David Andrieux

Pierre Gaspard

Sergio Ciliberto

Nicolas Garnier

Sylvain Joubaud

Artyom Petrosyan

Entropy production and time asymmetry in nonequilibrium fluctuations

Keywords: numbers: 05.70.Ln, 05.40.-a, 02.50.Ey

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Newton's equations ruling the motion of particles in matter are known to be time-reversal symmetric. Yet, macroscopic processes present irreversible behavior in which entropy is produced according to the second law of thermodynamics. Recent works suggest that this thermodynamic time asymmetry could be understood in terms similar as those used for other symmetry breaking phenomena in condensed matter physics. The breaking of time-reversal symmetry should concern the fluctuations in systems driven out of equilibrium. These fluctuations may be described in terms of the probabilities weighting the different possible trajectories of the systems. Albeit the time-reversal symmetry of the microscopic Newtonian dynamics says that each trajectory corresponds to a time-reversed one, it turns out that distinct forward and backward trajectories may have different probability weights if the system is out of equilibrium. For example, the probability for a driven Brownian particle of having a trajectory from a point A to a point B is different of having the same reverse trajectory from B to A.

This important observation can be further elaborated to establish a connection with the entropy production. We consider the paths or histories z = (z 0 , z 1 , z 2 , ..., z n-1 ) obtained by sampling the trajectories z(t) at regular time intervals τ . The probability weight of a typical path is known to decay as

P + (z 0 , z 1 , z 2 , ..., z n-1 ) ∼ exp(-nτ h) (1)
as the number n of time intervals increases [1,2,3,4].

The decay rate h is known as the entropy per unit time of dynamical systems theory [1,2,3,4]. This quantity characterizes the temporal disorder, i.e., the dynamical randomness of the stochastic process. We can compare (1) with the probability weight of the time-reversed path z R = (z n-1 , ..., z 2 , z 1 , z 0 ) in the nonequilibrium system with reversed driving constraints (denoted by the minus sign):

P -(z n-1 , ..., z 2 , z 1 , z 0 ) ∼ exp(-nτ h R ) . (2)
It can be shown that, out of equilibrium, the probabilities of the time-reversed paths decay faster than the probabilities of the paths themselves [5]. We can interpret this as a breaking of the time-reversal symmetry in the invariant probability distribution describing the nonequilibrium steady state, the fundamental underlying Newtonian dynamics still being time-reversal symmetric. The decay rate h R in Eq. ( 2) is called the time-reversed entropy per unit time and characterizes the dynamical randomness of the time-reversed paths [5,6]. The remarkable result is that the difference between both quantities h R and h gives the entropy production of irreversible thermodynamics

1 k B d i S dt = h R -h (3)
where k B is Boltzmann's constant [5,6,7,8,9]. In this sense, entropy production would find its origin in the time asymmetry of dynamical randomness. At equilibrium, the entropy production vanishes and both quantities h R and h are equal, as expected from the principle of detailed balance.

The purpose of the present Letter is to provide experimental evidence for the above connection in two nonequilibrium fluctuating systems. The first system is a Brownian particle in a fluid at inverse temperature β = (k B T ) -1 and bounded by a laser trap with a time-dependent potential V (x t , t) where x t is the particle position at time t. This potential is harmonic and driven at the velocity u so that V (x t , t) = (k/2)(x t -ut) 2 with stiffness k. The surrounding fluid is responsible for a friction force -αdx t /dt and the corresponding Langevin fluctuating force. In the experimental setup, k = 9.62 10 -6 kg s -2 and the relaxation time τ R = α/k = 3.05 10 -3 s. The second system is an electric circuit driven out of equilibrium by a current source which imposes the mean current I. The current in the circuit fluctuates because of the intrinsic thermal noise of the circuit (Nyquist noise) [10,11]. Specifically the circuit is composed of a capacitance C = 278 pF in parallel with a resistance R = 9.22 MΩ so that the time constant of the circuit is τ R = RC = 2.56 10 -3 s. This electric circuit and the driven Brownian particle, although physically different, are known to be formally equivalent by the correspondence α ↔ R, k ↔ 1/C and u ↔ I while x t corresponds to the charge q t inside the resistor at the time t [10,11]. The variables x t and q t are acquired at a sampling frequency 1/τ = 8192 Hz. In both experiments, the temperature is T = 298 K.

The proposed method is very closely related to the experiments recently undertaken to measure nonequilibrium fluctuation relations [11,12,13]. Here, we investigate the possibility to extract the dissipated heat and the entropy production from quantities -the so-called entropies per unit time -characterizing the dynamical randomness of the paths and the corresponding timereversed paths with also a reversal of the sign of the nonequilibrium constraint. The entropies per unit time can be calculated from two long time series with sufficient temporal and spatial resolutions, measured in two similar experiments but one driven with an opposite constraint, i.e., u → -u and I → -I. The experiment thus consists in recording a pair of long temporal series in each of the two different experimental setups.

We present the proposed method in the case of the Brownian particle in a moving trap. Following Ref. [14], the heat Q t dissipated along a random trajectory during a time interval t is given by

Q t = t 0 dx t ′ dt ′ F (x t ′ -ut ′ ) dt ′ . (4)
where F (x t , t) = -∂V (x t , t)/∂x t = -k(x t -ut) is the force exerted by the potential V (x t , t) of the trap. This dissipated heat is a random variable that has been shown to obey an extended fluctuation theorem due to his non-Gaussian tail [14]. After a long enough time, the system reaches a nonequilibrium steady state, in which the entropy production is related to the mean value of the dissipated heat according to

d i S dt = 1 T d Q t dt = αu 2 T . ( 5
)
Our aim is thus to show that one can extract the heat dissipated during an individual path by comparing the probabilities of this path with the time-reversed path having also reversed the displacement of the potential, i.e., u → -u.

We first make the change to the comoving frame with the minimum of the potential so that z ≡ x -ut. After initial transients, the system will reach a steady state characterized by a stationary probability distribution. As we are interested in the probability of a given succession of states corresponding to a discretization of the signal at small time intervals τ , a multi-time random variable is defined according to Z = [Z(t 0 ), Z(t 0 + τ ), . . . , Z(t 0 + nτ -τ )] which corresponds to the signal during the time period t -t 0 = nτ . For a stationary process their distribution do not depend on the initial time t 0 . From the point of view of probability theory, the process is defined by the n-time joint probabilities P σ (z; dz, τ, n) = Pr{z < Z < z + dz; σ} = p σ (z)dz, where p σ (z) is the probability density for Z to take the value z = (z 0 , z 1 , . . . , z n-1 ) at times t 0 + iτ for a nonequilibrium driving σ = u/|u| = ±1. In the overdamped case, the motion of the Brownian particle can be modeled as a Langevin equation with a Gaussian white noise. Thanks to the Markovian nature of this process, the joint probabilities can be decomposed into the products of the Green functions G(z i , z i-1 ; τ )dz i for i = 1, . . . , n. G(z, z 0 ; t) gives the probability density to be at position z at time t given that the initial position was z 0 [15,16]. To extract the dissipation occurring along a single trajectory, one has to look at the ratio of the probability of the forward path over the probability of the reversed path having also reversed the displacement of the potential. Indeed, taking the logarithm of this ratio and the continuous limit τ → 0, n → ∞ with nτ = t, we find ln

