent
�i, 1 � i � p. In this setting Sn(!) is a G-valued Z-cocycle of (
; �; �), i:e, for
m;n 2 Z:
Sm+n(!) = Sm (�n!) Sn(!);
with S0(!) = Id.
We denote by P (V ) the projective space of V and by x! g:x the projective
action of g 2 G on x 2 P (V ). A basic role in this ergodic context is played by
the skew product (
� P (V ), e�) and its e�-invariant measures with projection �.
Here e� is the extension of �:
e�(!; x) = (�!; g1(!):x):
On the other hand, a special situation, where � = �
Z is a product measure
and the random variables gk(!) are i:i:d, has already been deeply studied by
H. Furstenberg and H. Kesten. There, the basic object is the random walk Sn(!)
on G de�ned by �, and in particular the Markov chain on P (V ) with transition
kernel Q� de�ned by
Q�(x;A) =
Z
1A(g:x)d�(g):
The map e� considered above can be identi�ed with the shift transformation on
the path space of this Markov chain.
If supp� � G generates a large subgroup denoted by < supp� >, it was
observed by H. Furstenberg that the above Markov chain has nice properties of
contraction analogous to those of the iterates of a single positive matrix. For
example, if supp� is bounded and < supp� > is a dense subgroup of the unimo-
dular group SL(d;R), then kSn(!)k has exponential growth. This fact was used
as a key tool (for d = 2) by I. Goldsheid, S.A. Molcanov, L.A. Pastur in order to
prove the pure point spectrum property for the Schr�odinger operator with random
potential on the line. Motivated by this kind of consequence, and going a step
further, the author and A. Raugi, and then I. Goldsheid and G.AMargulis, showed
simplicity of the Lyapunov spectrum, (i:e; p = d), for the cocycle Sn(!), under
mild algebraic conditions on< supp� >. A basic fact, which can be used in a more
�exible way, is that the top Lyapunov exponent has multiplicity one. This is the
starting point for various nontrivial properties of the cocycle Sn(!). Then it is
clear that, under mild conditions on < supp� >, the asymptotic properties of
Sn(!) can be developed much further and applied to various probabilistic, ana-
lytic or geometrical questions. Furthermore, even in the i:i:d case, since "large"
subgroups play an important role, this topic cannot be considered as a simple
extension of Classical Probability Theory, from R
� to GL(d;R). Here we sketch
these developments and we restrict our survey to the case of Markov dependence
of the increments gk(k 2 Z). The emphasis is put more on the basic ideas than
458 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
On Contraction Properties for Products of Markov Driven Random Matrices
on the detailed results. We give a detailed exposition of the ideas in the i:i:d case
(Sects. 2�4), and we describe brie�y the required modi�cations for the Markovian
case (Sect. 5). We observe that this Markovian setting includes the case where
� is a Gibbs measure on
= AZ and g0(!) depends only on a �nite number of
coordinates. A few applications are described in Sect. 6 and references for other
topics are given. We describe now some notations used below.
For a Polish space E, the space of complex continuous functions on E will
be denoted C(E), and the space of continuous maps of E into itself by C(E;E).
The action of a map u on E will be denoted x ! u:x (x 2 E) if E is compact.
The space of probability measures on a Polish space F will be denoted M1(F ).
If � 2 M1(C(E;E)) and � 2 M1(E), we write � � � for the measure on E given
by ' !
R
'(g:x)d�(g)d�(x). A measure � 2 M1(E) is said to be �-stationary if
��� = �. In this context, we will consider the Markov chain on E with transition
kernel Q� de�ned by
Q�'(x) =
Z
'(g:x)d�(g);
where ' is a bounded Borel function on E.
The adjoint of u 2 End E, with respect to the given scalar product, will be
denoted u�. For g 2 GL(V ), we will also denote by g the corresponding projective
map on P (V ). The elements of P (V ) will be represented by vectors of unit length,
taken up to sign. In particular, for x 2 P (V ) and g 2 GL(V ), kgxk 2 R+ is well
de�ned. The wedge products over V will be denoted by ^kV (1 � k � d).
The Euclidean scalar product extends naturally to ^kV . The submanifold of
P (^2V ) corresponding to decomposable 2-vectors will be denoted by P2(V ). For
x 2 P (V ), x ^ y 2 P2(V ), g 2 G, we will consider the following cocycles:
�1(g; x) = Logkgxk; �2(g; x ^ y) = Logkg(x ^ y)k:
Also we will consider the submanifold P1;2(V ) � P (V )�P2(V ) of elements
� = (x; x ^ y) and the cocycle � de�ned by
�(g; �) = Log
kgx ^ gyk
kgxk2
:
For x; y 2 P (V ); we set Æ(x; y) = kx ^ yk: The unique probability measure on
P (V ), invariant under orthogonal maps will be denoted m, and the orthogonal
group of V by O(d). In addition to projective maps, we need also to consider
quasiprojective maps corresponding to nonzero endomorphisms of V . If u 2
EndV and x 2 P (V ), then u:x is well de�ned if x does not belong to the projective
subspace de�ned by Ker u, again denoted by Ker u. Then the quasiprojective
map u is de�ned and continuous outside Ker u. If � 2 M1(P (V )) satis�es
�(Ker u) = 0, then the push forward measure u:� is well de�ned. If F � G,
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 459
Y. Guivarc'h
we will denote by < F > (resp[F ]) the subgroup (resp subsemigroup) generated
by F . Their closures will be written < F >� and [F ]�, respectively. We will say
that a measure � on G has exponential moment and write � 2 M1:e(G) if there
exists c > 0 such thatZ
kgkcd�(g) +
Z
kg�1kcd�(g) < +1:
The unimodular group SL(V ) = SL(d;R) � G will be written G1. Occasionally
the projection of x 2 V on P (V ) will be denoted x, but in general we will take the
same notation for vectors and elements of the projective space. The same abuse
of notations will be made for subspaces.
2. Growth of Column Vectors
Let � be a probability measure on G1 = SL(d;R) and L2(V ) the Hilbert space
of square integrable functions with respect to Lebesgue measure on V . We say
that a subset S � GL(V ) is strongly irreducible if no nontrivial union of subspaces
of V is S-invariant. In particular strong irreducibility implies irreducibility.
Theorem 2.1. Let � 2M1(G1) and assume the closed subgroup < supp� >�
is strongly irreducible and unbounded. Let r(�) be the spectral radius of the con-
volution operator on L
2(V ) de�ned by �. Then r(�) < 1.
Corollary 2.2. Assume furthermore
R
Logkgkd�(g) < +1. Then there exists
a positive number �(�) such that
lim
n!+1
1
n
Z
Logkgkd�n(g) = �(�) �
2
d
Log
1
r(�)
> 0:
Furthermore: � � a:e, lim
n!+1
1
n
LogkSn(!)k = �(�) > 0, where Sn(!) = gn : : : g1.
Theorem 2.3. Assume that � satis�es the hypothesis of Th. 1, and further-
more
R
Logkgkd�(g) < +1. Then for every �xed v 2 V n f0g:
� � a:e; lim
n!+1
1
n
LogkSn(!)vk = �(�) > 0:
Also, 1
n
R
Logkgxkd�n(g) converges to �(�) uniformly on P (V ).
The proof of Th. 2.1 depends on the following lemmas.
Lemma 2.4. Assume that the subgroup � of G1 is strongly irreducible and
unbounded. Then no probability measure on P (V ) is �-invariant.
460 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
On Contraction Properties for Products of Markov Driven Random Matrices
P r o f. Assume � 2M1(P (V )) is �-invariant. Let gn 2 G1 with lim
n!+1
kgnk =
+1 and write un = gn
kgnk . Then det un = 1
kgnkd converges to zero. Since kunk = 1,
we can extract from un a convergent subsequence and assume lim
n!+1
un = u,
kuk = 1, det u = 0. Let W � P (V ) (resp W 0) be the projective subspace
associated with Ker u (resp Im u). We denote by �1 and �2 the restrictions
of � to W and P (V ) n W and write � = �1 + �2. We observe that u de�nes
a quasiprojective map, again denoted by u, of P (V ) n W into P (V ). Then we
have � = lim
n!+1
gn:� = u:�2+ lim
n!+1
gn:�1. Since P (V ) is compact, we can assume
that gn:�1 converges to � 01 concentrated on the subspace W1 = lim
n!+1
gn:W . It
follows that � = lim
n!+1
gn:� is concentrated on the union of W1 and W 0.
Let � be the set of subsets X of P (V ) which are �nite unions of projective
subspaces and which satisfy �(X) = 1. Since any decreasing sequence of elements
of � is asymptotically constant, � has a least element, which is X0 = \
X2�
X.
