Julien Trun

Stéphane Loisel

Ultimate ruin probability in discrete time with Bühlmann credibility premium adjustments

Keywords: Discrete-time ruin model, Bühlmann's model, light-tailed claims, large deviation, ultimate ruin probability, Lundberg coecient, path to ruin, heavy-tailed claims

In this paper, we consider a discrete-time ruin model where experience rating is taken into account. The main objective is to determine the behavior of the ultimate ruin probabilities for large initial capital in the case of light-tailed claim amounts. The logarithmic asymptotic behavior of the ultimate ruin probability is derived. Typical pathes leading to ruin are studied. An upper bound is derived on the ultimate ruin probability in some particular case. The inuence of the number of data points taken into account is analyzed, and numerical illustrations support the theoretical ndings. Finally, we investigate the heavy-tailed case. The impact of the number of data points used for the premium calculation appears to be rather dierent from the one in the light-tailed case.

Introduction

Computation of ruin probabilities is an important topic in actuarial science. Standard references include [START_REF] Asmussen | Ruin Probabilities[END_REF], [START_REF] Dickson | Insurance Risk and Ruin[END_REF], [START_REF] Grandell | Aspects of Risk Theory[END_REF], [START_REF] Kaas | Modern Actuarial Risk Theory Using R[END_REF] and [START_REF] Rolski | Stochastic Processes for Insurance and Finance[END_REF]. Even if the study of the ruin problem resulted in a huge amount of published articles (including in the closely related elds of queueing theory and dam and storage processes), some assumptions have rarely been questioned in the literature, until recently.

The majority of existing results require independent increments for the stochastic process describing the aggregate claim amounts led against the insurance company. However, a lot of practical issues violate this independence assumption, as the result of the implementation of experience rating mechanisms by the insurance company. In this case, the nancial result of a given calendar year then depends on the result(s) of one or several previous years.

In this paper, experience rating is taken into account. Allowing the premium amount to depend on past claims experience through an appropriate credibility mechanism is in line with insurance practice in most lines of business.

In the literature, only some authors have studied ruin problem with credibility dynamics. In [START_REF] Asmussen | On the ruin problem for some adapted premium rules[END_REF], the ruin model allows adapted premiums rules. More precisely, a continuous-time risk process is considered with compound Poisson claims, where the premium rate is exclusively calculated on the basis of past claims statistics, with no reference to the market. In practice, one would use a mixture between a quantity based on past claim experience and a market premium level. The easiest way to compute credibility adjusted premiums is to use Bühlmann's credibility model. This is what [START_REF] Tsai | Ruin probabilities: classical versus credibility[END_REF] have done in a discrete-time risk model. They have studied, by means of Monte-Carlo simulations, the impact of the number of past claim experience years taken into account for the calculation of the premiums on the ultimate ruin probabilities for light-tailed claims.

In this paper, we deal with a framework that is similar to the one of [START_REF] Tsai | Ruin probabilities: classical versus credibility[END_REF]. One of the main observation made by these authors is that for an insurance company with a large initial capital and light-tailed claims, the use of a bigger (nite) number of past claim experience years to calculate the premiums decreases the ultimate ruin probability. Among other results, until a certain number of past claim experience years, this observation is proved in this paper. We also show that the story is dierent for heavy-tailed claim amounts. [START_REF] Asmussen | Ruin Probabilities[END_REF] The model

Let us consider an insurance market made up of n portfolios, where portfolio j @j a I; : : : ; nA is characterized by independent and identically distributed annual aggregated claim amounts Y (j) k , with common distribution function F j . For a randomly chosen insurance company operating on this market, let p j be the probability that the portfolio held by this company is the portfolio j.

To motivate this setting, let us consider for example a simple market made up of P categories of policyholders, with n g "good" policyholders and n b "bad" policyholders, say.

Then a company with s min@n g ; n b A policyholders would have s C I possible portfolios, i.e. n a s C I. Furthermore, the probability p j (associated to portfolio j (j a I; : : : ; s C I), characterized by j I good policyholders) would be given by p j a s3 @s j C IA3@j IA3 We assume that the company uses credibility premiums in order to determine, on the basis of its own claim amounts, the pure premium corresponding to its unknown portfolio. In this setting, the model proposed in [START_REF] Bühlmann | Experience rating and credibility I[END_REF] and ( 1969) is largely used in practice. In this paper, we focus on this well-known credibility model.

Let us denote by Y k the annual claim amounts of the company. The a priori mean of the Y k 's is of course given by a EY k a n j=1 EY (j) p j ; (2.2) where Y (j) is a random variable with distribution function F j , while the a priori variance breaks up in two terms: 2 a VY k a a C ; Parameter a measures the part of the variance coming from the heterogeneity of the market, whereas parameter measures the part of hazard in the a priori variance.

As mentioned in [START_REF] Tsai | Ruin probabilities: classical versus credibility[END_REF], some casualty insurers only consider a nite number of most recent periods of claim experiences to renew the premium. Let us denote by m P N + fIg the so-called "horizon of credibility", which corresponds to the number of periods taken into account for the calculation of the premiums. So, depending on the choice of m, with the convention that 0 i=1 a H, the premium of the company in time k I is given by

C k;m a 2 @I z k;m A C z k;m k 1 i=max(k m;1) Y i min@m; k IA 3 @I C A; (2.6)
where z k;m a min@m; k IA a C min@m; k IA a (2.7) is Bühlmann's credibility factor and > H is the premium security loading. Notice that our premium rule depends on the market (via the parameters , a and ) contrary to [START_REF] Asmussen | On the ruin problem for some adapted premium rules[END_REF]. For simplicity, we have assumed that no historical data is available. This simplifying assumption does not inuence the asymptotic results derived in the sequel. Now, if the portfolio held by the insurer is portfolio j, then the dynamics of the insurer's surplus obeys to the equation

U (j) k;m a u C k i=1 C i;m k i=1 Y (j) i ;
(2.8)

where the C i;m 's are described by equation (2.6) with Y i a Y (j) i , and where u is the initial capital of the company. The corresponding ultimate ruin probability may be written as

(j)
m @uA a r h U (j) k;m < H for some k jU (j) 0;m a u i :

(2.9)

Immediately, it appears the existence of a subset of "bad" portfolios, denoting b m , such that for all j P b m , (j) m @uA a I for all u. Indeed, let us dene (j) m a @I C A@@I z m A C z m EY (j) A EY (j) I;

(

where z m a m a C m a :

(2.11)

Obviously, for m < I, (j) m corresponds to the average security loading of the premium for k > m while for m a I, (j) m a , which corresponds this time to the security loading for k 3 I. The subset b m is then dened as b m a fj a I; : : : ; n X (j) m Hg:

(2.12) Also, in the following, the complementary subset of ("not bad") portfolios is denoted by " b m , and we shall say that j P g if and only if EY (j) . Clearly, we have g & " b m .

The purpose of this paper is to investigate the behavior of the ultimate ruin probabilities (j) m @uA for large u in the case of light-tailed annual claim amounts. First, the logarithmic asymptotic behavior of (j) m @uA is derived. Then, typical pathes leading to ruin are studied.

Also, an upper bound on (j) m @uA is derived in some particular cases. Afterwards, we are able to determine, for each portfolio j, the inuence of "strategy" m on (j) m @uA for large u.

Next, whatever the portfolio j, optimal strategies, in a sense to be dened in the sequel, are deduced. As an illustration, numerical simulations are also performed. Finally, we also study the inuence of m in rst order on (j) m @uA in the heavy-tailed case. As we shall see, the conclusion is totally dierent than in the light-tailed case.

3

Limit results for light-tailed claims

In this Section, for simplicity, let us assume that for all j a I; : : : ; n, there exists r (j) > H such that Ee rY (j) exists for all H < r < r (j) and such that lim r3r (j) Ee rY (j) a I. Clearly, this assumption implies necessarily light-tailed claims.