P + (z; dz, τ, n) P -(z R ; dz, τ, n) = βu t 0 F (z t ′ ) dt ′ -β V (z t )-V (z 0 ) (6)
which is exactly the heat Q t in Eq. ( 4) expressed in the z variable and multiplied by the inverse temperature. This relation holds at the level of the paths of the particle. A similar relation but concerning the distribution of the work done on a time-dependent system has been derived by Crooks [17,18]. We emphasize that the reversal of u is essential for the logarithm of this ratio of probability to present a behavior growing linearly in time.

This calculation shows that the heat dissipated along a single trajectory can be calculated from a ratio of probabilities involving this trajectory and its time reversal with the supplementary condition that the sign of u be reversed as well. Now, due to the continuous nature in time and in space of the process, one has to consider (ǫ, τ ) quantities, i.e. quantities defined on cells of size ǫ and measured at time intervals τ . Therefore, let us consider the probability P + (Z m ; ǫ, τ, n) for the path to remain within a distance ǫ of some reference path Z m , made of n successive positions of the Brownian particle observed at time intervals τ for the forward process. The probability is obtained by searching for the recurrences of M such reference paths or patterns in the time series. Next, we can introduce the quantity P -(Z R m ; ǫ, τ, n) which is the probability for a reversed path of the reversed process to remain within a distance ǫ of the reference path Z m (of the forward process) for n successive positions. According to a numerical procedure proposed by Grassberger, Procaccia and others [1,2] the entropy per unit time can be estimated by the linear growth of the mean 'pattern entropy' defined as

H(ǫ, τ, n) = - 1 M M m=1 ln P + (Z m ; ǫ, τ, n) (7)
and, similarly,

H R (ǫ, τ, n) = - 1 M M m=1 ln P -(Z R m ; ǫ, τ, n) (8)
for the reversed process. The (ǫ,τ )-entropies per unit time h(ǫ, τ ) and h R (ǫ, τ ) are defined by the linear growth of the mean pattern entropy as a function of the time nτ of the sequences [1,2,4]. The mean entropy production in the nonequilibrium stationary state should thus be given by the difference between these two quantities:

1 k B d i S dt = lim ǫ→0 lim τ →0 [h R (ǫ, τ ) -h(ǫ, τ )] . (9)
It is important to note that the probability of the reversed paths are compared and averaged over the paths of the forward process in order to obtain Eq. ( 9). The entropy production is thus expressed as the difference of two usually very large quantities which increase for ǫ, τ going to zero with a scaling law in ǫ -2 as shown in Ref. [4,19]. Nevertheless, the difference remains finite and gives the entropy production. Equation ( 9) has the general form of large-deviation dynamical relationships recently derived in the context of nonequilibrium statistical thermodynamics [6]. In order to test experimentally Eq. ( 9), we have analyzed for a specific value of |u| a pair of time series up to 5 10 7 points each, one corresponding to the forward process and the other corresponding to the reversed process, having discarded first the transient evolution. depicts examples of paths z(t) for the Brownian particle in a moving optical trap.

For different values of ǫ between 5.6-11.2 nm, the mean pattern entropy ( 7) is calculated with the distance defined by taking the maximum among the deviations |Z(t) -Z m (t)| with respect to some reference path Z m for the times t = 0, τ, . . . (n -1)τ . The forward entropy per unit time h(ǫ, τ ) is evaluated from the linear growth of the mean pattern entropy (7) with the time nτ . The backward entropy per unit time h R (ǫ, τ ) is obtained similarly from the time-reversed pattern entropy (8). The difference of the two dynamical entropies gives the entropy production depicted in Fig. 2a for different driving speeds u. We get a good agreement with the theoretical prediction (5).

On the other hand, we have analyzed by the same method the time series of the RC electric circuit. We see in Fig. 2b that the entropy production obtained from the time series analysis of the RC circuit agrees very well with the known Joule law, which is a further confirmation of Eq. (9).

We also tested the possibility to extract the heat (4) dissipated along a single stochastic path by searching for the recurrences in the time series according to Eq. ( 6).

A randomly selected path as well as the corresponding heat dissipated are plotted in Fig. 3. We find a very good agreement so that the relation ( 6) is also verified at the level of single paths. At this level, the heat exchanged between the particle and the surrounding fluid can be positive or negative because of the molecular fluctuations. It is only by averaging over the forward process that the dissipated heat takes the positive value depicted in Fig. 2. The dissipated heat given by the logarithm of the ratio of the forward and backward probabilities according to Eq. ( 6) for different values of ǫ = k × 0.558 nm with k = 11, . . . , 20 in the range 6.1-11.2 nm. They are compared with the theoretical value (squares) calculated from Eq. ( 4). For small values of ǫ, the agreement is quite good for short time and within experimental errors for larger time.

In conclusion, we measured the entropy production by searching the recurrences of trajectories in the fluctuating dynamics of two nonequilibrium processes. The ex-periments we performed consisted in the recording of two long time series. The first one corresponds to a forward experiment while the other is measured from the same experimental setup except that the sign of the constraint driving the system out of equilibrium has been reversed. From these two time series, we are able to compute two dynamical entropies, the difference of which gives the entropy production. Moreover, we tested the possibility to extract the dissipated heat along a single random path. This shows that the entropy production arises from the breaking of the time-reversal symmetry in the probability distribution of the statistical description of the nonequilibrium steady state. Since the decay rates of the multitime probabilities of the forward and reversed paths characterize their dynamical randomness, the present results show that the thermodynamic entropy production finds its origin in the time asymmetry of the dynamical randomness.

FIG. 1 :

 1 FIG.1: Time series of typical paths z(t) for the Brownian particle in the optical trap moving at the velocity u for the forward process and -u for the reversed process with u = 4.24 10 -6 m/s.

Figure 1

 1

FIG. 2 :

 2 FIG.2:(a) Entropy production of the Brownian particle versus the driving speed u. The solid line is given by Eq. (5). (b) Entropy production of the RC electric circuit versus the injected current I. The solid line is the Joule law, diS/dt = RI 2 /T . The dots are the results of Eq. (9).