Since g:� = �, one has g:X0 = X0. This contradicts strong irreducibility of �.
P r o f of Theorem 2.1. We denote by �(�) the convolution operator on L2(V )
de�ned by �(�)(f)(v) =
R
f(g�1v)d�(g). Since every g 2 G1 preserves Lebesgue
measure, k�(�)fk2 � 1. Assume r(�) = 1 and let z 2 C be a spectral value of
�(�) with jzj = 1. Then, either lim
n!+1
k�(�)fn � fnk2 = 0 for some sequence
fn 2 L
2 (V ) with kfnk2 = 1, or Im (�(�) � zI) is not dense in L
2(V ). In the
second case, duality gives Ker(�(��) � zI) = f0g. Since �� satis�es also the
hypothesis we can only consider the �rst case.
Since j�(�)jfnj � jfnjj � j�(�)fn � fnj, we have also
lim
n!+1
k�(�)jfnj � jfnjk2 = 0; lim
n!+1
k�(�)jfnjk2 = 1:
Hence lim
n!+1
< �(�)jfnj; jfnj >= 1 = lim
n!+1
Z
< �(g)jfnj; jfnj > d�(g).
It follows that there exists a Borel subset S0 of supp� with �(S0) = 1, and
a subsequence nk such that < �(g)jfnk j; jfnk j > converges to 1, for every g 2 S0.
For the sake of brevity we write nk = n. The inequality
k�(g)jfnj
2
� jfnj
2
k1 � k�(g)jfnj � jfnjk2k�(g)jfnj+ jfnjk2 � 2k�(g)jfnj � jfnjk2
gives lim
n!+1
k�(g)jfnj
2
� jfnj
2
k1 = 0 for every g 2 S0.
We consider the probability measure �n = jfnj
2(v)dv on V and its projection
�n on P (V ). Then the above relation says that lim
n!+1
g:�n � �n = 0 in variation
norm, hence lim
n!+1
g:�n��n = 0 also in variation. Since P (V ) is compact, we can
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 461
Y. Guivarc'h
assume lim
n!1
�n = � in weak topology. In particular, for every g 2 S0 : g:� = �.
Since �(S0) = 1, S0 generates < supp� >�, hence g:� = � for any g 2< supp� >�.
Lemma 2.4 says that this is impossible.
P r o f of Corollary 2.2. We denote un =
R
Logkgkd�n(g). Since kgk � 1,
un � 0. The subadditivity of Logkgk implies um+n � um + un, hence 0 � un �
nu1 < +1.
It follows lim
n!+1
un
n
= Inf
n
un
n
=
� 0.
We consider the L2 -functions on V , f and 1C de�ned by
f(v) = Inf(1; kvk�Æ); C = fv 2 V ; 1 � kvk � 2g
with 2Æ > d and we normalize the Lebesgue measure dv on V such that vol C = 1.
The theorem gives lim sup
n!+1
j < �(�n)f; 1C > j1=n � r(�). On the other hand:
< �(�n)f; 1C > �
R
1C(v)
1
kg�1vkÆ d�
n(g) dv � 2�Æ
R
1
kg�1kÆ d�
n(g),
Log < �(�n)f; 1C >� �ÆLog2 � Æ
R
Logkg�1kd�n(g),
Æ lim inf
n!+1
1
n
R
Log kg�1kd�n(g) � �Logr(�),
lim inf
n!+1
1
n
R
Logkg�1kd�n(g) � 1
ÆLog
1
r(�)
.
Since Æ is arbitrary with Æ > d
2
, and r(�) = r(��), we get
� 2
dLog
1
r(�)
> 0.
The subadditivity of Logkgk implies that
LogkSm+n(!)k � LogkSm(!)k+ LogkSn Æ �
m(!)k;
hence we can apply the subadditive ergodic theorem to the sequence LogkSn(!)k:
1
n
LogkSn(!)k converges � � a:e and in L
1(
) to a constant �(�). It follows:
�(�) = lim
n!+1
1
n
Z
LogkSn(!)kd�(!) = lim
n!+1
1
n
Z
Logkgkd�n(g) =
> 0:
For the proof of Th. 2.3, we need the following lemmas.
Lemma 2.5. For any �xed c 2 R, the set W of elements v in V such that
� � a:e; lim sup
n!+1
1
n
LogkSn(!)vk � c
is a supp�-invariant subspace.
P r o f. We observe that if a; b > 0, then Log(a+ b) � 1 + Sup(Loga; Logb).
If v; v0 2 V , it follows
LogkSn(!)(v + v0)k � 1 + Sup(LogkSn(!)vk; LogkSn(!)v
0
k):
462 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
On Contraction Properties for Products of Markov Driven Random Matrices
Hence, the condition v; v0 2 W implies v + v0 2 W . Also the condition v 2 W
implies �v 2W for any � 2 R. It follows that W is a subspace of V .
We observe that Sn(!) = Sn�1(�!)g1(!). Hence, the condition v 2W implies
� � a:e, g1(!)v 2W .
The supp�-invariance of W follows
Lemma 2.6. Let m be the uniform measure on P (V ). For any u 2 EndV we
have Z
Logkuxkdm(x) � Logkuk � Log2:
P r o f. We use the polar decomposition of u: u = kak0 with k; k0 2 O(d),
a = diag(a1; : : : ; ad) with a1 � a2 � � � � � ad > 0 and kuk = a1. We write dk for
the normalized Haar measure on O(d). Then, since m is O(d)-invariant:Z
Logkuxkdm(x) =
Z
Logkake1kdk �
Z
Logja1 < ke1; e1 > jdk:
Z
Logkuxkdm(x) � Log a1 +
1
2�
Z 2�
0
Logjcos�j d� = Logkuk � Log2:
Lemma 2.7. Let � 2M1(P (V )) be �-stationary i:e
R
g:�d�(g) = �. Then:Z
Logkgxk d�(g)d�(x) = �(�):
P r o f. Let
� =
R
Logkgxkd�(g)d�(x). The �niteness of
� follows from
�-integrability of Logkgk. Since � is �-stationary, for any n 2 N:
n
� =
Z
Logkgxkd�n(g)d�(x) =
Z
LogkSn(!)xkd�(!)d�(x):
We observe that if f(!; x) is given by f(!; x) = Logkg1(!)xk, then
LogkSn(!)xk =
nX
1
f Æ ~�k(!; x):
Since jf(!; x)j � Logkg1(!)k, f is �
�-integrable and we can use the ergodic
theorem
�
� � a:e; lim
n!+1
1
n
LogkSn(!)xk =
Z
f(!; x)d�(!)d�(x) =
� :
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 463
Y. Guivarc'h
Then Lemma 2.5 and strong irreducibility of supp� imply that for every x 2 P (V ):
� � a:e; lim sup
n!+1
1
n
LogkSn(!)xk �
� :
In particular, the dominated convergence gives, for every x 2 P (V ):
lim
n!+1
Sup
x
1
n
Z
Logkgxkd�n(g) �
� ;
hence: limsup
n!+1
1
n
Z
Logkgxkdm(x)d�n(g) �
� . Using Lemma 2.6, we have
Z
Logkgkd�n(g) �
Z
Logkgxkdm(x)d�n(g) + Log2;
hence:
� � lim
n!+1
1
n
Z
Logkgkd�n(g) = �(�). Since
� � �(�), we conclude
� = �(�).
The following is a well known fact of Markov chain theory (see [9, 16]).
Lemma 2.8. Let G be a locally compact group, E be a compact metric G-space,
� 2M1(G); I �M1(E) the set of �-stationary measures on E, f a continuous
function on E such that �1(f) = �2(f), for every �1; �2 2 I. Then, with � 2 I:
� � a:e; lim
n!+1
1
n
nX
1
f(Sk(!):x) = �(f):
The convergence of 1
n
nX
1
Z
f(g:x)d�k(g) to �(f) is uniform on E.
We will use this lemma if E = P (V ) and f(x) =
R
Logkgxkd�(g). In the proof
of Th. 3 below we assume
R
Log2kgkd�(g) < +1.
P r o f of Theorem 3. We consider the Markov chain on P (V ) with transition
kernel Q�(x; :) = � � Æx, its space of trajectories
� P (V ), and the random vari-
ables Xk(!; x) = Logkgk(Sk�1:x)k, k � 1. Clearly, LogkSn(!)xk =
nX
1
Xk(!; x).