3.1

The case m < 1

3.1.1 Asymptotic behavior of (j) m @uA Theorem 3.1. Suppose that there exists a unique positive solution to the equation e r(1 zm)(1+) E h e rY (j) (1 (1+)zm) i a I;

(3.1)
which we denote (j) m . Furthermore, assume that for all k, the sum of the rst k annual losses Z k a k i=1 @Y (j) i C i;m A possesses a nite moment-generating function Ee rZ k for H < r < r 0 , with (j) m < r 0 . Hence, we have lim u3I I u ln (j) m @uA a (j) m :

(3.2)
Proof. As a consequence of the Gärtner-Ellis Theorem from large deviations (which is due to [START_REF] Glynn | Logarithmic asymptotics for steady-state tail probabilities in a single-server queue[END_REF], see also [START_REF] Nyrhinen | Rough descriptions of ruin for a general class of surplus processes[END_REF] and [START_REF] Müller | Asymptotic ruin probabilities for risk processes with dependent increments[END_REF]), one only has to show the two following properties:

(A1) m @rA Xa lim k3I 1 k ln Ee rZ k exists for H < r < r 0 , (A2) (j) m is the unique positive value such that m @ (j) m A a H. Hence, (A1) holds true. Concerning (A2), notice that m @rA a ln h m @rA, where h m @rA a e r(1 zm)(1+) E h e rY (j) (1 (1+)zm) i :

Notice that since Ee rY (j) exists for all H < r < r (j) , result (3.2) requires in fact only the existence and the uniqueness of (j) m , with the condition (j) m < r (j) .

The Lundberg coecient (j) m

Let us dene (j) m a @I z m A@I C A EY (j) @I @I C Az m A I:

(3.

3)

The equation (3.1) is equivalent to e r(1 (1+)zm) E[Y (j) ](1+ (j) m ) E h e r(1 (1+)zm)Y (j) i a I; (3.4) and consequently (j) m a (j) m I @I C Az m ; (3.5) where (j) m is the classical Lundberg coecient dened as the unique positive solution of the equation e r E[Y (j) ](1+ (j) m ) Ee rY (j) a I:

(3.6)
Now, from classical results, it is then obvious that the inequality @I >A (j) m > H is a necessary condition for the existence of (j) m . Let us prove that the inequality @I >A (j) m > H is equivalent to m < a and j P " b m .

Proposition 3.1.

@I >A (j) m > H D m < a and j P " b m :

(

Proof. @AA: If I > (j) m > H, then, obviously z m < 1 1+ , or equivalently m < a , and @I z m A@I C A > EY (j) @I @I C Az m A. Thus (j) m a @@I z m A C z m EY (j) A@I C A EY (j) I > EY (j) @I @I C Az m A C z m EY (j) @I C A EY (j) I a @I @I C Az m A C z m @I C A I a H: @@A: Since j P " b m , we have @@I z m ACz m EY (j) A@ICA > EY (j) . Thus, @I z m A@ICA > EY (j) @I @I C Az m A. Now, since m < a , z m < 1 1+ . So, (j) m < I and EY (j) @I @I C Az m A > H. Consequently I > (j) m a @I z m A@I C A EY (j) @I @I C Az m A I > EY (j) @I @I C Az m A EY (j) @I @I C Az m A I a H:

This result highlights the obvious fact that the existence of (j) m implies that j must belong to " b m , since (j) m @uA a I for j P b m .

Let us explain the reason why the horizon of credibility d a e, denoted by m (c) from now, constitutes a critical value for the existence of the Lundberg coecient and consequently for the logarithmic asymptotic behavior of (j) m @uA. From m a m (c) , the global impact of each Y (j) k on the surplus process for k > m, i.e. Y (j) k @I @I C Az m A, becomes negative, which is, of course, a drastic change in the nature of the insured risk. Also, b m a Y for all m ! m (c) .

More fundamentally, in rst order, the only eects selected of the credibility dynamics compared to the classical case m a H are a modication of the security loading and a decreasing of the claim amounts. Indeed, equations (3.4) and (3.5) teach us that (j) m is identical to that of the classical ruin model with a premium security loading equals to (j) m and annual claim amounts given by Y (j) k @I @I C Az m A. Therefore, in this associated classical ruin model, from the critical value m (c) , no ruin can be observed, which explains the reason why a Lundberg coecient does not exist in this case.

So, since Ee rY (j) exists for all H < r < r (j) with lim r3r (j) Ee rY (j) a I, it comes that the two following conditions are of course necessary but also sucient conditions to guarantee the existence and the uniqueness of (j) m : (C1) m < m (c) , (C2) j P " b m .

Let us recall that result (3.2) requires not only the existence and the uniqueness of (j) m but also the additional condition (j) m < r (j) . Firstly, we have the following property for the Lundberg coecient (j) m : Property 3.2. Assume that (j) m 1 and (j) m 2 exist, with H m 1 < m 2 . Then, (j) m 1 < (j) m 2 . Proof. Since z m 1 < z m 2 , we have (j) m 1 < (j) m 2 and consequently, by classical results, (j) m 1 < (j) m 2 . By (3.5), we deduce (j) m 1 < (j) m 2 .

Secondly, let us dene " m (j) a minfm < m (c) X (j) m exist and (j) m ! r (j) g:

(3.8)
If (j) m does not exist for all m < m (c) , then we put " m (j) a H. Also, if for all m < m (c) such that (j) m exists, we always have (j) m < r (j) , then we put " m (j) a m (c) . Hence, if we replace condition (C1) by the stronger condition m < " m (j) , we have now necessary and sucient conditions to guarantee the existence and the uniqueness of (j) m such that (j) m < r (j) , i.e. result (3.2).

Large deviations path to the ruin

Let us assume that m < " m (j) and j P " b m , i.e. that result (3.2) is valid. The goal here is to understand how ruin occurs for large u. More particularly, we want to determine the typical shape of a path leading to ruin when the initial capital of the company is large. To this end, let us use large deviations results of [START_REF] Glynn | Logarithmic asymptotics for steady-state tail probabilities in a single-server queue[END_REF]. First of all, for large u, we know that ruin occurs roughly at time u= H m @ (j) m A. Next, the cumulant generating function u= H m ( (j) m ) @rA of Z u= H m ( (j) m ) is asymptotically changed from u= H m ( (j) m ) @rA to

u= H m ( (j) m ) @r C (j) m A u= H m ( (j) m ) @ (j) m A a r@I C A u= H m ( (j) m ) i=1 @I z i;m A C u= H m ( (j) m ) i=1 ln E 4 e (r+ (j) m )Y (j) 1 ( z i+1;m + z i+2;m +¡¡¡+ z min (i+m;u= H m ( (j) m ));m )(1+) 5 E 4 e (j) m Y (j) 1 ( z i+1;m + z i+2;m +¡¡¡+ z min (i+m;u= H m ( (j) m ));m )(1+)
5 :

(3.9)

Consequently, given that ruin occurs, the claim size distribution Y (j) k is exponentially tilted by the time-dependent factor (j) m @kA, with

for k = 1; : :

: ; u= H m ( (j) m ) m 1 u= 0 m ( (j) m ) k m z m (1 + ) (j)
m ; for k = u= H m ( (j) m ) m + 1; : : : ; u= H m ( (j) m ) :

Hence, for large u, the drift of the path to ruin at time k (k a I; : : : ; u= H m @ (j) m A), denoted as m @kA, is given by m @kA a @I z m A@I C A @I @I C

Az m A E h Y (j) e (j) m (k) Y (j) i E h e (j) m (k) Y (j)
i < H:

(3.10) However, notice that since u 3 I, the time period @u= H m @ (j) m A m C I; u= H m @ (j) m AA where m @kA decreases with k is negligible compared with the time period @I; u= H m @ (j) m A mA where m @kA remains constant.