FIG. 3 :

 3 FIG.3: Measure of the heat dissipated by the Brownian particle along the forward and reversed paths of Fig.1. The trap velocities are ±u with u = 4.24 10 -6 m/s. We are searching for recurrences between the two processes. (a) Inset: A randomly selected trajectory in the time series. The probabilities of the corresponding forward (black) and the backward (red) paths for ǫ = 8.4 nm. These probabilities present an exponential decrease modulated by the fluctuations. (b) The dissipated heat given by the logarithm of the ratio of the forward and backward probabilities according to Eq. (6) for different values of ǫ = k × 0.558 nm with k = 11, . . . , 20 in the range 6.1-11.2 nm. They are compared with the theoretical value (squares) calculated from Eq. (4). For small values of ǫ, the agreement is quite good for short time and within experimental errors for larger time.

Acknowledgments. This research is financially supported by the F .N .R .S . Belgium and the "Communauté française de Belgique" (contract "Actions de Recherche Concertées" No. 04/09-312).


Abdelhamid Hassairi
email: abdelhamid.hassairi@fss.rnu.tn

Criteria of convergence of a non ordinary random continued fractions on a symmetric cone

Keywords: Symmetric matrices, Jordan algebra, symmetric cone, division algorithm, continued fractions. AMS Classification : 40A15, 17C65, 60B20

Criteria of convergence of a non ordinary random continued fractions on a symmetric cone

Abdelhamid Hassairi

Introduction

Besides the well known theory of real continued fractions and its applications, many multivariate versions have been introduced to bring answers to some needs which have arisen in different areas. The most important seem to be the ones defined on some matrix spaces. Since there is no single way to define a matrix inverse, there exist many ways to define continued fractions with matrix argument.

A first natural way is to assume invertibility and use the classical matrix inverse to define a ratio of matrices. A second way is based on the partial matrix inverse. Another way uses the generalized inverse considered to be more efficient in matrix continued fraction interpolation problems. For details about these different matrix continued fractions and their applications, we refer the reader to [START_REF] Zhaoa | Matrix-valued continued fractions[END_REF] and to the references within. It is worth pointing out that for the applications, the convergence of a matrix continued fraction is usually needed. However even if in many cases, a three term recurrence relation is available, because of the non-commutativity, it is difficult to derive applicable convergence criteria. The present work is motivated by probability problems concerning the characterization of some distributions on the cone of positive symmetric matrices or more generally on any symmetric cone such as the beta-hypergeometric distribution. The results will be given in the general setting of a Jordan algebra and its symmetric cone, however to make the paper accessible for a reader who is not familiar with the theory of Jordan algebra, we have emphasized on the case of symmetric matrices. Of course in these spaces, we can't use the multiplication by the inverse to define a ratio and we need to use a suitable division algorithm. We mention here that, for the proof of a characterization result concerning the Wishart distribution on a symmetric cone, Evelyne Bernadac [START_REF] Bernadac | Random continued fractions and inverse Gaussian distribution on a symmetric cone[END_REF] has used an ordinary continued fraction defined with a construction process based on the inversion without any use of multiplication or ratio. Assuming the independence of the random variables and some conditions on their distributions, she has given a criteria of convergence for the ordinary continued fraction. As a continuation of this work, we first use a notion of quotient based on a division algorithm to define in its more general form, a continued fraction K(x n /y n ) where (x n ) and (y n ) are two sequences of elements of the symmetric cone. The ordinary continued fraction considered in [START_REF] Bernadac | Random continued fractions and inverse Gaussian distribution on a symmetric cone[END_REF] corresponds to the case where x n = e, for all n, where e is the identity element of the algebra. We then show that, as in the classical matrix case (see [START_REF] Kacha | Convergence of matrix continued fractions[END_REF]), any non ordinary continued fraction is equivalent to an ordinary one. This result has a mathematical interest, but it has no implication on the convergence of the non ordinary random continued fraction. In fact, when the sequences in the non ordinary fraction are constituted of independent random variables, the corresponding ordinary continued fraction has not this property. Accordingly, the main part of the paper is devoted to establish a criteria of convergence for the non ordinary random continued fractions of the form K(x n /e), where (x n ) is a sequence of independent random variables valued in a symmetric cone. This kind of non ordinary random continued fractions with the ordinary ones arise in the characterization of many important distributions on symmetric matrices or more generally on any symmetric cone.

Symmetric cones

It is well known that any symmetric cone is associated to a Jordan algebra, so that in order to present our results in their most general form, it is necessary to review some facts concerning Jordan algebra and their symmetric cones. Our notations are the ones used in [START_REF] Faraut | Analysis on symmetric cones[END_REF] or in [START_REF] Hassairi | Riesz exponential families on symmetric cones[END_REF]. Let us recall that a Euclidean Jordan algebra is a Euclidean space V with scalar product < x, y > and a bilinear map

V × V -→ V, (x, y) -→ x.y
called Jordan product such that, for all x, y, z in V , i) x.y = y.x, ii) < x, y.z >=< x.y, z >, iii) there exists e in V such that e.x = x, iv) x.(x 2 .y) = x 2 .(x.y), where we used the abbreviation x 2 = x.x. A Euclidean Jordan algebra is said to be simple if it does not contain a nontrivial ideal. For the algebra of symmetric matrices, the Jordan product is defined by x.y = 1 2 (xy + yx) where xy is the ordinary product of the matrices x and y and the scalar product is defined by < x, y >= trace(xy). Now, to each Euclidean simple Jordan algebra V , we attach the set of Jordan squares Ω = {x 2 ; x ∈ V }.

Its interior is denoted Ω.

In general, Ω is a symmetric cone, i.e., a convex cone which is i) self dual, i.e., Ω = {x ∈ V ; < x, y > > 0 ∀y ∈ Ω \ {0}}.

ii) homogeneous, i.e., the group G(Ω) of linear automorphisms of Ω acts transitively on Ω. Observe that when V is the algebra of symmetric matrices, Ω (resp. Ω) denotes the cone of symmetric positive definite (resp. nonnegative) matrices. By analogy with the case where V = R and Ω =]0, ∞[, for x and y in the cone Ω, we write 0 < x < y if y -x ∈ Ω. With this, one can talk about an increasing or decreasing sequence in Ω. Given a Euclidean simple Jordan algebra V , we denote by G the connected component of the identity in G(Ω). If V is the space of (r, r)-symmetric matrices and GL(R r ) is the group of invertible matrices, The elements of G(Ω) are the maps g : V -→ V such that there exists a in GL(R r ) with

g(x) = axa * ,
where a * is the transpose of a. For two automorphisms g 1 and g 2 in G, we write g 1 g 2 the composition of g 1 and g 2 , and for g 1 , ..., g n in G, we write n i=1

g i = g 1 ...g n .
Throughout, we will also use the following notations for (x, y, z) ∈ V : L(x)y = xy P (x)y = 2x(xy) -x 2 y (quadratic representation)

x y = L(xy) + [L(x), L(y)] = L(xy) + L(x)L(y) -L(y)L(x).