We �x x 2 P (V ) and we denote by Fn the �-�eld on
generated by the
random variables Sk(!):x, 0 � k � n. Then we have E (Xk )jFk�1j = f(Sk�1:x),
hence the sequence Yk = Xk � f(Sk�1:x) is the sequence of increments of the
martingale Zn =
nX
1
Yk.
464 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
On Contraction Properties for Products of Markov Driven Random Matrices
If we assume
R
Log2kgkd�(g) < +1, then Sup
k�1
E (jYk j
2) �
Z
Log2kgkd�(g)
< +1.
Hence the law of large numbers for martingales gives ��a:e; lim 1
n
nX
1
Yk = 0.
Using Lemma 2.7, we conclude that for any �-stationary measure �, �(f) =R
Logkgxkd�(g)d�(x) = �(�).
Then Lemma 2.8 implies
� � a:e; lim
n!+1
1
n
nX
1
f(Sk:x) = �(�).
From the convergence of 1
n
nX
1
Yk to zero, we get
� � a:e; lim
n!1
1
n
LogkSn(!)xk = �(�).
The last assertion is a direct consequence of Lemma 2.8.
Remarks.
a) We have used the condition
R
Log2kgkd�(g) < +1 instead of
R
Logkgkd�(g)
< +1. A re�nement of the above argument gives the complete result (see [9]).
It can also be obtained as a consequence of Oseledets' multiplicative ergodic theo-
rem (see [22]).
b) Strong irreducibility of < supp� > have been used only in order to get
lim
n!+1
1
n
LogkSn(!)vk > 0. The proof above shows that under irreducibility of
< supp� > one gets, for every v 2 V n f0g,
� � a:e lim
n!+1
1
n
LogkSn(!)vk = lim
n!+1
1
n
LogkSn(!)k = �(�) � 0:
c) A typical example with < supp� > irreducible but not strongly irreducible
is G = SL(2;R); � = 1
2
(Æa + Æb) with a = diag(�; 1�); � > 1, b =
�
0 �1
1 0
�
.
Then �(�) = 0.
d) Theorem 2.3 was obtained in [9] by a di�erent argument. Here it is a con-
sequence of Th. 2.1 which can be considered as a special case of the main result
of [7].
e) For a corresponding result where independence of increments is replaced by
markovian dependence with spectral gap, see [39].
3. Uniqueness of Stationary Measures and Contraction
Properties
Here we consider the group G = GL(V ), its action on P (V ), and a probability
measure � 2M1(G). In order to state the results we give some de�nitions.
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 465
Y. Guivarc'h
De�nition 3.1. An element g 2 GL(V ) is said to be proximal if one can write
V = Rvg � V <
g ; gvg = �(g)vg ; j�(g)j = lim
n!+1
kgnk1=n; gV <
g = V <
g ;
and the spectral radius of g on V <
g is strictly less than j�(g)j.
De�nition 3.2. A subsemigroup S � GL(V ) is said to satisfy condition i:p if
S is strongly irreducible and S contains a proximal element.
De�nition 3.3. A probability measure � 2 M1(P (V )) is said to be proper if
for every proper projective subspace H � P (V ) one has �(H) = 0.
De�nition 3.4. A sequence gn 2 G is said to satisfy the contracting property
towards z 2 P (V ) if one has lim
n!+1
gn:m = Æz. (Where m is the uniform measure
on P (V )).
Theorem 3.5. Assume that the closed subsemigroup of G generated by supp�
satis�es condition i:p. Then, there exists a measurable map z from
to P (V ),
de�ned � � a:e such that
g1:(zo�) = z:
The map z is unique mod � and
� � a:e; Æz(!) = lim
n!+1
g1 � � � gn:m:
The Markov operator de�ned by x! � � Æx has a unique stationary measure � on
P (V ) and � is the law of z(!). The measure � is proper.
Corollary 3.6. Let z�(!) be de�ned by Æz�(!) = lim
n!+1
g�1 � � � g
�
n:m and assume
x =2 Kerz�(!). Then, if Sn(!) = gn � � � g1:
� � a:e; lim
n!+1
kSn(!)xk
kSn(!)k
= j < z�(!); x > j; lim
n!+1
kSn(!)x ^ Sn(!)yk
kSn(!)xk2
= 0:
If furthermore y =2 Kerz�(!):
� � a:e; lim
n!+1
Æ(Sn(!):x; Sn(!):y)
Æ(x; y)
= 0:
For �xed !, the above convergences are uniform if x; y vary in a compact subset
of P (V ) nKerz�(!).
The proof of Th. 3.5 depends of the following lemmas.
466 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
On Contraction Properties for Products of Markov Driven Random Matrices
Lemma 3.7. Assume � 2M1(P (V )) is proper and gn 2 G is a sequence such
that lim
n!+1
gn:� = Æz Then gn has the contraction property towards z.
P r o f. We can assume that gn converges to a quasiprojective map u, i:e, for
H � P (V ) a projective subspace and x =2 H : lim
n!+1
gn:x = u:x.
Using dominated convergence, we get that for any ' 2 C(P (V )):
lim
n!+1
Z
'(gn:x)d�(x) = (u:�)(') = '(z):
It follows u:x = z if x =2 Keru. Using again dominated convergence, we get:
u:m = Æz , hence lim
n!+1
gn:m = Æz .
Lemma 3.8. Assume that [supp�]� is strongly irreducible. Then every
�-stationary measure on P (V ) is proper.
P r o f. Let � be a �-stationary measure on P (V ).
We consider the set H of projective subspaces H � P (V ) such that �(H) > 0
and H has minimal dimension with respect to this condition. We observe that,
if H;H 0 2 H and H 6= H 0, then �(H \ H 0) = 0. It follows that for every
" > 0: H" = fH 2 H; �(H) � "g is �nite. Hence, there exists H0 2 H with
�(H0) = Supf�(H);H 2 Hg and the set H0 of such subspaces H0 is �nite.
On the other hand, the equation �(H) =
R
(g:�)(H)d�(g) implies g�1H0 2 H0,
�� a:e for any H0 2 H0, hence (supp�) (H0) = H0. This contradicts the strong
irreducibility assumption. Hence H = �, i:e; � is proper.
Lemma 3.9. Let ' 2 C(P (V )) and denote for (!; �) 2
�
, ! = (gk)k2N,
� = (
k)k2N : fn(!) = (g1 � � � gn:�)('), f rn(!; �) = (g1 � � � gn:
0 � � �
r:�)(').
Then, if r is �xed, �
� � a:e lim
n!+1
f rn(!; �)� fn(!) = 0.
P r o f. We denote by Fn the �-�eld on
generated by g1(!) � � � gn(!).
Since � is �-stationary: E(fn+1 jFn) = fn, i:e, fn is a martingale. It follows
that fn and fn+r � fn are orthogonal, i:e, E ((fn+r � fn)
2) = E(f2n+r )� E (f2n ).
Then, for any m > 0
mX
n=1
E(fn+r � fn)
2
� 2rj'j21:
The convergence of the series
1X
n=1
E((fn+r � fn)
2) follows. Since
E((fn+r � fn)
2) =
Z
jf rn(!; �)� fn(!)j
2d�(!)d�(�);
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 467
Y. Guivarc'h
we get the convergence �
��a:e of the series
1X
n=1
jf rn(!; �)�fn(!)j
2. In particular,
the assertion of the lemma follows.
P r o f of Theorem 3.5. We have observed above that for any
' 2 C(P (V )); fn(!) is a martingale. Taking ' in a countable dense subset of
C(P (V )), we get that there exists �! 2M1(P (V )) de�ned � � a:e such that
� � a:e; lim
n!+1
g1 � � � gn:� = �!:
In the same way we get, using Lem. 3.5,
�
� � a:e; lim
n!+1
g1 � � � gn
0 � � �
r:� = lim
n!+1
g1 � � � gn:� = �!:
Hence
� � a:e; lim
n!+1
g1 � � � gn
:� = �!
for every
2 [supp�]�. Let nk(!) be a subsequence such that g1 � � � gnk converges
to a quasiprojective map �!. Since � and
:� are proper
�!:(
:�) = �!:� = �!:
LetH! be the kernel of �!,
1 a proximal element of [supp�]�, with attractive �xed
point x. Using the strong irreducibility of [supp�]�, we can �nd
0 2 [supp�]�
such that
0:x =2 H!. Then, taking
=
0
n
1 (n 2 N), we get: lim
n!+1
0
n
1 :� = Æ
:x.
The continuity of �! outside H! gives �nally
�!:� = �! = �!:(
:Æx) = Æ�!