This analysis highlights that ruin is caused by a succession of small claims, followed by a succession of moderately large claims. (j) m @uA In this subsection only, we take into account a possible claim experience of h years say, i.e.

Upper bound on

(Y (j) h+1 ; Y (j) h+2 ; : : : ; Y (j) 0 ), because, this time, as we see in the following, it will impact the result. Hence, equation (2.6) becomes

C k;m a 2 @I z k;m A C z k;m k 1 i=max(k m;1 h) Y i min@m; h C k IA 3 @I C A; (3.11)
with z k;m a min@m; h C k IA a C min@m; h C k IA a :

(3.12)

Let b a min@m; hA, and let us denote (j) m @u; y 0 ; : : : ; y 1 b A a r h U (j) k;m < H for some k jU (j) 0;m a u; Y (j) 0 a y 0 ; : : : ; Y where U k a U (j) k;m C @I C AY (j) k @" z k+1;m C ¡ ¡ ¡ C "

z k+m;m A C Y (j) k 1 @" z k+1;m C ¡ ¡ ¡ C " z k+m 1;m A C ¡ ¡ ¡ C Y (j) max(k m+1;1 h) @" z k+1;m C ¡ ¡ ¡ C "
z m+max(k m+1;1 h);m A, u a U 0 , and T a inffkjU (j) k;m < Hg, with " z k;m a z k;m min(m;h+k 1) . Proof. First, we note that U k a U (j) k;m C @I C AY (j) k @" z k+1;m C ¡ ¡ ¡ C "

z k+m;m A C Y (j) k 1 @" z k+1;m C ¡ ¡ ¡ C " z k+m 1;m A C ¡ ¡ ¡ C Y (j) max(k m+1;1 h) @" z k+1;m C ¡ ¡ ¡ C " z m+max(k m+1;1 h);m A a U (j) k 1;m C C k;m Y (j) k C@I C AY (j) k @" z k+1;m C ¡ ¡ ¡ C " z k+m;m A C Y (j) k 1 @" z k+1;m C ¡ ¡ ¡ C " z k+m 1;m A C ¡ ¡ ¡ C Y (j) max(k m+1;1 h) @" z k+1;m C ¡ ¡ ¡ C " z m+max(k m+1;1 h);m A a U (j) k 1;m C @I z k;m A@I C A C z k;m @I C A k 1 i=max(k m;1 h) Y (j) i min@m; k C h IA Y (j) k C@I C AY (j) k @" z k+1;m C ¡ ¡ ¡ C " z k+m;m A C Y (j) k 1 @" z k+1;m C ¡ ¡ ¡ C " z k+m 1;m A C ¡ ¡ ¡ C Y (j) max(k m+1;1 h) @" z k+1;m C ¡ ¡ ¡ C " z m+max(k m+1;1 h);m A a U (j) k 1;m C @I C AY (j) k 1 @" z k;m C ¡ ¡ ¡ C " z k+m 1;m A C Y (j) k 2 @" z k;m C ¡ ¡ ¡ C " z k+m 2;m A C ¡ ¡ ¡ C Y (j) max(k m;1 h) @" z k;m C ¡ ¡ ¡ C " z m+max(k m;1 h);m A C@I z k;m A@I C A Y (j) k @I @I C A@" z k+1;m C ¡ ¡ ¡ C " z k+m;m AA a U k 1 C @I z k;m A@I C A Y (j)
k @I @I C A@" z k+1;m C ¡ ¡ ¡ C " z k+m;m AA:

Hence, if we denote by ¨k the sigma-algebra generated by fY (j) i ; i a I; P; : : : ; kg, we have E h e (j) m Ûk j¨k 1 i a e (j) m Ûk 1 e (j) m (1 z k;m )(1+) E h e (j) m Y (j) (1 (1+)( z k+1;m +¡¡¡+ z k+m;m )) i :

Now, let us dene h k;m @rA a e r(1 z k;m )(1+) E h e rY (j) (1 (1+)( z k+1;m +¡¡¡+ z k+m;m )) i :

Obviously, for k ! m h C I, h k;m @rA a h m @rA. Since z k 1;m z k;m and " z k 1;m ! " z k;m , we clearly have h k 1;m @rA h k;m @rA h m @rA. Consequently, by (3.1), h k;m @ (j) m A h m @ (j) m A a I.

Thus, we get E h e (j) m Ûk j¨k 1 i a e (j) m Ûk 1 h k @ (j) m A e (j) m Ûk 1 :

So, the process fe (j) m Ûk g is a super-martingale.

Let w be a positive integer. By the Optional Stopping Theorem (considering the stopping time T w a minfT; wg),we get e (j) m @u; y 0 ; : : : ; y 1 b A:

Remark Inequality (3.14) becomes an equality when h ! m. Indeed, in this case, for k ! I, we have h k;m @ (j) m A a h m @ (j) m A a I. Thus, the process fe (j) m Ûk g is a martingale.

Furthermore, the random variables e (j) m Ûw I T >w pointwise converge to zero for w 3 I and are bounded by I since U w > H for w < T .

3.1.5

The case m ! " m (j) and j P " b m Now, let us consider the case m ! " m (j) and j P " b m . For the next result, let us denote (j) m @uA by (j) m; @uA, (j) m by (j) m; , (j) m by (j) m; , (j) m by (j) m; and (j) m by (j) m; to reveal explicitly the dependence in . Obviously, notice that (j) m; increases with . Hence, we know by classical results that (j) m; also increases with and consequently (j) m; by (3.5) (?;1) < r (j) (j) m; (?;2) . Then, we have lim u3I sup I u ln (j) m; @uA r (j) :

(3.17)

Proof. Obviously, there exists (?;1) < "

(?;2) such that (j) m; a r (j) . For all (?;1) < " , we have lim u3I sup I u ln (j) m; @uA lim u3I I u ln (j) m; @uA a (j) m; :

Letting 4 "

, we obtain (3.17).

As a particular, let us assume that Y (j) $ Exp@ j A. In this case, Ee rY (j) exists all for r < j and is equal to j j r . Clearly, r (j) a j . Now, since lim u3I I u ln rU (j) 1;m < H a j (3.18) and that (j) m @uA ! rU (j) 1;m < H, result (3.17) becomes lim u3I inf I u ln (j) m @uA a lim u3I sup I u ln (j) m @uA a j :

(3.19)

Conclusion

In conclusion, on the one hand, if j P b m then (j) m @uA a I for all u. On the other hand, if j P " b m , then for m < " m (j) , the logarithmic asymptotic behavior of (j) m @uA is given by equation (3.2) while for @IA > m ! " m (j) , the logarithmic asymptotic behavior of (j) m @uA is this time characterized by equation (3.17).

3.2

The case m = 1

Clearly, with z m a I, i.e. m a I, the results of the previous section have no meaning. This is the reason why this important case has to be treated separately. which we denote (j) I . If we assume that for all k, Ee rZ k exists for H < r < r 0 (with (j) I < r 0 ), where Z k a k i=1 @Y (j) i C i;I A, we have lim u3I I u ln (j) I @uA a (j) I :

(3.21)

Proof. Again, it suces to show that I @rA Xa lim k3I 1 k ln Ee rZ k exists for H < r < r 0 , and that (j) I is the unique positive value such that I @ (j) I A a H. We have ln Ee rZ k a r@I C A ln Ee rY (j) @1+(1+)ln@ ai ak+ AA a 1 0 dx ln E h e rY (j) (1+(1+) ln x) i :

It appears that result (3.21) requires only the existence and the uniqueness of (j) I (since the condition (j) I < r (j) is always fullled).