An element c of V is said to be idempotent if c 2 = c. It is a primitive idempotent if furthermore c = 0 and is not the sum t + u of two non-null idempotents t and u such that t.u = 0.

A Jordan frame is a set {c 1 , ..., c r } of primitive idempotents such that r i=1 c i = e and c i .c j = δ ij c i , for 1 ≤ i, j ≤ r. The size r of such a frame is a constant called the rank of V . For any element x of a Euclidean simple Jordan algebra, there exist a Jordan frame (c i ) 1≤i≤r and (λ 1 , ..., λ r ) in R r such that x = r i=1 λ i c i . The real numbers λ 1 , λ 2 , ..., λ r depend only on x. They are called the eigenvalues of x and this decomposition is called its spectral decomposition. The trace and the determinant of x are then respectively defined by tr(x) = r i=1

λ i and ∆(x) = r i=1 λ i .
If c is a primitive idempotent of V , the only possible eigenvalues of L(c) are 0, 1 2 and 1. The corresponding eigenspaces are respectively denoted by V (c, 0), E(V, 1 2 ) and V (c, 1) and the decomposition

V = V (c, 0) ⊕ V (c, 1 2 ) ⊕ V (c, 1)
is called the Peirce decomposition of V with respect to c. Suppose now that (c i ) 1≤i≤r is a Jordan frame in V and for 1 ≤ i, j ≤ r, let

V ij = V (c i , 1) = Rc i if i = j V (c i , 1 2 ) ∩ V (c j , 1 2 ) if ieqj.
Then (See [START_REF] Faraut | Analysis on symmetric cones[END_REF] Th.IV.2.1) we have V = ⊕ i≤j V ij and the dimension of V ij is, for i = j, a constant d called the Jordan constant. It is related to the dimension n and the rank r of V by the relation n = r + r(r -1) d 2 . When V is the algebra of symmetric matrices, d = 1. If c is an idempotent and if z is an element of V (c, 1 2 ),

τ c (z) = exp(2z c)
is called a Frobenius transformation. It is an element of the group G.

Given a Jordan frame (c i ) 1≤i≤r , the subgroup of

G T =    τ c 1 (z (1) ) • • • τ c r-1 (z (r-1) )P r i=1 a i c i , a i > 0, z (j) ∈ r k=j+1 V jk   
is called the triangular group corresponding to the Jordan frame (c i ) 1≤i≤r .

It is an important result (see [START_REF] Faraut | Analysis on symmetric cones[END_REF], p.113) that for an element y in the symmetric cone Ω there exists a unique element in the triangular group T such that y = t(e).

To define a notion of "quotient" in the symmetric cone Ω, we usually introduce a division algorithm which is a measurable map g from Ω into G such that, for all y ∈ Ω, g(y)(y) = e. With the algorithm g, the "quotient" of an element x by y is defined as g(y)(x). The most usuel division algorithms are the one corresponding to the quadratic representation and the one corresponding to the Cholesky decomposition. In fact, one can define the quotient of x by y as P (y -1 2 )x. On the other hand, given that for each y in Ω, there exists a unique t in the triangular group T such that y = t(e), we define the multiplication of x by y as

π(y)(x) = t(x),
and the map y → t -1 from Ω into G realizes a division algorithm so that the "quotient" of x by y is defined as

π -1 (y)(x) = t -1 (x).
We also set

π * (y)(x) = t * (x) and π * -1 (y)(x) = t * -1 (x).
The definition of π(y) and π * (y) may be extended to y ∈ -Ω, by setting π(-y) = -π(y) and π * (-y) = -π * (y).

Consider the set

Str(V ) = {g ∈ GL(V ) ; P (g(x)) = gP (x)g * }.
It is shown in [START_REF] Faraut | Analysis on symmetric cones[END_REF] page 150 that G(Ω) is a subgroup of Str(V ). Hence we have in particular that for t in the triangular group T ,

P (t(e)) = tP (e)t * = tt * . (1)
If y = t(e), then

P (y) = π(y)π * (y). ( 2
)
When Ω is the cone of positive definite symmetric matrices, the Cholesky decomposition of an element y of Ω, consists in writing y in a unique manner as y = tt * , where t is a lower triangular matrix with strictly positive diagonal, and we have

π(y)(x) = txt * π -1 (y)(x) = t -1 xt * -1 , and P (y)(x) = yxy,
where the last product is the ordinary product of matrices.

In this case, ( 2) is easily established.

Continued fraction in a symmetric cone

We use the division algorithm defined by the Cholesky decomposition. Let (x n ) n≥1 and (y n ) n≥0 be two sequences in Ω. The continued fraction denoted

K(x n /y n ) or y 0 + x 1 y 1 , x 2 y 2 , ...
is an expression whose the nth convergent

R n = y 0 + x 1 y 1 , ..., x n y n
is defined in the following recursive way.

x 1 y 1 = π * -1 (x 1 )y 1 -1 = π(x 1 ) y -1 1 x 1 y 1 , ..., x n+1 y n+1 = π * -1 (x 1 ) y 1 + x 2 y 2 , ..., x n+1 y n+1 -1 = π(x 1 ) y 1 + x 2 y 2 , ..., x n+1 y n+1 -1
.

Note that to get R n+1 from R n , we replace y n by y n + π * -1 (x n+1 )y n+1 -1 .

When x n = e, for all n, we say that the continued fraction K(e/y n ) is an ordinary continued fraction. Its kth convergent is given by

R k = y 0 + y 1 + y 2 + y 3 + ... + (y k ) -1 -1 ... -1 -1 . (3)
As mentioned in the introduction, the construction process of this continued fraction uses only the inversion without any use of the product or of the division algorithm.

Next, we show that any continued fraction K(x n /y n ) is equivalent to an ordinary continued fraction K(e/a n ).

Proposition 3.1 The continued fraction denoted K(x n /y n ) is equivalent to the continued fraction K(e/a n ) where

a 0 = y 0 a 2k = π(x 1 )π * -1 (x 2 )...π(x 2k-1 )π * -1 (x 2k )y 2k (4)
a 2k+1 = π * -1 (x 1 )π(x 2 )...π(x 2k )π * -1 (x 2k+1 )y 2k+1 . (5)
Proof. The proof is performed by induction. We will use the fact that

π -1 (u)v -1 = (π * (u)v) -1 or equivalently π * (u)v -1 = (π -1 (u)v) -1 (6) Since x 1 y 1 = π * -1 (x 1 )y 1 -1 ,
and

x 1 y 1 , x 2 y 2 = π * -1 (x 1 ) y 1 + x 2 y 2 -1 = π * -1 (x 1 ) y 1 + π * -1 (x 2 ) y 2 -1 -1 = π * -1 (x 1 )y 1 + π * -1 (x 1 ) π * -1 (x 2 ) y 2 -1 -1 = π * -1 (x 1 )y 1 + π(x 1 )π * -1 (x 2 ) y 2 -1 -1 .
We have that

a 0 = y 0 a 1 = π * -1 (x 1 )y 1 a 2 = π(x 1 )π * -1 (x 2 ) y 2 .
Therefore ( 5) is true for k = 0 and ( 4) is true for k = 1. Suppose that