:x:
This shows that �! is � � a:e a Dirac measure Æz(!), and, furthermore
�!(P (V ) nH!) = z(!). In particular,
� � a:e; lim
n!+1
g1 � � � g1:� = �!:� = Æz(!):
This convergence implies
� � a:e; z(!) = g1:z(�!);
and furthermore � is the law of z(!). Also we have E (Æz(!) jFnj = g1 � � � gn:�.
Using Lemma 3.8, we know that � is proper. Then Lemma 3.7 gives that
g1 � � � gn has the convergence property towards z(!), hence
� � a:e; lim
n!+1
g1 � � � gn:m = Æz(!):
468 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
On Contraction Properties for Products of Markov Driven Random Matrices
This relation de�nes z(!) independently of �. Since � is the law of z(!), � is
unique as a �-stationary measure.
If z0(!) is a solution � � a:e of the equation z0 = g1:(z
0 Æ �) and � 0 is the law
of z0, we have, using the independence of g1 and z0 Æ �: � 0 = � � � 0.
From above, we have � 0 = �. Also E (Æz0 (!)jFn) = g1 � � � gn:�
0 and from the
martingale convergence theorem
� � a:e; Æz0(!) = lim
n!+1
g1 � � � gn:�
0:
Since � 0 = �, we get z0 = z � � a:e.
For the proof of Cor. 3.6 we need the following.
Lemma 3.10. Assume gn 2 G is such that g�n has the contraction property
towards z� 2 P (V ). Then, for any x; y 2 P (V ), with x =2 Kerz�:
lim
n!+1
kgnxk
kgnk
= j < z�; x > j; lim
n!+1
kgnx ^ gnyk
kgnxk2
= 0:
Furthermore, the sequence
Æ(gn:x;gn:y)
Æ(x;y)
converges uniformly to zero if x; y vary in
a compact subset of P (V ) nKerz�.
P r o f. We use the polar decomposition G = KA+K : gn = knank
0
n with
kn; k
0
n 2 K = O(d), an 2 A+. Then the convergence of g�n:m to z� implies
a
(2)
n = o(a1n); lim
n!+1
k
0�1
n :e1 = z�.
If x =
dX
i=1
xiei, we get
kgnxk
2 =
dX
i=1
jain < k0nx; ei > j
2
� ja1n < k0nx; e1 > j
2.
Since kgnk = a1n
lim
n!+1
kgnxk
2
kgnk2
= lim
n!1
j < k
0�1
n e1; x > j
2 + lim
n!+1
X
i>1
�
ain
a1n
�2
< k0nx; ei >
2
= j < z�; x > j
2:
Also kgnx ^ gnyk
2 =
X
i<j
(aina
j
n)
2
j < k0n(x ^ y); ei ^ ej > j
2.
It follows
kgnx ^ gnyk � da(1)n a(2)n kx ^ yk; kgnxk � a(1)n j < k0nx; e1 > j;
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 469
Y. Guivarc'h
jgnyk � a(1)n j < k0ny; e1 > j;
kgnx ^ gnyk
kx ^ yk kgnxk2
� d
a
(2)
n
a
(1)
n
1
j < k0nx; e1 > j2
:
Since lim
n!+1
j < k0nx; e1 > j = j < z�; x > j 6= 0 and a
(2)
n = 0(a
(1)
n ), we get
lim
n!+1
kgnx ^ gnyk
kgnxk2
= 0:
Also
kgnx^gnyk
kgnxk kgnyk kx^yk � d a
(2)
n
a
(1)
n
1
j<k0nx;e1><k0ny;e1>j .
Since x; y =2 Kerz�:
lim
n!+1
Æ(gn:x; gn:y)
Æ(x; y)
= 0:
Since lim
n!+1
j < k0nx; e1 >< k0ny; e1 > j = j < z�; x > j j < z�; y > j is bounded
from below on a compact C of P (V ) n Kerz�, the convergence to j < z�; x >
< z�; y > j is uniform on C.
P r o f of Corollary 3.6. We observe that if a semigroup S satis�es i:p, then
the semigroup S� satis�es also i:p. Then the theorem implies the convergence
� � a:e; lim
n!+1
g�1 � � � g
�
n:m = Æz�(!):
If Sn(! = gn � � � g1, we have S
�
n(!) = g�1 � � � g
�
n. The theorem implies that S�n(!)
has the contracting property towards z�(!), hence the corollary follows from
Lemma 3.10.
R e m a r k. The weak convergence of measures to a Dirac measure, stated
in Th. 2.5, plays an important role in various questions, in particular in the
superrigidity of lattices in semisimple groups (see [10, 32]), as well as in com-
pacti�cations of symmetric spaces (see [24]). The proof given here is borrowed
from [21].
4. Angles of Column Vectors: Exponential Decrease
Here we consider the wedge product ^2V generated by the decomposable
2-vectors x ^ y (x; y 2 V ). A natural scalar product on ^2V is given
by < x ^ y; x0 ^ y0 >= det
�
< x; x0 > < x; y0 >
< y0; y0 > < y; y0 >
�
. The angle �(x; y) between
x and y is given by sin�(x; y) =
kx^yk
kxk kyk . Here we are interested by the angle
�(Sn(!)x; Sn(!)y). We denote by P2(V ) the projection on P (^2V ) of the cone
470 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
On Contraction Properties for Products of Markov Driven Random Matrices
of decomposable 2-vectors. We note that Æ(x; y) = sin�(x; y) de�nes a distance Æ
on P (V ). We represent an element of P2(V ) by a 2-vector x^ y with kx^ yk = 1.
Then we will write �2(g; x ^ y) = Logkgx ^ gyk.
Also we consider the compact space P1;2(V ) of contact elements � = (x; x^y),
where kxk = kx ^ yk = 1, and the cocycle on P1;2(V ) �(g; �) = Log
kgx^gyk
kgxk2 .
This cocycle can be interpreted as an in�nitesimal coe�cient of expansion of the
projective map g, at x in the direction of (x ^ y).
Here we will assume that Logkgk and Logkg�1k are �-integrable. Also we
assume that the semigroup supp�� satis�es condition i:p.
Theorem 4.1. Assume � 2 M1(G) is such that Logkgk and Logkg�1k are
�-integrable, and [supp�]� satis�es condition i:p. We denote
1 = lim
n!+1
1
n
Z
Logkgkd�n(g) ;
2 = lim
n!+1
1
n
Z
Logkg ^ gkd�n(g):
Then
2 < 2
1.
Corollary 4.2. lim
n!+1
Sup
kxk=kyk=1
1
n
Z
Log
kgx ^ gyk
kgxk2
d�n(g) =
2 � 2
1 < 0:
Corollary 4.3. Assume � 2 M1(G) has an exponential moment, i:e,R
kgkcd�(g) < +1,
R
kg�1kcd�(g) < +1 for some c > 0. Then, for " su�ciently
small, there exists �(") < 1 such that
lim
n!+1
Sup
x;y2P (V )
�Z
Æ"(g:x; g:y)
Æ"(x; y)
d�n(g)
�1=n
= �(") < 1:
For a continuous function ' on P (V ), we write
j'j = Supj'(x)j
x2P (V )
; [']" = Sup
x6=y
j'(x) � '(y)j
kx ^ yk"
:
We denote by H"(P (V )) the space of "-Hoelder functions on P (V ), i:e,
H"(P (V )) = f' 2 C(P (V )); [']" < +1g ;
and we observe that H"(P (V )) is a Banach space for the norm
k'k = j'j+ [']":
If t 2 R, we consider the operator P it on C(P (V )) de�ned by (P it')(x) =R
kgxkit'(g:x)d�(g). Then P it de�nes a bounded operator on H"(P (V )).
Then we have
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 471
Y. Guivarc'h
Corollary 4.4. With the notations of Cor. 4.3, there exists C � 0 such that
for any ' 2 H"(P (V )) and t 2 R:
[P it']" � �(")[']" + jtjCj'j:
In particular 1 is an isolated spectral value of P and if t 6= 0 the spectral radius
of P it is strictly less than one.
For the proof of Th. 4.1 we will need the following lemmas.
Lemma 4.5. There exists C > 0 such that for any u 2 EndV ,
Logku ^ uk �
Z
Logkux ^ uykdm2(x ^ y) + C;
where m2 is the uniform measure on P2(V ).