Equation (3.20) and so (j) I does not depend on market parameters , a and . It means that for u 3 I, the logarithmic asymptotic behavior of (j) m @uA is insensitive to the initial premium and to the speed with which premiums C k;I become Y (j)

1 +¡¡¡+Y (j) k 1 k 1 @I C A.
As a particular case, let us consider Y (j) $ N i=1 U i , where N is Poisson distributed with parameter and the U i 's are i.i.d. random variables. We assume that N and the U i 's are independent. We have Ee rY (j) We recognize the equation (2.2) in [START_REF] Asmussen | On the ruin problem for some adapted premium rules[END_REF]. In fact, this is not surprizing since (j) I is insensitive to market parameters , a and .

The Lundberg coecient (j) I

Since m a I, we know that (j) m a > H for all j, i.e. b m a Y.

Proposition 3.3. Assume that HH I @rA > H. Then, (j) I exists and is unique.

Proof. Obviously, we have I @HA a H. Denoting Y (j) @I C @I C A ln xA by Z x , we have H I @rA a since > H, and H I @rA becomes positive for large r. Now, since HH I @rA > H by assumption, the proof is over.

Large deviations path to the ruin

As previously, we aim at knowing how ruin occurs for the case m a I. Of course, the ruin occurs roughly at time u= H I @ (j) I A. Now, the cumulant generating function u= H I ( (j) I ) @rA of Z u= H I ( (j) I ) is asymptotically changed from u= H I ( (j) I ) @rA to u= H I ( (j) I ) @r C (j) I A u= H I ( (j) I ) @ (j) I A a r@I C A (3.24)

Consequently, given that ruin occurs, the claim size distribution Y (j) k is this time exponentially tilted by the factor (j) I @kA, where (j) I @kA a (3.26)

The factor (j) I @kA is negative as long as k < u= H I @ (j) I Ae 1 1+ .

We learn that this time, ruin is the consequence of atypically small claim amounts in a rst time, followed by atypically large claim amounts in a second time. This result is similar to those obtained in [START_REF] Asmussen | On the ruin problem for some adapted premium rules[END_REF] Impact of the insurer's strategy m on (j) m (u) for large u 3.3.1 Comparison of (j) m @uA and (j) m+1 @uA for large u (with m < I) Let us assume that the insurer's strategy is to take an horizon of credibility m < I. The objective is to determine for large u if it is benecial to take one year more to renew the premium. In other words, the objective is to compare (j) m @uA and (j) m+1 @uA for large u.

For j P b m , we have the following Proposition: Proposition 3.4.

(j) m+1 @uA (j) m @uA a I for all u:

(3.27)

Proof. It suces to show that b m+1 b m . Let j P b m+1 . By denition, (j) m+1 H. Let h@xA a @I xA C xEY (j) . We have h H @xA a EY (j) > H since (j) m+1 H means EY (j) ! @I C A@@I z m+1 A C z m+1 EY (j) A and > H. Thus, since z m+1 > z m , we have (j) m a @I C

A h(zm) E[Y (j) ] I < @I C A h(z m+1 ) E[Y (j) ] I a (j) m+1
H, or equivalently, j P b m .

This means that if j P b m , it is benecial to pass from an horizon of credibility m to an horizon m C I, whatever the initial capital u. Now, for j P " b m , it could seem a priori that the comparison of (j) m @uA and (j) m+1 @uA for large u diers according to the involved portfolio j. Indeed, two opposite a priori behaviours seem to take shape. If j P g, then h H @xA H and consequently (j) m ! (j) m+1 , which seems to be in favor of (j) m @uA (j) m+1 @uA whatever u and in particular for large u. But, on the other hand, if j P " b m n g, then h H @xA > H and consequently (j) m < (j) m+1 , which seems this time to be in favor of the inequality (j) m+1 @uA (j) m @uA whatever u. For m < " m (j) , we have the next Proposition: Proposition 3.5. There exists a positive constant u 0 such that for all u ! u 0 , we have (j) m+1 @uA (j) m @uA:

(3.28)

Proof. We know that j P " b m and m < " m (j) . Since b m+1 b m , j P " b m+1 . Hence, we have to consider two cases.

Firstly, let us assume that m C I < " m (j) . By Property 3.2, our result is proved.

Secondly, let us assume that m C I ! " m (j) . Hence, we know that lim u3I I u ln (j) m @uA a (j) m and lim u3I sup I u ln (j) m+1 @uA r (j) ;

and since (j) m < r (j) , this completes the proof.

Thus, even if j P g and hence (j) m ! (j) m+1 , (3.28) holds for large u. In fact, this is not surprizing. With light-tailed claims, the event ruin occurs after many atypically large claims.

At most m is large, at most the premium reacts favorably to this large claims, i.e. (j) m+1 becomes larger than (j) m in such circumstances. This facts are conrmed by our analysis of the path leading to ruin. Indeed, it teaches us that the ruin is caused by a succession of many atypically large claims more dangerous as m is large since (j) m increases with m.

For m ! " m (j) , we can nothing conclude from this kind, since the only result we have is equation (3.17).

In conclusion, assuming that the strategy of the company is m, if j P b m , then it is always benecial for large u to increase by I the horizon of credibility. Furthermore, if j P " b m , for m < " m (j) , the same conclusion comes while for m ! " m (j) , we cannot certify that this is also the case.

Sub-optimal strategies m for large u

Let us investigate sub-optimal strategies m for large u. Denition 3.5. A strategy m 1 is said to be sub-optimal compared to a strategy m 2 if for all portfolio j, we have (j) m 2 @uA (j) m 1 @uA for u 3 I. Denition 3.6. A strategy m 1 is said to be sub-optimal if there exists a strategy m 2 such that m 1 is sub-optimal compared to m 2 .

We have the next Proposition: Proposition 3.6. m < min fj: m (j) >0g " m (j) like m a I are sub-optimal.

Proof. In view of Propositions 3.4 and 3.5, it is clear that m < min fj: m (j) >0g " m (j) is suboptimal. For m a I, it is also obvious since (j) I < r (j) .

As we have seen, for a given portfolio j, increasing m decreases the number of "bad portfolios" and provides better premium adaptation to scenarios leading to ruin, at least as long as m < " m (j) . An extension of this reasoning to all m would allow to think that all m < I is sub-optimal compared to m a I. However, Proposition 3.6 shows that this is not the case. In fact, as u 3 I, all m < I becomes negligible compared to the ruin time T . Hence, the premiums will be able to react "quickly" (relatively to T ) and "signicantly" (at least for m "large" but nite) to a succession of large claim amounts, whatever the time where this claims occur and whatever the claim amounts observed before. But for m a I, obviously, this is not the case anymore. Indeed, as time passes, the premium reaction will become less and less eective to nish by not be able anymore to thwart several important successive claim amounts. This phenomenon is even truer if the claim amounts observed initially are small. This is well highlighted by the analysis of the path leading to ruin.

A probable extension of Proposition 3.6 would be then the following: m a I is suboptimal and all m < I is sub-optimal compared to m C I.

Numerical illustration

Let us consider an insurance market made up of Q portfolios, i.e. n a Q. Let us assume that Y (j) (j a I; P; Q) are Exponentially distributed, with mean I= j and variance I= 2 j . We then have a @I= 1 Ap 1 C@I= 2 Ap 2 C@I= 3 Ap 3 , a a @I= 1 A 2 p 1 C@I= 2 A 2 p 2 C@I= 3 A 2 p 3 and a @I= 2 1 Ap 1 C @I= 2 2 Ap 2 C @I= 2 3 Ap 3 . Let us assume that I= 1 a Q=R, I= 2 a I, I= 3 a S=R, p 1 a p 2 a I=Q and a H:I. We have j a I; P P g and j a Q P b m as long as m < QH.