R 2k = a 0 + a 1 + a 2 + a 3 + ... + (a 2k ) -1 -1 ... -1 -1
.

with the a i , 0 ≤ i ≤ 2k, defined by ( 5) and ( 4). In particular

a 2k = π(x 1 )π * -1 (x 2 )...π(x 2k-1 )π * -1 (x 2k )y 2k .
To get R 2k+1 from R 2k , we replace in a 2k , y 2k by y 2k + π * -1 (x 2k+1 )y 2k+1 -1 , that is we replace a 2k by

a 2k + π(x 1 )π * -1 (x 2 )...π(x 2k-1 )π * -1 (x 2k ) π * -1 (x 2k+1 )y 2k+1 -1 -1
.

Using (6), this can be written

a 2k + π * -1 (x 1 )π(x 2 )...π * -1 (x 2k-1 )π(x 2k )π * -1 (x 2k+1 )y 2k+1 -1 -1
.

Hence

R 2k+1 = a 0 +   a 1 + a 2 + a 3 + ... + a 2k + (a 2k+1 ) -1 -1 -1 ... -1   -1
.

with a 2k+1 = π * -1 (x 1 )π(x 2 )...π * -1 (x 2k-1 )π(x 2k )π * -1 (x 2k+1 )y 2k+1 .
A similar reasoning is used for R 2k+2 .

Convergence of a non ordinary continued fraction

In this section, we suppose that (x n ) n≥1 and (y n ) n≥0 are sequences of independent random variables defined on the same probability space and valued in the cone Ω. As we have mentioned, a criteria of convergence for the ordinary random continued fraction K(e/y n ) has been given in [START_REF] Bernadac | Random continued fractions and inverse Gaussian distribution on a symmetric cone[END_REF]. In what follows, we give a criteria of convergence for the non ordinary random continued fraction K(x n /e). For the sake of simplicity, x 1 e , x 2 e , ..., x k e will be written [x 1 , x 2 , ..., x k ] which is the notation usually used for the ordinary continued fraction given in (4). This should not cause any confusion, the continued fractions are completely different, the results and their proofs are completely different although there are some similarities. The continued fraction [x 1 , x 2 , ..., x k ] which we will study in what follows is then defined by

[x 1 ] = x 1 and [x 1 , x 2 , ..., x k ] = π(x 1 )(e + [x 2 , ..., x k ]) -1 . ( 7
)
We first prove the following result:

Proposition 4.1 Let (x n ) n≥1 be a sequence in Ω. Then for k ≥ 1, (-1) k+1 ([x 1 , x 2 , ..., x k ] -[x 1 , x 2 , ..., x k+1 ]) is in Ω.
Proof. The proof uses induction on k.

The property is true for k = 1, in fact

[x 1 ] -[x 1 , x 2 ] = x 1 -π(x 1 ) (e + x 2 ) -1 = π(x 1 ) e -(e + x 2 ) -1 = π(x 1 ) (e + x 2 -e)(e + x 2 ) -1 ,
which is in (-1) 2 Ω. Suppose that the property is true for k.

Then [x 1 , ..., x k+1 ] -[x 1 , ..., x x+2 ] = π(x 1 ) (e + [x 2 , ..., x k+1 ]) -1 -π(x 1 ) (e + [x 2 , ..., x k+2 ]) -1 = π(x 1 ) (e + [x 2 , ..., x k+1 ]) -1 -(e + [x 2 , ..., x k+2 ]) -1
Using the induction hypothesis, we have that

[x 2 , ..., x k+1 ] -[x 2 , ..., x k+2 ] ∈ (-1) k+1 Ω,
so that (e + [x 2 , ..., x k+1 ]) -(e + [x 2 , ..., x k+2 ]) ∈ (-1) k+1 Ω.

To finish the proof, we use the fact that for all x and y in Ω,

(x -y ∈ Ω) ⇔ x -1 -y -1 ∈ -Ω .

Now, Denoting

w k = (-1) k+1 ([x 1 , x 2 , ..., x k ] -[x 1 , x 2 , ..., x k+1 ]) , (8)
it is easy to see that

[x 1 , x 2 , ..., x n ] = x 1 + n-1 k=1 (-1) k w k . (9)
Thus the continued fraction [x

F k (x 1 , ..., x k+2 ) = [x 1 , ..., x k ] -1 -[x 1 , ..., x k+1 ] -1 -1 + [x 1 , ..., x k+1 ] -1 -[x 1 , ..., x k+2 ] -1 -1 . ( 10
)
Then we have

F 1 = π(x 1 )π * -1 (x 2 )π * -1 (x 3 )(e), (11)
F 2 (x 1 , x 2 , x 3 , x 4 ) = -π(x 1 )π * -1 (x 2 )π * -1 (x 3 ) P (e + π * (x 3 )(e)) π * -1 (x 4 )(e), ( 12
)
and for all k ≥ 3,

F k (x 1 , ..., x k+2 ) = (-1) k+1 π(x 1 ) k-1 i=2 π * -1 (x i )P (e + [x i+1 , ..., x k ]) π * -1 (x k )π * -1 (x k+1 )P (u k ) π * -1 (x k+2 )(e), (13)
where

u k = u k (x 1 , ..., x k+1 ) = e + π * (x k+1 )   e + k-3 i=0 (-1) i+1   i j=0 π * (x k-j )P (e + [x k-j , ..., x k ]) -1   (e + [x k-i , ..., x k ])   (14) Proof. F 1 = F 1 (x 1 , x 2 , x 3 ) = [x 1 ] -1 -[x 1 , x 2 ] -1 -1 + [x 1 , x 2 ] -1 -[x 1 , x 2 , x 3 ] -1 -1 = π * -1 (x 1 )(e) -π * -1 (x 1 )(e + x 2 ) -1 + π * -1 (x 1 )(e + x 2 ) -π * -1 (x 1 ) e + π * -1 (x 2 )(e + x 3 ) -1 -1 = -π * -1 (x 1 )(x 2 ) -1 + π * -1 (x 1 )(x 2 ) -π * -1 (x 1 )π(x 2 ) (e + x 3 ) -1 -1 = -π(x 1 )π * -1 (x 2 )(e) + π(x 1 )π * -1 (x 2 ) e -(e + x 3 ) -1 -1 = π(x 1 )π * -1 (x 2 ) e -(e + x 3 ) -1 -1 -e = π(x 1 )π * -1 (x 2 )π * -1 (x 3 )(e)
Now to calculate