P r o f. We proceed as in Lemma 2.6, i:e, we write u = kak0 with k, k0 2 O(d),
a = diag(a1; : : : ; ad). Then we getZ
Logkux^uykdm2(x^y) � Logku^uk+
Z
Logj < x^y; e1^ e2 > jdm2(x^y):
Hence it su�ces to show that the integral I in the right-hand side
is �nite. We consider the unit sphere of ^2V , its algebraic submanifold V2 =
f(x ^ y) 2 ^2V ; kx ^ yk = 1g, and we denote by em2 its normalized Riemannian
measure. Clearly,
I =
Z
Log j < x ^ y; e1 ^ e2 > jdem2(x ^ y):
Since the map x ^ y ! j < x ^ y; e1 ^ e2 > j2 is a polynomial map, there exists
an integer r > 0 and c > 0 such that
fm2fx ^ y 2 V2; j < x ^ y; e1 ^ e2 > j
2
� tg � ctr:
Then the push forward of fm2 on [0; 1] by this map has a density f which satis�es
tf(t) � ctr=2. Then
Z
Logj < x^y; e1^ e2 > jdm2(x^y) =
1Z
0
(Logt)f(t)dt �
1Z
0
tr=2(Logt)
dt
t
> �1;
since r > 0.
472 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
On Contraction Properties for Products of Markov Driven Random Matrices
Lemma 4.6. For any �-stationary measure � on P2(V )Z
Logkgx ^ gykd�(g)d�(x ^ y) �
2:
The sequence 1
n
R
Logkgx^gykd�n(g)dm2(x^y) converges to
2. For any cluster
value � of the sequence
1
n
n�1X
0
�k
!
�m2, one has
2 =
Z
Logkgx ^ gykd�(g)d�(x ^ y):
P r o f. Let �n = 1
n
n�1X
0
�k, � 2M1(P2(V )) and
In(�) =
Z
�2(g; x ^ y)d�
n(g)d�(x ^ y):
Using the cocycle identity for �2:
In(�) = In�1(�) +
Z
f(x ^ y)d(�n�1 � �)(x ^ y);
with f(x ^ y) =
R
�2(g; x ^ y)d�(g). Hence, 1
nIn(�) = (�n � �)(f). If � = � is
�-stationary,
1
n
In(�) = �(f) =
Z
�2(g; x ^ y)d�(g)d�(x ^ y):
Since In(�) �
R
Logkg ^ gkd�n(g), the �rst assertion follows.
If � = m2, Lemma 4.5 gives
�
C
n
+
1
n
Z
Logkg ^ gkd�n(g) �
In(m2)
n
�
1
n
Z
Logkg ^ gkd�n(g);
hence lim
n!+1
In(m2)
n
=
2.
Also, from above 1
n In(m2) = (�n �m2)(f). Since f is continuous,
lim
n!+1
�n �m2 = �(f) =
Z
Logkgx ^ gykd�(g)d�(x ^ y):
Hence
2 =
R
Logkgx ^ gykd�(g)d�(x ^ y).
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 473
Y. Guivarc'h
Lemma 4.7. Let (X;T; �) be a measured dynamical system with �
�nite T -invariant, f an integrable function. Then, if
�� a:e; lim
n!+1
n�1X
0
f Æ T k = �1;
then
R
f(x)d�(x) < 0.
For the proof of this statement see [15].
P r o f of Theorem 4.1. Using Lemma 2.7, we know that for any �-stationary
measure � on P (V ) Z
Logkgxkd�(g)d�(x) =
1:
On the other hand, Lemma 4.6 gives
R
Logkgx ^ gykd�(g)d�(x ^ y) =
2,
where � is a cluster value of the sequence �n �m2.
We consider the compact space P1;2(V ). Clearly, G acts on P1;2(V )
and the maps � ! x and � ! x ^ y and are G-equivariant. It follows from
Markov�Kakutani theorem that there exists on P1;2(V ) a �-stationary measuree� which has projection � on P2(V ). Its projection � on P (V ) satis�es as above:R
Logkgxkd�(g)d�(x) =
1. HenceZ
Log
kgx ^ gyk
kgxk2
d�(g)de�(�)
=
Z
�2(g; x ^ y)d�(g)d�(x ^ y)� 2
Z
�1(g; x)d�(g)d�(x) =
2 � 2
1:
In particular, there exists a �-stationary measure � on P1;2(V ) such thatZ
�(g; �)d�(g)d�(�) =
2 � 2
1:
On the other hand, every �-stationary measure �0 on P1;2(V ) satis�esR
�(g; �)d�(g)d�0(�) �
2 � 2
1. This follows from the fact that the projections
�01; �
0
2, on P1(V ) and P2(V ) respectively, satisfyZ
�(g; x)d�(g)d�01(x) =
1;
Z
�2(g; x ^ y)d�(g)d�
0
2(x ^ y) �
2
in view of Lemmas 2.7 and 4.6. Using this property we see that we can assume �
to be extremal �-stationary in the formula
R
�(g; �)d�(g)d�(�) =
2 � 2
1.
We consider the transformation b� on
� P1;2(V ) de�ned by
b�(!; �) = (�!; g1(!):�);
474 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
On Contraction Properties for Products of Markov Driven Random Matrices
the function f(!; �) = �(g1(!); �) and the measure �
� on
�P1;2(V ). We ob-
serve that
�P1;2(V ) is the space of trajectories of the Markov chain on P1;2(V )
with transition kernel R�(�; :) = � � � . Since � is �-stationary extremal, �
� isb�-invariant and ergodic. Since jf(!; �)j � 2Logkg1k + 2Logkg�11 k, it follows that
f is �
� integrable.
On the other hand, the cocycle property for �(g; �) implies
nX
1
f Æ b�k(!; �) = �(Sn(!); �) = Log
kSn(!)x ^ Sn(!)yk
kSn(!)xk2
:
We are going to use Cor. 3.6 with Æz�(!) = lim
n!+1
g�1 ; : : : ; g
�
n:m.
Using Theorem 3.5 and Lemma 3.8, we see that the law of z�(!) 2 P (V ) gives
measure 0 to any projective subspace. In particular, if x 2 P (V ) is �xed, the
condition < z�(!), x >= 0 is sati�ed � � a:e. In other words, using Cor. 3.6
�
�� a:e; lim
n!+1
�(Sn(!); �) = �1:
From above, this implies
�
�� a:e; lim
n!+1
nX
1
(f Æ b�k)(!; �) = �1:
Then, using Lemma 4.7Z
f(!; �)d�
�(�) =
Z
�(g; �)d�(g)d�(�) < 0; i:e
2 < 2
1:
P r o f of Corollary 4.2. We denote un = Sup
x^y2P2(V )
Z
�2(g; x ^ y)d�
n(g) and
we observe that using the cocycle identity for �2: um+n � um + un:
Also, un �
R
Logkg ^ gkd�n(g), hence lim sup
n!+1
un
n
�
2. Furthermore, by
subadditivity of un, the sequence
un
n converges. It follows
lim
n!+1
Sup
x^y2P2(V )
1
n
Z
�2(g; x ^ y)d�
n(g) �
2:
Furthermore Lemma 4.6 implies that there exists x ^ y 2 P2(V ) such that
lim
n!+1
1
n
Z
Log�2(g; x ^ y)d�
n(g) =
2:
Hence lim
n!+1
1
n
Sup
kx^yk=1
Z
Log�2(g; x ^ y)d�
n(g) =
2.
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 475
Y. Guivarc'h
Using Th. 2.3 and the uniform convergence of 1
n
R
Logkgxkd�n(g) to
1,
the statement follows
P r o f of Corollary 4.3. We denote
un(") = Sup
x;y
Z
Æ"(g:x; g:y)
Æ"(x; y)
d�n(g):
Using Schwarz inequality:
un(") � Sup
kxk=kyk=1
Z �
kgx ^ gyk
kgxk2kx ^ yk
�"
d�n(g) = Sup
kxk=kx^yk=1
Z
e"�(g;�)d�n(g):
We observe that
e"� � 1 + "�+ "2�2e"�; j�(g; �)j � 2Log(kgk kg�1k):
Using u2eu � e3juj, we get for 0 � " � "0:
(�2e"�)(g; �) �
1
"20
(kgk kg�1k)6"0 ; un(") � 1 + "
Z
�(g; �)d�n(g) + "2In
with In = 1
"20
R
(kgk kg�1k)6"0d�n(g) < +1. Now we observe that um+n(") �
um(")un(") for m, n 2 N.
It follows: lim
n!+1
(un("))
1=n = Inf
k
(uk("))
1=k . Hence, in order to show
lim
n!+1
(un("))
1=n
� �(") < 1, it su�ces to �nd k 2 N with uk(") < 1.
Using Cor. 4.2, we have
lim
n!+1
1
n
Sup
�
Z
�(g; �)d�n(g) =
2 � 2
1 < 0;
hence we can �x k 2 N such that Sup
�
Z
�(g; �)d�k(g) = ck < 0.