The critical horizon of credibility m (c) a PSH. For our numerical purpose, let us compare strategies m a H; P; IH; PSH; IHHH and I. Of course, for all u, (3) m @uA a I for m a H; P and IH. Carrying out IHH HHH simulations on an horizon of IH HHH time periods, we obtain the results below, also shown on 0:764% 0:540% 0:386% 0:260% 0:200% 0:148% 0:086% 0:060% 0:046% 0:036% m = 2 0:608% 0:408% 0:276% 0:214% 0:150% 0:088% 0:054% 0:042% 0:018% 0:010% m = 10 0:450% 0:308% 0:220% 0:148% 0:094% 0:046% 0:020% 0:010% 0:006% 0:000% m = 250 0:478% 0:318% 0:236% 0:160% 0:094% 0:050% 0:024% 0:008% 0:005% 0:000% m = 1000 0:478% 0:318% 0:236% 0:160% 0:094% 0:050% 0:024% 0:008% 0:004% 0:000% m = I 0:478% 0:318% 0:236% 0:160% 0:094% 0:050% 0:024% 0:008% 0:004% 0:000% 8:688% 7:016% 5:686% 4:618% 3:746% 3:018% 2:462% 1:970% 1:598% 1:328% m = 10 2:866% 2:016% 1:452% 1:004% 0:738% 0:538% 0:374% 0:280% 0:188% 0:134% m = 250 2:190% 1:360% 0:860% 0:554% 0:380% 0:224% 0:140% 0:088% 0:060% 0:038% m = 1000 2:170% 1:304% 0:823% 0:528% 0:367% 0:203% 0:128% 0:084% 0:058% 0:037% m = I 2:286% 1:395% 0:865% 0:530% 0:365% 0:202% 0:127% 0:084% 0:058% 0:037% 18 20 m = 0 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% m = 2 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% m = 10 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% m = 250 99:992% 99:648% 96:894% 88:370% 73:964% 58:720% 42:814% 29:850% 20:238% 13:378% 8:816% m = 1000 99:994% 99:790% 97:490% 89:582% 75:532% 57:010% 41:046% 28:186% 18:682% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% m = 2 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% m = 10 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% 100:000% m = 250 5:714% 3:660% 2:440% 1:566% 1:034% 0:704% 0:486% 0:330% 0:240% 0:142% m = 1000 4:708% 2:862% 1:784% 1:048% 0:644% 0:412% 0:250% 0:146% 0:098% 0:038% m = I 4:708% 2:862% 1:784% 1:048% 0:644% 0:412% 0:250% 0:146% 0:098% 0:038% On the one hand, asymptotically, whatever the portfolio j, we see that the larger m is, the smaller (j) m @uA is, even for m ! " m (j) @ m (c) A. This conrms our theoretical ndings and intuition, namely that (also for m ! min fj: m (j) >0g " m (j) ) m is sub-optimal compared to m CI.

On the other hand, for small initial capital u, a bigger value of m may this time increase in certain cases the ultimate ruin probability. Obviously, this observation is not unexpected.

Remark Clearly, the numerical illustrations can not highlight the fact that m a I is sub-optimal.

Limit results for heavy-tailed claims

We have seen that in the case where the Lundberg coecient exists, adjusting premium with credibility may increase this one. As typical sample paths for which ruin occurs exhibit a sequence of moderately large claims, there is quite often enough time to compensate early losses with credibility-adjusted future premiums. On the opposite, in the heavy-tailed case, it is often said that ruin is likely to be caused by one large claim (see for example Theorems 1.1 and 1.2 in [START_REF] Asmussen | Large deviations results for subexponential tails, with applications to insurance risk[END_REF] in the classical risk model). It is logical to think that credibility plays a less important role for heavy-tailed claim amount distributions.

In this Section, we focus on the regular variation case.

We note F P . The convergence is uniform on each subset y P y 0 ; IA (H < y 0 < I).

In the light-tailed case, we have studied the inuence of m on the Lundberg coecient (j) m a lim u3I I u ln (j) m @uA:

(4.2)

In our heavy-tailed case, the equivalent quantity to be studied is parameter (j) m a lim u3I I ln u ln (j) m @uA:

(4.3)
As we consider the case F j P (j), and since credibility adjustments involve implicitly that the two rst moments of claim amounts exist, we restrict to the case (j) > P. Now, for m < m (c) , we shall prove that, if j P " b m , then m does not modify the value of (j) m , which is in accordance with our intuition. Theorem 4.2. Suppose that the Y (j) k 's are regularly varying with index (j) > P. If m < m (c) and j P " b m , i.e. > EY (j) @I z m A C z m EY (j) I; (4.4) we get (j) m a lim u3I I ln u ln (j) m @uA a (j) I: where c is the (continuous-time) premium income rate, is the intensity of the Poisson process, F W is the cumulative distribution function of the individual claim amounts and W is the expected individual claim amount. In the (j) -regularly varying case with (j) > P (this means that I F W @xA $ x (j) as x 3 I;

this corresponds to cont @uA $ c W I (j) I u (j) +1 : This result may be adapted to the discrete-time model, with constant I-period premium income and independent, identically distributed (aggregate) claim amounts on each period, with (j) -regularly varying distribution: if the safety loading is positive, for (j) > I, there exists C > H such that the (discrete-time) probability of ruin disc @uA satises disc @uA $ Cu (j) +1 as u 3 I. To show this, one might for example use arguments developed in [START_REF] Rullière | Another look at the Picard-Lefèvre formula for nite-time ruin probabilities[END_REF] and [START_REF] Lefèvre | On nite time ruin probabilities for classical risk models[END_REF].

To show that there exist C 1 > H and a function 1 such that (j) m @uA ! 1 @uA and 1 @uA $ C 1 u (j) +1 as u 3 I; let us consider a modied model, in which future earned premiums due to each claim are anticipated and earned at the claim instant: if claim amount Y (j) k occurs at time k, the sum of future credibility premiums associated to Y (j) k corresponds to Y (j) k @I C A z i;m a z m , we have from Embrechts and Veraverbeke (1982) that the ruin probability 1 @uA in the modied model satises 1 @uA $ C 1 u (j) +1 for some C 1 > H as u 3 I as long as the safety loading is positive in the modied model, which is guaranteed by condition (4.4).

To show that there exist C 2 > H and a function 2 such that (j) m @uA 2 @uA and 2 @uA $ C 2 u (j) +1 as u 3 I; let us rewrite (j) m @uA as (j) m @uA a r h WI i m; U (j) i;m < H or Wi > m; U (j) i;m < H i : This means that (j) m @uA r h WI i m; U (j) i;m < H i C r h Wi > m; U (j) i;m < H i :

The rst term is O@u (j) A and so o@u (j) +1 A. where a @I C A@I z m A EY (j) @I z m @I C AA > H from Condition (4.4).

Hence, since for any sequences of nonnegative r.v.'s @X i A i!1 and @Z i A i!1 , and for any sequences of nonnegative numbers @a i A i!1 and @b i A i!1 , we have rWi ! I; X i C Z i > a i C b i rWi ! I; X i > a i C rWj ! I; Z j > b j ; we get that r h Wi > m; U (j) i;m < H The second term is equivalent to C 3 u (j) +1 as u 3 I for some constant C 3 > H, because it corresponds to a ruin probability with claim size distribution in RV @ (j) A and with a positive Y (j) l > u=P C P i 5 r Wi > m; mY (j) i > u=P C P @i mA ! a r Wi > H; mY (j) i > u=P C P i ! :

This may be rewritten as (3) m (u)

  j) (1 ( z i+1;m + z i+2;m +¡¡¡+ z min (i+m;k);m )(1+)) rZ k a r@I z m A@I C A C ln E h e rY (j) (1 (1+)zm) i :

3 .

 3 Assume that m < " m (j) and j P " b m . Then (j) m @u; y 0 ; : : : ; y 1 b A e

3. 2 . 1

 21 Asymptotic behavior of(j) I @uA Let us state an analogue of Theorem 3.1. Theorem 3.4. Suppose that there exists a unique positive solution to the equation

  a e (e r 1) . So, the equation (3.20(1+(1+) ln x) £ a H:(3.23)

  for his continuous time compound Poisson model. A typical path to ruin is shown in Figure 3.1.