F 2 = = F 2 (x 1 , x 2 , x 3 , x 4 ) = [x 1 , x 2 ] -1 -[x 1 , x 2 , x 3 ] -1 -1 + [x 1 , x 2 , x 3 ] -1 -[x 1 , x 2 , x 3 , x 3 ] -1 -1 , we write [x 1 , x 2 ] = [X 1 ] with X 1 = π(x 1 ) (e + x 2 ) -1 ,
and we write

[x 1 , x 2 , x 3 ] = [X 1 , X 2 ] with X 2 = -π * (e + x 2 ) -1 π(x 2 ) x 3 (e + x 3 ) -1 , finally, we write [x 1 , x 2 , x 3 , x 4 ] = [X 1 , X 2 , X 3 ] , with X 3 = π -1 (e + π * (x 3 )(e)) (x 4 ).
Thus using (11), we have

F 2 (x 1 , x 2 , x 3 , x 4 ) = F 1 (X 1 , X 2 , X 3 ) = π(X 1 )π * -1 (X 2 )π * -1 (X 3 )(e).
Replacing, we get

F 2 (x 1 , x 2 , x 3 , x 4 ) = -π(x 1 )π (e + x 2 ) -1 π -1 (e + x 2 ) -1 π * -1 (x 2 )
π * -1 (x 3 ) π (e + π * (x 3 )(e)) π * (e + π * (x 3 )(e)) π * -1 (x 4 )(e).

Given that

π (e + π * (x 3 )(e)) π * (e + π * (x 3 )(e)) = P (e + π * (x 3 )(e)) , we obtain

F 2 (x 1 , x 2 , x 3 , x 4 ) = -π(x 1 )π * -1 (x 2 )π * -1 (x 3 ) P (e + π * (x 3 )(e)) π * -1 (x 4 )(e).
Therefore ( 13) is true for k = 2. Suppose that it true for k. Then with the same reasoning, we have that

F k+1 = F k+1 (x 1 , ..., x k-1 , x k , x k+1 , x k+2 , x k+3 ) = F k (x 1 , ..., x k-1 , X k , X k+1 , X k+2 ), with X k = π(x k ) (e + x k+1 ) -1 , X k+1 = -π * (e + x k+1 ) -1 π(x k+1 ) x k+2 (e + x k+2 ) -1 ,
and

X k+2 = π -1 (e + π * (x k+2 )(e)) (x k+3 ). Inserting this in F k (x 1 , ..., x k-1 , X k , X k+1 , X k+2 ) as defined in (13), we have in particular that [x i+1 , ..., x k-1 , X k ] is nothing but [x i+1 , ..., x k , x k+1 ]. We also have π * -1 (X k )π * -1 (X k+1 ) = -π * -1 (x k )π * -1 (e + x k+1 ) -1 π -1 (e + x k+1 ) -1 π * -1 (x k+1 )π * -1 (x k+2 ) π (e + π * (x k+2 )(e)) = -π * -1 (x k )P (e + x k+1 ) π * -1 (x k+1 )π * -1 (x k+2 ) π (e + π * (x k+2 )(e)) ,
and that π * -1 (X k+2 )(e) = π * (e + π * (x k+2 )(e)) π * -1 (x k+3 )(e)

Thus

F k+1 = F k+1 (x 1 , ..., x k-1 , x k , x k+1 , x k+2 , x k+3 ) = F k (x 1 , ..., x k-1 , X k , X k+1 , X k+2 ) = (-1) k+1 π(x 1 ) k i=2 π * -1 (x i )P (e + [x i+1 , ..., x k+1 ]) π * -1 (x k+1 )π * -1 (x k+2 ) π (e + π * (x k+2 )(e)) P (u k (x 1 , ..., x k-1 , X k , X k+1 )) π * (e + π * (x k+2 )(e)) π * -1 (x k+3 )(e).
Denoting

g = π (e + π * (x k+2 )(e)) P (u k (x 1 , ..., x k-1 , X k , X k+1 )) π * (e + π * (x k+2 )(e)) ,
It remains to verify that

g = P (u k+1 (x 1 , ..., x k-1 , x k , x k+1 , x k+2 )) .
In fact,

u k (x 1 , ..., x k-1 , X k , X k+1 ) = e -π -1 (e + π * (x k+2 )(e)) π * (x k+2 ) π * (x k+1 )π (e + x k+1 ) -1 e -π * (e + x k+1 ) -1 π * (x k )P (e + [x k , x k+1 ]) -1 (e + [x k , x k+1 ]) + k-3 i=1 (-1) i+1 π * (e + x k+1 ) -1 π * (x k ) P (e + [x k , x k+1 ]) -1   i j=1 π * (x k-j )P (e + [x k-j , ..., x k+1 ]) -1   (e + [x k-i , ..., x k+1 ])   .
Using the fact that π(y)P (x)π * (y) = P (π(y)x), and π * (x -1 )(x) = e we obtain that

g = P e + π * (x k+2 )(e) -π * (x k+2 ) π * (x k+1 )π (e + x k+1 ) -1 e -π * (e + x k+1 ) -1 π * (x k )P (e + [x k , x k+1 ]) -1 (e + [x k , x k+1 ]) + k-3 i=1 (-1) i+1 π * (e + x k+1 ) -1 π * (x k ) P (e + [x k , x k+1 ]) -1   i j=1 π * (x k-j )P (e + [x k-j , ..., x k+1 ]) -1   (e + [x k-i , ..., x k+1 ]))   = P e + π * (x k+2 )(e) -π * (x k+2 ) π * (x k+1 )P (e + x k+1 ) -1 ((e + x k+1 )) +π * (x k+2 ) π * (x k+1 )P (e + x k+1 ) -1 π * (x k )P (e + [x k , x k+1 ]) -1 (e + [x k , x k+1 ]) + k-3 i=1 (-1) i+2 π * (x k+2 ) π * (x k+1 )P (e + x k+1 ) -1 π * (x k ) P (e + [x k , x k+1 ]) -1   i j=1 π * (x k-j )P (e + [x k-j , ..., x k+1 ]) -1   (e + [x k-i , ..., x k+1 ]))   .
Setting i ′ = i + 1 and j ′ = j + 1, with some standard calculation, this is noting but P (u k+1 (x 1 , ..., x k-1 , x k , x k+1 , x k+2 )).

We now coin a useful lemma.