Then uk(") � 1 + ck"+ "2Ik < 0, if " is su�ciently small.
The statement follows.
P r o f of Corollary 4.4. The inequality follows from Cor. 4.3 and a simple
computation. The spectral gap property is a consequence of the spectral theorem
of [26]. The fact that r(P it) < 1 if t 6= 0 follows also from this theorem and
Prop. 6.7 (see [20, 25]).
R e m a r k. The fact that, under condition i:p, the "ergodic Lemma" 4.7
allows to deduce quantitative information from weak convergence of measures on
projective spaces, as in Th. 3.5, was observed in [15]. Under stronger conditions,
Th. 4.1 implies simplicity of Lyapunov spectrum for Sn(!) in the i:i:d case (see
476 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
On Contraction Properties for Products of Markov Driven Random Matrices
Sect. 6). This property has played an important role in the study of pure point
spectrum for Schr�odinger operators on the line if d = 2 [13] and more generally in
the strip [5], as well as for the study of propagation in inhomogeneous waveguides
([38]). The observation in [15] has been developed in [21] and [12]. In [12], it was
observed that condition i:p can be checked from the Zariski density of [supp�]
in G1. The relations between condition i:p and Zariski density of [supp�] in the
context of semi-simple real algebraic groups were studied in [34]. For another
approach to proximality properties see [1].
5. Contraction Properties for Transitive Markov Systems
1) De�nitions.
Let (X; d) be a compact metric space, bX = C(X;X) the semigroup of con-
tinuous maps of X into itself. We endow bX with the Borel structure de�ned by
uniform convergence, and we write a:x for the action of a 2 bX on x 2 X. We con-
sider a class of Markov operators on X de�ned as follows. Let � be a positive
Radon measure on bX, q(x; a) a nonnegative continuous function on X � bX such
that for every x in X,
R
q(x; a)d�(a) = 1.
For a 2 bX , we denote
q(a) = Sup
x2X
q(x; a).
Then we consider the Markov transition kernel Q on X:
Q'(x) =
Z
q(x; a)'(a:x)d�(a);
where ' 2 C(X). Clearly, Q preserves C(X). We denote
= bXN and for
! = (ak)k2N and n 2 N, we write qn(x; !) =
n
�
k=1
q(sk�1(!):x; ak), where sk(!) =
ak � � � a1, s0(!) = Id. Then, for x 2 X, we de�ne a probability measure Qx on
by
Qx(A1 � � � � �An) =
Z
A1�����An
qn(x; !)d�
n(!);
where Ai, 1 � i � n, is a Borel subset of bX . Also, if � 2M1(X), we write Q� =R
Qxd�(x). The shift transformation on
is denoted �, i:e, �(!) = (a2; a3; : : :),
where ! = (a1; a2; : : :) 2
. If � isQ-invariant, then Q� is �-invariant. We observe
that X �
can be identi�ed with the space of trajectories of the Markov chain
de�ned by Q. If (x; !) 2 X �
, a trajectory can be written as the sequence
(sk:x)k2N , hence the shift e� on X �
is given by e�(x; !) = (s1:x; �!). We will
summarize the data (X; q; �) by (X; q
�).
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 477
Y. Guivarc'h
De�nition 5.1. We say that (X; q
�) is a transitive Markov system (t:M:s)
on X if:
a) For every a 2 supp�, Inf
x2X
q(x; a) > 0.
b) In the variation norm on M1(
), Qx depends continuously on x 2 X.
c) The equation Qh = h, with h 2 C(X) implies h = cte.
Condition b seems to be very restrictive. However it is satis�ed in various
situations (see below). If � is Q-stationary, the above conditions imply that Q� is
independent of � and �-ergodic, hence the main role below will be played by Q�,
not by � itself. It is easy to see that, if every [supp�]-orbit is dense, conditions
a,b imply condition c.
2) Some examples.
a) Product measures.
If q(x; a) = q(a), then Qx is the product measure Qx = (q�)
N , hence condi-
tion b is satis�ed
b) Doeblin condition.
If supp� is equal to the set bX of constant maps, then q(x; a) = q(x; y) with
fyg = a:X. If q(x; y) > 0, conditions a,b,c are satis�ed
c) Quantum measurements (see [29]).
We consider the vector space W = C
d , with the usual scalar product and the
vector space H of Hermitian operators on W , H+ � H the cone of nonnegative
operators and we denote q(x; g) = Trg�xg if x 2 H+ n f0g and g 2 G = GL(W ).
Let X = fx 2 H+;Trx = 1g and ~g be the transformation of X de�ned by
~g:x = g�xg
Tr(g�xg)
. If � = fa1; a2; � � � ; apg is a �nite subset of G with
pX
i=1
a�i ai = Id,
we have
pX
i=1
q(x; ai) = 1, hence we can consider the following Markov operator Q
on X:
Q'(x) =
pX
i=1
q(x; ai)'(eai:x):
If � =
pX
i=1
Æai , and [supp�] = [�] satis�es the complex version of condition i:p
(see [17]), then (X; q
�) is a t:M:s. This example can also be considered in the
framework of [23], and of example d below, with s = 1. We can de�ne a norm k:k1
on H as follows. Since x 2 H is conjugate to diag (�1; : : : ; �d) with �i 2 R, we can
write kxk1 =
dX
i=1
j�ij. Then we consider the representation � of G in H de�ned
by �(g)(x) = g�xg and write q(x; g) = k�(g)xk1. Then X can be considered as
478 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
On Contraction Properties for Products of Markov Driven Random Matrices
a part of the complex projective space P (H) and eg as the restriction to X of the
projective map de�ned by �(g).
The corresponding algebraic framework was developed in ([29]).
d) Mellin transforms on GL(V ).
Let G = GL(V ), � 2M1(G) be as in Sects. 3, 4 and assume [supp�] satis�es
condition i:p. We �x a norm v ! kvk on V , and we consider the function (g; v)!
kgvks (s � 0) onG�V . We assume � 2M1;e(G) i:e
R
(kgkc+kg�1kc)d�(g) < +1,
for some c > 0. Then the following function
k(s) = lim
n!+1
�Z
kgksd�n(g)
�1=n
is well de�ned, strictly convex and analytic on ([0, c[) (see [23]). We consider also
the positive operator P s on C(P (V )) de�ned by
P s'(x) =
Z
kgxks'(g:x)d�(g):
Then, there exists a unique positive continuous function es on P (V ) such that
P ses = k(s)es. If we de�ne qs(x; g) =
kgxks
k(s)
es(g:x)
es(x)
we observe that
R
qs(x; g)d�(g))
= 1. As shown in [23], conditions a, b, c are satis�ed by (X; qs
�), i:e (X; qs
�)
is a t:M:s. The function k(s) can be considered as a kind of Mellin transform of
� and is useful in the study of various limit theorems of Probability Theory for
products of random matrices. This is the case for large deviations (see [31]) and
for Cramer estimates of �uctuation theory (see [23, 14]) and below. We observe
that the expression of the operator P s de�ned above is reminiscent of the transfer
operators of thermodynamic formalism (see below). If there exists a closed convex
cone sent into its interior by supp�, then this analogy can be made precisely.
However, in general, it is not possible to distinguish a region of attractivity for
all the maps g 2 supp�, hence a deeper analysis is needed (see [23]).
e) Gibbs measures (see [37]).
Let A be �nite set,
= AN
�
,
� = A�N ; f(!) a Holder function on AZ, �
the shift on AZ. If x 2
�, a 2 A, we de�ne x:a by juxtaposition, and we have
an action of A on
� by continuous maps. We can write f uniquely as
f = f� + 'o� � '+ c;
where c 2 R, '; f� are Holder, f� depends only on the component of ! in
�
and
X
a2A
q(x; a) = 1, where q(x; a) = exp f�(x:a). The transfer operator Q on
�
de�ned by
Q'(x) =
X
a2A
q(x; a)'(x:a)
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 479
Y. Guivarc'h
has a unique stationary measure � and the Gibbs measure on AZ, de�ned by the
potential f , is the unique �-invariant measure on AZ with projection � on
�.
If � is a counting measure on A and X =
�, (X; q
�) is a t:M:s. Then the
probability Qx on
is the conditional law of ! 2
, given x 2
� .
3) Harmonic kernels and contraction properties ([23]).
Here we consider a t:M:s (X; q
�), and a Borel map � from supp� � bX
to GL(V ). If ! = (ak)k2N 2 bXN , we denote gk = �(ak); Sn(!) = gn � � � g1,
sn(!) = an � � � a1 2 bX . We want to construct an analogue of the martingale of
Sect. 3.