Figure 3

 3 Figure 3.1:

Figures 4. 1

 1

  to 4.3.

(4. 5 )

 5 Proof. In the compound Poisson risk model with sub-exponential claims,[START_REF] Embrechts | Estimates for the probability of ruin with special emphasis on the possibility of large claims[END_REF] have shown that the (continuous-time) classical ruin probability @uA with initial surplus u ! H (without credibility premium adjustment) behaves as u 3

  m a @I z k;m A@I C A:For each sample path, if ruin occurs in the modied model, then it occurs as well in the initial model as credibility adjusted premiums are the same in both models but are received later in the modied model. Since from k a m C I, k+m i=k+1 "

4 Wi

 4 safety loading (from (4.4)). The rst term r

Figure 4 . 2 :

 42 Figure 4.2: Numerical results for(2) m(u)

Figure 4 . 3 :

 43 Figure 4.3: Numerical results for(3)

Let us conclude in saying that, on the one hand, if j P b m then (j) m+1 @uA (j) m @uA a I by Proposition 3.4 (clearly also valid for the heavy-tailed case), but on the other hand, this time, if j P " b m , then an increasing of the horizon of credibility has asymptotically no impact on (j) m @uA in rst order, at least for m < m (c) .


Zhangzhung and Qiangic languages

Guillaume Jacques, Université Paris Descartes, CRLAO Zhangzhung (ZZ), a dead Sino-Tibetan language only known by fragmentary sources, has no living descendant. The area of the former Tibetan empire is home to many non-Tibetan languages, some of which could be related to ZZ, and therefore be of tremendous importance for interpreting ZZ data. However, the huge diversity of Sino-Tibetan (ST) languages, and the poor accessibility of data on many non-literary languages, makes it difficult for specialists of Old Tibetan philology to evaluate etymological claims regarding the ZZ vocabulary.

The aim of the present paper is to evaluate the degree of relationship between ZZ and Qiangic languages. Several scholars, such as [START_REF] Hummel | 1974-5 Materialen zu einem Wörterbuch der Źaṅ-źuṅ-Sprache[END_REF], have proposed to locate the origin of ZZ in Eastern Tibet rather than in Western Tibet, using some linguistic comparative data. This hypothesis, if true, would be of far-reaching consequences for the study of Tibetan History. However, this paper will show that the evidence is rather limited.

Methodology

In this paper, we present all ZZ words with possible Qiangic etymologies (drawing data from Tangut and Rgyalrong). 1 However, finding cognates is not sufficient in itself to prove that two languages of the same language family belong to the same subbranch: according to August Leskien's principle, only common innovations are valid evidence.

In order to prove that ZZ is closely related to Qiangic, or to LB-Qiangic 2 , we must not only find isoglosses between ZZ and these languages (words unfound in other branches of ST), but also prove that these isoglosses are innovations, not retentions from proto-ST.

Concerning the problem of the internal classification of ST languages, it should be stressed that since no common innovation has been found between all ST languages outside of Chinese, there is not a shred of evidence to talk of a 'Tibeto-Burmese' subgroup. Chinese seems to be just one of the thirty more branches of ST, and deserves no special place in the Stammbaum of the family.

Morphology

1 The Rgyalrong data are from [START_REF] Jacques | Phonologie et morphologie du japhug (rGyalrong)[END_REF][START_REF] Jacques | Phonologie et morphologie du japhug (rGyalrong)[END_REF], and the Tangut reconstruction is based on Gong [START_REF] Hwangcherng | Collected Papers on Tangut Philology[END_REF]. We indicate the number in Li (1997)'s dictionary for each Tangut character to facilitate crosschecking. We use the following abbreviations: Bu Burmese, Jpg Japhug, Jg Jingpo, LB Lolo-Burmese, MC Middle Chinese, OC Old Chinese, Si Situ, Skt Sanskit, ST Sino-Tibetan, Tb Tibetan, Ta Tangut, Zb Zbu, ZZ Zhangzhung. 2 It is probably that Qiangic languages, LB and Naxi form a clade in the ST family, though data are still insufficient to prove it.

Not much is known about ZZ morphology because of the lack of extensive texts in this language, and this paper is limited to the lexical evidence.

However, an interesting grammatical morpheme is the negative prefix kV-, found in words such as: ZZ Tb This form is extremely puzzling, as ST languages usually do not have negative prefixes beginning with a velar stop. If one could find a ST language with a velar negative prefix, this would be an interesting clue to classify ZZ within the ST family.

Vocabulary

Here is a list of possible cognates between ZZ and Qiangic languages. Many of the words presented here are pan-ST, and thus of little help for classifying ZZ within the family. We found only two potential examples of common innovations between ZZ and Qiangic languages (examples 1 and 5) and two possible lexical isoglosses unfound outside Qiangic (examples 8 and 9), though all are problematic and might be errors.

Loanwords from Tb and Skt have not been included. Besides, we also discarded roots about which it is unclear whether they are cognates or loanwords, such as dug 'poison', mig 'eye' etc as they are identical with Tb.

We have tried to avoid presenting spurious comparisons in the present paper. [START_REF] Hummel | 1974-5 Materialen zu einem Wörterbuch der Źaṅ-źuṅ-Sprache[END_REF] found more examples than us between ZZ and Ta, but some of his proposed cognates are not reliable. For instance, he compares ZZ <wer> (=rgyal) 'king, victor' to Ta 'wu-(tsu)', in fact not a real Ta form, just the Mandarin reading of the characters 兀卒 that are used in Chinese historical texts to transcribe Ta 晃虻 #5306 #510 ŋwər 1 dzjwɨ 1 'emperor'. The first syllable of the Ta compound means 'blue, sky', while the second one 虻 dzjwɨ 1 means 'lord'; This syllable is clearly unrelated to ZZ <wer>. He also compares ZZ <tal> (=lcags) 'iron' with MC 铁 thet, although this becomes impossible if the OC reconstruction *hllik is taken into account.

Although we tried not to repeat such mistakes, errors may still have crept into this list of examples and some of the hypotheses proposed here might be eventually proven to be wrong.

The examples are listed following the order of the Tibetan alphabet, with the Tibetan gloss for each ZZ word in brackets.

ku ra (=khyi 'dog')

Some ZZ scholars have interpreted this word as derived from Skt kukkura-or a Pkt equivalent thereof (Hoffmann 1972: 196), others have proposed cognates in various ST languages (Hummel 1986:12, Nishi andNagano 2001:21).

The vocalism of the first syllable <ku> is consistent with the usual reconstruction *kwi proposed for this root in proto-Tb (in Tb, medial *-w-changes to -y-before front vowels). The second syllable, in turn, could be compared with the na element found in many ST languages, including Tamangic and Rgyalrong. This would suggest that a lenition *VnV > VrV would have taken place at some stage in the evolution of proto-ZZ. The existence of this lenition is proven by the word <gu-ra> (=yon-tan) 'virtue' [START_REF] Haarh | The Zhang-zhung Language[END_REF], Martin 2004: 30), an obvious loanword from Skt guṇ a.

It seems that the structure of the ZZ word is identical to Jpg khɯ-na : the first syllable is cognate to the pan-ST root in Tb khyi, OC 犬 *kkhwir, Ta 蹟 #1200 khjwɨ 1.30, while the second syllable is cognate to Ta #573 漉 na 1.17.

To have exactly the same compound in both Rgyalrong and ZZ would be a strong evidence of a common innovation between ZZ and Qiangic, although such a compound could also have been formed independently recreated in two branches of ST. Of course, it is also possible that Hoffmann's hypothesis is correct and that this word is a corruption of an Indic word, in which case any comparison with ST languages would be irrelevant.

rko / sko (=lus, gzugs 'body')

This word is cognate with Tb sku 'body' [START_REF] Haarh | The Zhang-zhung Language[END_REF], Stein 1971: 248) and OC 躯*qqho > khju, Jpg tɯ-skhrɯ, Ta #860 囚 kwər 1.84. Its pan-ST nature makes it useless for the purpose of language classification.

skod / skos (=so 'tooth')

This etymon is possibly related to Tb so 'tooth (Matisoff 2001:174, Nishi andNagano 2001:19 discuss this etymon). Tb so regularly comes from proto-Tb *swa.