Lemme 4.4 For all x 1 , ..., x k in Ω, the vector

H = e+ k-3 i=0 (-1) i+1   i j=0 π * (x k-j )P (e + [x k-j , ..., x k ]) -1   (e + [x k-i , ..., x k ]) is in Ω. Denoting for 1 ≤ i ≤ k -3, A i =   i j=0 π * (x k-j )P (e + [x k-j , ..., x k ]) -1   (e + [x k-i , ..., x k ]) ,
we have that

H = e -π * (x k )P (e + x k ) -1 (e + x k ) + 1≤i≤k-3, i odd (A i -A i+1 ), with A k-2 = 0. Now e -π * (x k )P (e + x k ) -1 (e + x k ) = e -π * (x k ) (e + x k ) -1 = e -π -1 (x k ) (e + x k ) -1 = e -π -1 (x k )(e) + e -1 ∈ Ω.
On the other hand, we have for i odd,

A i -A i+1 =   i j=0 π * (x k-j )P (e + [x k-j , ..., x k ]) -1   (e + [x k-i , ..., x k ]) -π * (x k-(i+1) ) e + x k-(i+1) , ..., x k -1 =   i j=0 π * (x k-j )P (e + [x k-j , ..., x k ]) -1   (e + [x k-i , ..., x k ]) -π * (x k-(i+1) ) e + π(x k-(i+1) ) (e + [x k-i , ..., x k ]) -1 -1 =   i j=0 π * (x k-j )P (e + [x k-j , ..., x k ]) -1   (e + [x k-i , ..., x k ]) - π -1 (x k-(i+1) )(e) + (e + [x k-i , ..., x k ]) -1 -1 , which is clearly in Ω. Thus H is in Ω. ! Theorem 4.5
The sequence (w k ) defined in ( 8) is decreasing.

Proof. Form the definition, we have that

[e, x 1 , x 2 , ..., x k ] = π(e) (e + [x 1 , x 2 , ..., x k ]) -1 , so that [e, x 1 , x 2 , ..., x k ] -1 = e + [x 1 , x 2 , ..., x k ] .
Hence

w -1 k = (-1) k+1 ([x 1 , x 2 , ..., x k ] -[x 1 , x 2 , ..., x k+1 ]) -1 = (-1) k+1 [e, x 1 , x 2 , ..., x k ] -1 -[e, x 1 , x 2 , ..., x k+1 ] -1 -1 w -1 k -w -1 k+1 = (-1) k+1 [e, x 1 , x 2 , ..., x k ] -1 -[e, x 1 , x 2 , ..., x k+1 ] -1 -1 -(-1) k+2 [e, x 1 , x 2 , ..., x k+1 ] -1 -[e, x 1 , x 2 , ..., x k+2 ] -1 -1 = (-1) k+1 [e, x 1 , x 2 , ..., x k ] -1 -[e, x 1 , x 2 , ..., x k+1 ] -1 -1 + [e, x 1 , x 2 , ..., x k+1 ] -1 -[e, x 1 , x 2 , ..., x k+2 ] -1 -1 = (-1) k+2 F k+1 (e, x 1 , ..., x k+2 ) = (-1) k+1 (-1) k+2 k-1 i=1 π * -1 (x i )P (e + [x i+1 , ..., x k ]) π * -1 (x k )π * -1 (x k+1 )P (u k+1 (e, x 1 , ..., x k+1 )) π * -1 (x k+2 )(e) Therefore w -1 k -w -1 k+1 = - k-1 i=1 π * -1 (x i )P (e + [x i+1 , ..., x k ]) π * -1 (x k )π * -1 (x k+1 ) P (u k+1 (e, x 1 , ..., x k+1 )) π * -1 (x k+2 )(e). (15)
As for all v in the cone Ω, the automorphisms π(v), and P (v) are in the group G, they conserve Ω, we deduce that

w -1 k -w -1 k+1 ∈ -Ω,
which is equivalent to

w k -w k+1 ∈ Ω.
Hence the sequence (w k ) is decreasing.

Next, we give an other important intermediary result. Using the notation above, for a sequence (x n ) in the cone Ω, we define the following element of G:

Q k = k-1 i=1
π * -1 (x i )P (e + [x i+1 , ..., x k ]) π * -1 (x k )π * -1 (x k+1 )P (u k+1 (e, x 1 , ..., x k+1 ))

(16) Note that according to (15), we have

w -1 k+1 -w -1 k = Q k (x -1 k+2 ). ( 17
)
Proposition 4.6 Let y be an element of Ω. Then for all k ≥ 2,

Q * k (y) ≥ π -1 (x 1 )(y)
Proof. We have that

Q * k = P (u k+1 (e, x 1 , ..., x k+1 )) π -1 (x k+1 )π -1 (x k ) k-1 i=1 P (e + [x k+1-i , ..., x k ]) π -1 (x k-i )
Using the fact that P (x) = π(x)π * (x), we obtain that Q * k is equal to the composition of the following elements of G:

π * (e + [x 2 , ..., x k ]) π -1 (x 1 ), (18)
then the automorphisms

π * (e + [x k+1-i , ..., x k ]) π -1 (x k-i )π (e + [x k-i , ..., x k ]) , (19)
for i = 1, ..., k -2, then π -1 (x k )π (e + [x k ]) , (20)
and P (u k+1 (e, x 1 , ..., x k+1 )) π -1 (x k+1 ).

For the terms in (19), we have

π * (e + [x k+1-i , ..., x k ]) π -1 (x k-i )π (e + [x k-i , ..., x k ]) = π * (e + [x k+1-i , ..., x k ]) π -1 (x k-i )π e + π(x k-i ) (e + [x k+1-i , ..., x k ]) -1 = π * (e + [x k+1-i , ..., x k ]) π π -1 (x k-i )(e) + (e + [x k+1-i , ..., x k ]) -1 = π e + π * (e + [x k+1-i , ..., x k ]) π -1 (x k-i )(e) ,
where in the last equality, we have used the fact that π * (x)(x -1 ) = e.

Concerning the term in (20), we have

π -1 (x k )π (e + [x k ]) = π π -1 (x k ) (e + [x k ]) = π e + π -1 (x k ) (e) .
Finally, for (21), according to the definition of u k see ( 14), and to Lemma 4.4, we have that u k+1 (e, x 1 , ..., x k+1 ) can be written in the form

u k+1 (e, x 1 , ..., x k+1 ) = e + π * (x k+1 )(e + v),
where v is an element of Ω. Thus

P (u k+1 (e, x 1 , ..., x k+1 )) π -1 (x k+1 ) = P (e + π * (x k+1 )(e + v)) π -1 (x k+1 ) = π (e + π * (x k+1 )(e + v)) π * (e + π * (x k+1 )(e + v)) π -1 (x k+1 ) = π (e + π * (x k+1 )(e + v)) π * π * -1 (x k+1 ) (e + π * (x k+1 )(e + v)) = π (e + π * (x k+1 )(e + v))
π * e + π * -1 (x k+1 )(e) + v .