De�nition 5.2. Assume (X; q
�) is a t:M:s, and x! �x is a Markov kernel
from X to P (V ). We say that �x is an �-harmonic kernel if:
a) x! �x is continuous in variation;
b) x! �x satis�es the equation �x =
R
q(x; a)�(a):�a:xd�(a).
It follows from this de�nition that the sequence of measures �n(!; x) =
�(a1) � � ��(an):�sn(!):x is a Qx-martingale for any x 2 X.
Theorem 5.3. Let (X; q
�) be a t:M:s and � a Borel map from supp� to
GL(V ) such that [�(supp�)] satis�es condition i:p.
Then there exists z(!) 2 P (V ) de�ned Qx� a:e such that lim
n!+1
g1 � � � gn:m =
Æz(!).
Furthermore, the Qx-law of z(!) is the unique �-harmonic kernel and
! ! z(!) is the unique Borel map which satis�es
Qx � a:e; g1(!):z(�(!)) = z(!):
This theorem can be applied to the � dual� function of �(a); i:e(�(a))�
since the semigroup [��(supp�)] satis�es also condition i:p. It gives, in turn,
information on the product Sn(!), using Lem. 3.10.
Corollary 5.4. For every x 2 P (V ) and v; v0 =2 Ker z�(!), one has the
Qx � a:e convergences
lim
n!+1
g�1 � � � g
�
n:m = Æz�(!);
lim
n!+1
kSn(!)vk
kSn(!)k
= j < z�(!); v > j;
lim
n!+1
Æ(Sn(!):v; Sn(!):v
0)
Æ(v; v0)
= 0:
For �xed !, the last convergences are uniform on every compact subset
of P (V ) n Ker z�(!). Furthermore, if f 2 C(P (V )), the sequence fn(x; v) =R
f(Sn(!):v)dQx(!) is equicontinuous on X � P (V ).
480 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
On Contraction Properties for Products of Markov Driven Random Matrices
We consider now the situation of example d. We �x s � 0, we assume
that � 2 M1(G) satis�es
R
kgkcd�(g) < +1, we denote if s 2 [0; c[, k(s) =
lim
n!+1
(
Z
kgksd�n(g))1=n and we consider the positive operator P s on C(P (V ))
de�ned by
P s'(x) =
Z
kgxks'(g:x)d�(g):
We assume that [supp�] � G = GL(V ) satis�es the condition i:p; hence there
exists a unique normalized positive and continuous function es on P (V ) such that
P ses = k(s)es.
Then we write qs(x; g) =
kgxks
k(s)
es(g:x)
es(x)
and we consider the t:M:s (P (V ); q
�).
We denote by Qs
x the Markov measure on
= GN de�ned by qs and �. Here we
consider the function ��(g) = g� 2 GL(V ) and apply the above corollary to this
situation. In particular, we denote z�s(!) the point of P (V ) de�ned by
Qs
x � a:e; Æz�s (!) = lim
n!+1
g�1 � � � g
�
n:m:
Then we can compare Qs
x and Qs
y in terms of z�s(!), as follows.
Corollary 5.5. For every x; y 2 P (V ) the Markov measures Qs
x and Qs
y on
GN are equivalent and
dQs
x
dQs
y
(!) =
����< z�s (!); x >
< z�s (!); y >
����s es(y)
es(x)
:
In particular, for the laws �sx and �sy of z�s (!) we have
d�s
x
d�s
y
(z) =
���<z;x><z;y>
���s es(y)
es(x)
.
4) Angles of column vectors: exponential decrease.
Here we give a quantitative version of the contraction property studied in
Sect. 4 and in the above paragraph. We consider the t:M:s (X; q
�), a Borel
map � from supp� � bX into GL(V ) and we assume the �niteness of the integralsR
Logk�(a)kq(a)d�(a) and
R
Logk�(a)�1kq(a)d�(a).
We denote by � a stationary measure on X and by Q� the corresponding
Markov measure on GN . With the above notations we de�ne
q
1 = lim
n!+1
1
n
Z
LogkSn(!)kdQ�(!);
q
2 = lim
n!+1
1
n
Z
LogkSn(!) ^ Sn(!)kdQ�(!):
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 481
Y. Guivarc'h
Theorem 5.6. Let (X; q
�) be a t:M:s, � a Borel map of bX into GL(V ) such
that [supp�(�)] satis�es condition i:p. We assume the �niteness of the integralsR
Logk�(a)kq(a)d�(a) and
R
Logk�(a)�1kq(a)d�(a).
Then the sequence
Sup
(x;v;v0)2X�P (V )�P (V )
Z
Log
Æ(Sn(!):v; Sn(!):v
0)
Æ(v; v0)
dQx(!)
converges to
q
2 � 2
q
1 < 0.
In the special case of a t:M:s associated with a Gibbs measure we have
Corollary 5.7. Assume A is a �nite set, � is a Gibbs measure on
= AN
de�ned by a Holder potential, � a Borel map from A to GL(V ) such that the semi-
group [�(A)] satis�es condition i:p. Then one has the inequality
q
2 < 2
q
1 , where
q
1 = lim
n!+1
1
n
Z
LogkSn(!)kd�(!),
q
2 = lim
n!+1
1
n
Z
LogkSn(!) ^ Sn(!)kd�(!).
In the situation of example d above, under exponential moment and i:p con-
ditions (see Subsects. 3 and 4 above), we can develop, following [23], a spectral
analysis of the operators P s(s � 0) on the space H"(P (V )) of "-Holder functions.
For a subset S � G we write
1(S) = lim
n!+1
1
n
Log Sup fkgk; g 2 Sng. This gives
in particular
Corollary 5.8. With the above hypothesis and notation above, the operator
P s on H"(P (V )), de�ned by
P s'(v) =
Z
kgvks'(g:v)d�(g);
has spectral radius k(s). It has the unique normalized eigenfunction es and eigen-
measure �s:
P ses = k(s)es; P s�s = k(s)�s; jesj1 = 1; �s(es) = 1;
where es > 0.
For " small, one has the direct sum decomposition P s = k(s)(�s
es + Rs),
where Rs commutes with P s and has spectral radius rs(") < 1. The function k(s)
is analytic on [0; c[ and k0(0) =
1. If c =1, then lim
s!+1
Logk(s)
s
=
1(supp�).
Let Qs be the Markov operator de�ned by Qs' = 1
k(s)es
P s(es') and �n;s(") =
Sup
v;v0
R Æ"(Sn(!):v;Sn(!):v
0)
Æ"(v;v0)
dQs
v(!).
Then, for " small, we have lim
n!+1
(�n;s("))
1=n = �s(") < 1.
In particular, if s is �xed, the resolvent (�I�Qs)�1 has a simple pole at � = 1
and is holomorphic in the domain f� 2 C ; j�j > rs("); � 6= 1g.
482 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
On Contraction Properties for Products of Markov Driven Random Matrices
6. On Some Consequences
1) Lyapunov spectrum.
Let (
; �; �) be a measured dynamical system where � is a �-invariant and er-
godic probability measure, � a Borel function from
to G = GL(V ). We assume
that the functions Logk�(!)k and Logk��1(!)k are �-integrable. We write for
i 2 N, �(�i!) = gi(!) and we consider the product Sn(!) = gn(!) � � � g1(!) 2 G.
In general, if v 2 V , the asymptotic behaviour of Sn(!)v is described by the
multiplicative ergodic theorem of V.I. Oseledets ([33]). For a recent detailed proof
of this result see ([36]). A more elementary approach is to consider the quantities
(1 � i � d):
1 = lim
n!+1
1
n
Z
LogkSn(!)kd�(!);
2 = lim
n!+1
1
n
Z
LogkSn(!) ^ Sn(!)kd�(!);
i = lim
n!+1
1
n
Z
Logk ^i Sn(!)kd�(!);
where the limits of the quantities under the integrals exist a:e by the subadditive
ergodic theorem. The following result (see [35]) allows to de�ne the so-called
Lyapunov spectrum of Sn(!).
Theorem 6.1. Assume (
; �; �) and �(!) are as above. Then we have the con-
vergence
� � a:e; lim
n!+1
1
2n
Log(S�nSn) = ^(!);
where ^(!) is a symmetric endomorphism of V .
The spectrum of ^(!) is constant � � a:e, and of the form (�1; �2; : : : ; �p),
where �1 > �2 > � � � > �p, and �j , 1 � j � p, has multiplicity mj > 0. Each �j is
called a Lyapunov exponent of Sn(!), and �1 is called the top Lyapunov exponent.