Cognates are found everywhere in the ST family, in particular Jpg tɯ-ɕɣa, Ta #169 吸 ɕjwi 1.10.

The ZZ spelling is puzzling, but it could be an attempt at rendering a sound unfound in Tb, such as the velar fricative [ɣ]. If we assume a ZZ form such as *sɣo or *sɣwa, a spelling with a velar stop *<sko> or *<sgo> would have been the only possible way of representing this sound using the Tb alphabet. We have no explanation for the final consonants. Hummel (1986:12) believes that ZZ <skod/skos> ought to be compared to Ta #39 慣 kowr 2.82 'tooth'. It is also a possibility, though the Ta word is isolated even within the Qiangic group. This etymology is mutually incompatible with the former one (contra Hummel, who treats all three words Tb so, Ta kowr and ZZ <skod/skos> as cognates).

khri (=gru 'corner')

The gloss 'corner, tip' is found in Martin (2004: 25). This word could be related to Jpg ɯ-rkɯ 'corner, border', itself a cognate of Tb gru 'corner'. The distribution of this root in the ST is too wide to use it for classifying ZZ. The orthographic <i> is probably to be understood as a central vowel (the Tb dialect on the basis of which the orthography was conceived was perhaps like Khams and Amdo dialects).

5. lgyam (pra lgyam dug = phra rgyas dug) Martin (2004: 37) translates <lgyam> as 'wide', but he insists that this entry is 'rather dubious'.

The Tb term phra-rgyas corresponds to Skt anuśaya-'propensity' (classical Skt 'regret', from anu√śī 'to lie along, to adhere closely to'). However, the morphological structure of the Tb word is not directly modeled after its Skt equivalent. The concept of anuśaya-is the subject of the whole fifth chapter of Vasubandhu's Abhidharmakośa (La Vallée-Poussin 1925:1-118). An explanation of the meaning of anuśaya-is given in śloka 39:

(1) De la Vallée Poussin (1925:78-79) translates this passage in the following way: « Ils sont atomiques (phra ba, aṇu-) ; ils adhèrent (rjes vbrel,4 anu√gam); ils se nourrissent (rgyas vgyur, anu√śī) de deux manières (rnam gnyis, dvidhā) ; ils lient continuellement (rjes su vbrang, anu√bandh), c'est pourquoi (de yi phyir, tasmāt) on les nomme anuśaya-». The Tb word 'phra ba' translates Skt aṇu-'atom of matter' (Nominative plural aṇavas), and rgyas vgyur corresponds to anu√śī 'to lie along, to adhere closely to, to follow upon' (Present 3pl Middle anuśerate), here translated by Poussin as 'se nourrir' (to feed).

Tb rgyas vgyur is not a literal translation of the Skt form: 5 rgyas-pa has two distinct meanings: 'to increase' and 'large, wide'. Therefore, rgyas vgyur could be translated either as 'to grow' or 'to become wide'. De la Vallée Poussin (1925:78, n.ii.) analyzes Tb phra-rgyas as 'atomique-étendu', considering here rgyas to mean 'wide', not 'to increase'. However, one of the Chinese translation of this term, 細增 'tiny-increase' would suggest that rgyas means 'to increase', not 'wide'.

Therefore, it is not straightforward to decide whether rgyas should be understood as 'to increase' or as 'wide' in this expression, and the ZZ syllable <lgyam> can be interpreted either way. If 'wide' is indeed the intended meaning of <lgyam>, it can be compared to two roots found in Qiangic and LB languages meaning 'wide' or 'broad' (Jacques 2004:244): proto-Jpg *rtljom, Jpg rɟum Si rdʑam wide, broad (of a piece of clothes)' proto-Jpg *ljam, Jpg jom, Si jam, Zb laḿ 'wide (of a place)'.

These two roots are distinct in proto-Rgyalrong, but from the point of view of historical phonology, both are potentially comparable with Tangut 蛟 #4874 and 寛 #34 low 2.47 and Zaiwa lam 51 . This root is probably derived from the ST root meaning 'to fathom, to search, to stretch', found in OC 覃 dom < *llɨm, 寻 zim < *s-lɨm, Tb 'dom-pa 'a fathom', lam 'path'. The semantic development to 'wide' is an example of a common innovation in Qiangic and LB languages, as it is not found in other ST languages. The fact that ZZ shared this common innovation would be an argument for postulating a closer relationship with the Qiangic-LB branch than to the Bodic one. However, it should be stressed that all these hypotheses rest on the assumption that <lgyam> translates rgyas in the sense 'wide', which is not at all certain.

The ZZ word spelled <lgyum> or <lgyu> meaning 'road' [START_REF] Haarh | The Zhang-zhung Language[END_REF], Hoffman 1972, Martin 2004:38, Pasar et al. 2008: 49) is related to the same root, but this word is not discussed in this paper, as it is absent from Qiangic languages. 6The ZZ cluster spelled <lgy> (see the discussion in Hoffman 1972) does not necessarily represent a triple consonant cluster. It might be an attempt at representing a lateral palatal *ʎ, or it could be the result of a metathesis from a cluster such as *k-lj-. Alternatively, the -g-could be an epenthetic consonant, in the same way as -gin Tb words such as brgyad < *p-rjat [START_REF] Fang-Kuei | Tibetan glo ba 'dring[END_REF]. None of these hypotheses are easy to test given our limited knowledge of the ZZ lexicon, but all have to be taken into consideration.

du (=sprin 'cloud')

This form 'cloud' is reminiscent of the Qiangic and LB root found in Jpg zdɯm 'cloud', Ta #2738 咥 djɨ̣ j 2.55, Bu tim. The loss of final -m reminds of the free alternation between the two forms <lgyu> and <lgyum> for 'road' in ZZ, and it is possible that final -m was lost in some contexts. This etymology is very tentative.

ni (=mi 'man'), ne (=me 'fire'), ma-ning (=ming 'name')

These three words are synonyms and cognates of Tb mi 'man', me 'fire' and ming 'name' respectively. These ubiquitous roots are found everywhere in the ST, including Rgyalrong (Jpg tɯrme, smi, tɤrmi) and Tg (#4574 耄 mjɨ 1.30 'the other one', #4408 粐 məə 1.31 'fire', #2639 剳 mjiij 2.35 'name').

The presence of a dental instead of a labial initial is certainly due to a process of palatalisation before front vowel. The spelling <ma-ning> is possibly an attempt at representing *mniŋ by a Tb scribe who spoke a dialect where OT mn-clusters where simplified to n-.

pe brag / se brag (=srog gcod 'kill')

In this form, the first syllable <pe/se> corresponds to Tb srog 'life', and the second brag to Tb gcod 'cut'. This syllable <brag> can be compared to Jpg phaʁ 'cut, break' and Tg #4007 猖 pha 1.17, #4459 綛 bja 2.17. ZZ would have added an -r-infix to this verbal root.

ma thun / mang thun (=sha 'meat')

This word 'meat, flesh' [START_REF] Haarh | The Zhang-zhung Language[END_REF][START_REF] Stein | La langue Źaṅ-źuṅ du Bon organisé[END_REF]:243, Martin 2004:105) is potentially comparable to a root exclusively found in Rgyalrongic languages, Jpg tɤmth mth mth mthɯ ɯ ɯ ɯm m m m 'cooked meat'. A serious problem with this etymology is the difference in final consonant -n vs. -m. If this comparison is genuine, it would be a very important isogloss between ZZ and Rgyalrongic, but it is likely to be a coincidence.

mu / dmu (=mkha', gnam, dbyings, gnas 'sky')

The ZZ word for 'sky' (Hummel 1972:14, Martin 2004:111, Pasar et al. 2008:184) is cognate with the Qiangic and LB root found in Jpg tɯm m m mɯ ɯ ɯ ɯ 'sky, rain ', Tg #3513 朿 mə 1.27 'sky', Bu mui C 'rain'.

mu zhi (=lto 'phye che 'snake'), mur (=klu 'nâga')

<mu zhi> translates Skt Mahoraga 'The Great Serpent' (Hummel 1974-5:515, Martin 2004:117). 7 The second syllable <zhi> is comparable to Tb che or chen 'big' (wer zhi = rgyal chen, tru zhi = rin chen etc), so that the first one <mu> can be equated with lto 'phye 'serpent'. <mu> is to be compared to the ZZ word for Nāga <mur>, which comes from the pan-ST root 'serpent' (Stein 1971: 246) Tb sbrul < *s-mrul,8 OC 虺 *hmmɨlʔ, Bu mrwe B (unrelated to the root 'bug ', Tb 'bu).