Having done this, we see that to get Q * k (y), we first apply π -1 (x 1 ) to y, then we apply automorphisms which are of the form π(e + z) or π * (e + z) or P (e + z), with z ∈ Ω. Using the fact that for all z and y in Ω, we have that π(e + z)(y) -y ∈ Ω, π * (e + z)(y) -y ∈ Ω and P (e + z)(y) -y ∈ Ω, in other word, π(e + z)(y) > y, π * (e + z)(y) > y and P (e + z)(y) > y, we conclude that, given y in Ω, we have that for all k ≥ 2,

Q * k (y) > π -1 (x 1 )(y).
Corollary 4.7 For all y in Ω, there exists a positive random variable C such that for all k,

Q * k (y) > C Proof. As Q * k (y) > π -1 (x 1 )(y), then (Q * k (y)) 2 > (π -1 (x 1 )(y)) 2 . In fact (Q * k (y)) 2 -(π -1 (x 1 )(y)) 2 = (Q * k (y) -π -1 (x 1 )(y))(Q * k (y) + π -1 (x 1 )(y)) ∈ Ω.
Therefore tr((Q * k (y)) 2 ) > tr((π -1 (x 1 )(y)) 2 ), which means that Q * k (y) > π -1 (x 1 )(y) . Thus it suffices to take C = π -1 (x 1 )(y) . Besides the general facts that we have established above, for the proof of the main result of the paper concerning the convergence of a random non ordinary continued fraction, we need the following probability result which also appears in [START_REF] Bernadac | Random continued fractions and inverse Gaussian distribution on a symmetric cone[END_REF]. We give it with a slightly different proof.

Proposition 4.8 Let (A k ) k≥1 be an increasing sequence of σ-fields, and let A k be in A k , for k ≥ 1. Suppose that there exit K > 1, 0 < k 0 < K, and a random variable α which is A k 0 -measurable and valued in (0, 1) such that for all k ≥ K,

E 1 A k+1 |A k ≤ α. Then P   k≥K A k   = 0 Proof. Let us show by induction on p ≥ 1 that E 1 A K ...1 A K+p |A k 0 ≤ α p a.s. For p = 1, E 1 A K 1 A K+1 |A k 0 = E 1 A K 1 A K+1 |A K |A k 0 = E 1 A K 1 A K+1 |A K |A k 0 ≤ αE (1 A K |A k 0 ) a.s.
≤ α a.s..

Now suppose that

E 1 A K ...1 A K+p |A k 0 ≤ α p a.s.. Then E 1 A K ...1 A K+p+1 |A k 0 = E 1 A K ...1 A K+p+1 |A K+p |A k 0 = E 1 A K ...1 A K+p 1 A K+p+1 |A K+p |A k 0 ≤ αE 1 A K ...1 A K+p |A k 0 a.s.
≤ αα p = α p+1 a.s..

Having shown that

E 1 A K ...1 A K+p |A k 0 ≤ α p a.s.,
we just need to take the expectation and let p → ∞ to get the result.

We are now in position to state and prove the main result of the paper.

Theorem 4.9 Let (x k ) k≥1 be a sequence of independent random variables in the cone Ω with distributions absolutely continuous with respect to the Lebesgue measure and such that for all k, the complementary in Ω of the support of x -1 k is bounded. Suppose that there exists p ≥ 1, such that for all i ≥ 1,

L(x i ) = L(x i+p ).
Then the continued fraction [x 1 , ..., x k ] is almost surely convergent.

Proof. With the results what we have established above, we need only to show that under these assumptions, the sequence (w k ) defined in (8) converges almost surely to 0, or equivalently that for all y in Ω, the sequence ( w -1 k , y ) k≥1 diverges almost surely. in order to do so, we will show that Given that w -1 k+1 -w -1 k = Q k (x -1 k+2 ), with Q k defined in (16) we have

A k+1 = Q k (x -1 k+2 ), y ≤ ε = x -1 k+2 , Q * k (y) ≤ ε = x -1 k+2 , Q * k (y) Q * k (y) ≤ ε Q * k (y)
.

Let S be the unit sphere in V , and define for a random variable z, h z (a) = sup θ∈S P (| z, θ | ≤ a) , then we have that

E 1 A k+1 |A k = P x -1 k+2 , Q * k (y) Q * k (y) ≤ ε Q * k (y) |A k ≤ h x -1 k+2 ε Q * k (y) , because Q * k (y) is A 2 -measurable ≤ h x -1 k+2 ε C . Denoting α = max h x -1 1 ε C , ..., h x -1 p ε C
, then according to the assumptions on the distributions of the x i and on the support of x -1 i , we have that α < 1, and that E 1 A k+1 |A k ≤ α.

Therefore using Proposition 4.8 with k 0 = 2, we deduce that

P   k≥n A k   ,
which is the desired result.

Application Consider the beta distribution of the second kind β

p,q on Ω given by β (2) p,q (dx) = (B Ω (p, q)) -1 (∆(x)) p-r+1 2 (∆(e + x)) -(p+q) 1 Ω (x)dx, where B Ω (p, q) is the multivariate beta function defined by B Ω (p, q) = Γ Ω (p)Γ Ω (q) Γ Ω (p + q) , with Γ Ω (p) = (2π) Let (W n ) n≥1 and (W ′ n ) n≥1 be two independent sequences of independent random variables with respective distributions β Then we have that • (x n ) n≥1 is a sequence of independent random variables in the cone Ω with distributions absolutely continuous with respect to the Lebesgue measure.

• For all k, the support of x -1 k is Ω, its complementary in Ω is bounded. • For all i ≥ 1, L(x i ) = L(x i+2 ). According to Theorem 4.9, the continued fraction [x 1 , ..., x k ] is almost surely convergent. As the convergence of a continued fraction doesn't depend on the initial vector, we deduce that the continued fraction [e, x 1 , ..., x k ] is also almost surely convergent. This result is used in a forthcoming paper concerning the matrix beta-hypergeometric distribution. References

  A k be the σ-field generated by the random variables x 1 , ..., x k+1 , and define for ε > 0,A k = w -1 k -w -1 k-1 , y ≤ ε . Then A k ∈ A k ,and we have

  -(k -1)/2).

( 2 )

 2 b,a ′ , and consider the random continued fraction [x 1 , ..., x k ] as defined in (7) such that x 2n+1 = W n+1 for n ≥ 0 and x 2n = W ′ n for n ≥ 1.

  1 , x 2 , ..., x n ] converges if and only if the alternating w k converges. For the convergence of an alternating series in the algebra V , we have the following result which appears in[START_REF] Bernadac | Random continued fractions and inverse Gaussian distribution on a symmetric cone[END_REF]. Proposition 4.2 Let (u n ) n≥1 be a sequence in Ω. If (u n ) is decreasing and con-Let (x n ) n≥1 be a sequence in Ω, and denote

	+∞
	series Next, we state and prove a theorem which is crucial for the study of the convergence +∞ k=1 (-1) k u k is convergent of the alternating series +∞ k=1 (-1) k w k . (-1) verges to 0, then the alternating series Theorem 4.3
	k=1

k