Clearly, �1 =
1. If m1 = 1, then �2 =
1 +
2, hence �2 � �1 =
2 � 2
1 < 0
controls the exponential decay of � ((Sn(!)v, Sn(!)v
0), where v; v0 are "typical"
vectors. Conversely, if
2 � 2
1 < 0, then m1 = 1.
These facts allow to translate Th. 5.6 into
Theorem 6.2. [15]. Assume (X; q
�) is a t:M:s and the Borel map � from bX
to G is such that the integrals
R
Logk�(a)kq(a)d�(a) and
R
Logk��1(a)kq(a)d�(a)
are �nite, and [�(supp�)] satis�es condition i:p. Then the top Lyapunov expo-
nent of
Sn(!) = gn(!) � � � g1(!)
has multiplicity 1.
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 483
Y. Guivarc'h
In order to deal more generally with the irreducibility and proximality ques-
tions, it is convenient to recall the
De�nition 6.3. Let U be a subset of G, I(U) the set of real polynomials in
the coe�cients of g and (detg)�1 which vanishes on U . Then
U�Z = fg 2 G;8P 2 I(U); P (g) = 0g
is called the Zariski closure of U .
The Zariski topology on G is de�ned by its closed sets, i:e sets U with U =
U�Z . If U is a semigroup, then U�Z is a closed Lie subgroup of G with a �nite
number of connected components. An important fact observed in [12] and [34]
is that, if S � G is a subsemigroup, a proximal element exists in S i� such
an element exists in S�Z . Taking this into account, Th. 5.2 gives the following
extension of an important result of [12].
Corollary 6.4. Assume (X; q
�) is a t:M:s and [�(supp�)]�Z contains
SL(V ). Then the Lyapunov spectrum of Sn(!) is simple, i:e each Lyapunov
exponent has multiplicity one.
For some applications of geometrical character (see, for example, [17, 18, 8])
it is convenient to have "intrinsic" forms of the above. Then we consider a semi-
simple algebraic group G, de�ned over R. We denote by GR the group of its real
points and we use as a tool the Zariski topology on GR. We assume G to the
Zariski connected. For a Lie subgroup L of GR we denote its Lie algebra by the
calligraphic letter L. We consider a maximal connected subgroup A � G such that
Ad A is diagonal, a maximal compact subgroup K and the polar decomposition
G = KA
+
K, where A+ is an open Weyl chamber of A and A+ is its closure.
If d 2 A, write Log d for the unique element of A such that expLogd = d.
We write, if g 2 G; g = kd(g)k0, with d(g) 2 A
+
, k; k0 2 K, and we �x a norm on
A � G.
Let (X; q
�) be a t:M:s, � a Borel function from bX to G such that the integralR
kLogd(�(u))kq(u)d�(u) is �nite and write Sn(!) as Sn(!) = kndn(!)k
0
n with
kn; k
0
n 2 K, dn(!) 2 A
+
. Then, using the subadditive ergodic theorem, we can
de�ne the "Lyapunov vector" L(�) 2 A
+
by
� � a:e; lim
n!+1
1
n
Logdn(!) = L(�):
In particular, in the i:i:d case we write L(�) for L(�).
484 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
On Contraction Properties for Products of Markov Driven Random Matrices
Theorem 6.5. [12]. Assume (X; q
�) is a t:M:s, � a Borel function frombX to GR. Assume [�(supp�)]�Z = GR.
Then L(�) 2 A+.
We observe that the condition L(�) 2 A+ can be satis�ed under much weaker
conditions than [�(supp�)]�Z = GR, in particular, if G has a complex structure.
For a detailed study see [17], and for an extension to local �elds see [22].
2) Some limit theorems.
Assume here that � 2 M1(G) and Sn(!) is the product of random matrices
Sn(!) = gn � � � g1. We are interested by re�nements of the results in the above
paragraphs, i:e, by re�nements of the law of large numbers for Sn(!). Hence we
have to consider a possible degeneracy of limiting laws. It turns out that if dim
V > 1, such degeneracies can be avoided if we assume geometric conditions like
Zariski density of [supp�] or condition i:p for [supp�]. We recall that if d = 1, these
degeneracies depend on arithmetic conditions on supp�. We begin by developing
some results of this type and we formulate them for a general semigroup � instead
of a semigroup of the form [supp�].
De�nition 6.6. For a proximal element g 2 GL(V ) = G we denote �(g) =
Logr(g), where r(g) is its spectral radius. For a semi-group � � G, we denote by
�(�) the set of its proximal elements.
Then we have the
Proposition 6.7 ([25]). Assume � � G satis�es condition i:p. Then �(�(�))
generates a dense subgroup of R.
If GR is as in the above paragraph, and g 2 GR, we need to consider other
notions of proximality related to the actions on the �ag spaces of G.
De�nition 6.8. Assume g 2 GR and write L(g) = lim
n!+1
1
n
Logd(gn) 2 A
+
.
We say that g is �ag proximal if L(g) 2 A+. For a semigroup � � GR, we denote
by �prox the set of its �ag proximal elements.
Then we have [17].
Theorem 6.9. Assume � is a Zariski-dense subsemigroup of GR. Then
L(�prox) generates a dense subgroup of A.
The following is an analogue of the classical renewal theorem, where
:
V is the
factor space of V by �Id.
Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 485
Y. Guivarc'h
Theorem 6.10. Assume dim V > 1 and � 2 M1(G) is such thatR
Logkgkd�(g) and
R
Logkg�1kd�(g) are �nite, [supp�] satis�es condition i:p
and
�(�) = lim
n!+1
1
n
Z
Logkgkd�n(g) > 0:
Then, for every v 2
:
V nf0g, the potential
1X
0
�k � Æv is �nite and
lim
v!0
1X
0
�k � Æv =
1
�(�)
�
`;
where � is the unique �-stationary measure on P (V ) and ` = dr
r is the radial
Lebesgue measure on R
�
+ .
The multidimensional analogues of this result can be applied to dynamical
problems like density of orbits for the groups of automorphism action on tori
([19, 25]).
In the case �(�) < 0 and
1(supp�) > 0, there exists � > o such that
k(�) = 1, as explained in Sect. 5. In particular, we have the so-called Cramer
estimate as the consequence of Cor. 5.8.
Theorem 6.11. With the above notation, assume [supp�] satis�es condition
i:p,
1 < 0,
1 (supp�) > 0 and
R
kgkcd�(g) +
R
kg�1kcd�(g) < +1 for some
c > 0. Let � 2]0; c[ be de�ned by k(�) = 1. Then, for every v 2 V n f0g,
the sequence of functions t��f! 2
; sup
n2N
kSn(!)vk > tg converges to a positive
function on P (V ) proportional to e�(v).
This result allows to study the tail of stationary solutions of a�ne recursions
on R
d of the form Xn+1 = An+1Xn +Bn+1, where (An; Bn) 2 Aff(Rd) are i:i:d
([28, 14]).
Furthermore, the existence of such tails allows to obtain fractional expansions
of Lyapunov exponents near critical points for some classes of products of random
matrices (see [6] for d = 2). Near a point � 2 M1(G) such that [supp�] satis�es
condition i:p, the top Lyapunov exponent is in general nondi�erentiable, but only
Hoelder (see [30]).
For the Gaussian behaviour of LogkSn(!)vk we refer to [3, 16, 20, 27, 38].
The convergence to the Gaussian law can also be studied in the context of i:i:d
random variables taking values in a semi-simple group of the form GR, as in
Subsect. 1. In the notations of Th. 5 we have
Theorem 6.12. Assume � 2M1(GR) satis�es
R
exp ckLogd(g)kd�(g)< +1,
for some c > 0, and [supp�]�Z = GR. Then
1p
n
(Logd(Sn)� nL(�)) converges in
law to a Gaussian law on A with full dimension.
486 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4
On Contraction Properties for Products of Markov Driven Random Matrices
R e m a r k s. This theorem extends the result of [11], which was stated
for GR = SL(d;R). The proof is based on the spectral properties of �ag space
analogs of the Fourier operators P it(t 2 R) from Sect. 4. The fullness of the
Gaussian law is a consequence of Th. 6.9 (see [17]).
A special case of interest for Mathematical Physics is GR = Sp(2n;R).
We observe that the exponential moment condition is not necessary for the
validity of Th. 6.12. One can expect that a 2-moment condition is su�cient.
The method used for the proof of Th. 6.5, i.e the construction of a suitable
martingale as in Th. 5.3, remains valid in more general settings. For examples of
such results see [4, 2].
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