Cognate roots seem to be found in Qiangic languages: Jpg qapri, Ta #80 巌 phio 2.43, though no nasal initial appears there.

li / le (=rlung 'wind')

This word is clearly related to the pan-ST root found in Bu le B 'wind and Jp bunglí 'breeze' (Matisoff 2001:165). This root is attested in Qiangic: Jpg qale, Ta #2302 名 ljɨ 1.29.

sha 'bal (=sta re 'axe')

This ZZ word for 'axe' [START_REF] Nyima | The early spread of Bon[END_REF]:16, Martin 2004:156, Pasar et al. 2008: 259) could be tentatively compared to the ST etymon found in OC 斧 *paʔ, Jpg tɯrpa rpa rpa rpa, #5203 躅 wjị 1.67, under the assumption that a phonetic change -al → -a occurred in the Tb dialect of the person who wrote this ZZ word. This etymology is hypothetical.

shin (=shes pa 'to know')

This word is found in many compounds (Martin 2004:158) such as <kun shin> = kun shes, <ti shin> = rnam shes, <nges de shin> = rang rig. It is related to the ST root 'know' found in Tb shes, Bu si A , Jpg sɯs. The final consonant -n could be a nominalizing suffix, as this ZZ form appears mostly in abstract words. <shin> would mean 'the knowledge' rather than the verb 'to know'.

shin / shin ni / shin tun (=mchin pa 'liver')

The ZZ word for 'liver' (Stein 1971:237) belongs to a well-known ST cognate set, comprising Tb mchin-pa, Bu a-sany C , Jpg tɯmtshi, Ta #5273 逞 sji 2.10 (Matisoff 2001:170).

sli (=zla-ba 'moon')

This word is only cited in Pasar et al. (2008:277). It is clearly related to the pan-ST root for 'moon' found in Tb zla (OTb also sla).

Nevertheless, its unusual front vocalism is interesting, because it suggests that the donor language underwent a 'brightening' change *-a → -i (to use Matisoff 2004's term) like Qiangic languages Ta #2814 嚠 lhjị 2 'moon' or Prinmi ɬí. Nevertheless, brightening in this word is also found in the non-Qiangic language Mtshosna monpa le 53 'month'. It is also difficult to explain why brightening did not occur in the word 'axe' (13).

hrang (=rta 'horse')

ZZ scholars have noted that this word is relatable to a root widespread in the ST family (Stein 1971:153): Bu mrang C , Jpg mbro, Ta #764 字 rjijr 1.74, and possibly OC *mmraʔ. However, this is not a genuine ST etymon, and must be a (relatively) late Wanderwort, as the horse was unknown in Eastern Asia before the second millennium BC.

Numerals

The ZZ numerals from one to ten are all attested (see Table 2), but it is not always clear whether these are corruptions of Tb forms or genuine ZZ etyma. Three comments can be made on the ZZ numerals.

First, the etyma 'one' and 'two' differ from Tb by not having the regular palatalisation *ti > ci and *ni > nyi. 9 Second, the numeral 'seven' is the root found in most ST languages (for instance, Jpg kɯɕnɯs), unlike Tb, where an innovative form bdun of unclear etymology replaced the original numeral. Third, the numeral for 'ten' is <cu> and is suspiciously similar to Tb bcu: this root is not widespread in the ST family.

ZZ

Conclusion

9 Syllables such as ni are extremely rare in Tb, and there is little doubt that a palatalization of dentals occurred in proto-Tb. There are many examples of dental stops with i, though (mthil, gtib etc), and the exact conditioning factors of palatalization are not entirely clear.

In this paper, we have found very little evidence of a special relationship between ZZ and the Qiangic languages. The most interesting examples, such as etyma #1 <ku ra> 'dog', #5 <lgyam> 'wide?', #8 <brag> 'cut' and #9 <ma thun> 'meat', are all problematic and might be coincidences. Other potential cognates are found throughout the whole ST family and are not usable as an argument that ZZ was related to Qiang and Ta as [START_REF] Hummel | 1974-5 Materialen zu einem Wörterbuch der Źaṅ-źuṅ-Sprache[END_REF] has suggested. The relationship of some ZZ etyma with Almora languages of Western Tibet seems more convincing [START_REF] Stein | La langue Źaṅ-źuṅ du Bon organisé[END_REF]). This would be strong evidence against the hypothesis that ZZ originated in Eastern Tibet.

Several possible ZZ sound changes have been identified: the palatalisation of m to n before front vowel (7), the lenition of n to r between two vowels (1), the loss of final -m (the conditioning context is unclear, examples 5 and 6) and change -a → -i in open syllables ( 16).

Due to the fragmentary nature of our ZZ data, the hypotheses presented in this paper must all be considered as tentative and subject to revision if new data on the ZZ language are discovered in the future.

Table 1 :

 1 Examples of the ZZ negative prefix.

Table 2 :

 2 Numerals 1-10.

		Tb	Jpg
	tig	gcig	tɤɣ	1
	ne, nis, ni	gnyis	ʁnɯs	2
	sum	gsum	χsɯm	3
	bing, bing-nga	bzhi	kɯβde	4
	nga	lnga	kɯmŋu	5
	drug	drug	kɯtʂɤɣ	6
	snis, sne, sni, snel, snes	bdun	kɯɕnɯs	7
	gyad	brgyad	kɯrcat	8
	gu dun	dgu	kɯngɯt	9
	cu	bcu	sqi	10

The Skt text comes from the Digital Sanskrit Buddhist Canon database: http://www.uwest.edu/sanskritcanon/dp/

De la Vallée Poussin reconstructs anu√sañj.

We would have expected something like *rjes su nyal ba.

However, it is found in many ST languages, including Bu lam C , Tb lam, Jg lam etc. A special phonetic change must have occur in this ZZ word, given the back rounded vocalism.

The Tb lto 'phye is a mistranslation from Skt uraga-, which literally means 'the one who moves (-ga) on his chest (uras-)' (= serpent): Tb lto means 'belly' not 'chest'. This error resulted from a confusion of the first part of the compound with Skt udara-'belly'. A form such as *udara-ga 'the one who moves on his belly' would have been possible and semantically plausible as a metaphor for 'serpent'.

Proto-Tb *m becomes denasalized between s-and -r-. The Tb smr-clusters (smra, ngur-smrig etc) are probably derived from proto-Tb *sǝ-mr-.


M Jamet

I.-S Yu

T Devillers

A Barski

P Bayle-Guillemaud

C Beigne

J Rothman

V Baltz

J Cibert

Growth of (Ge,Mn) nanocolumns on GaAs(100): the role of morphology and co-doping on magnetotransport

Keywords: numbers: 75.50.Pp, 73.50.-h, 75.47.-m, 75.75.-c

Changing the morphology of the growing surface and the nature of residual impurities in (Ge,Mn) layers -by using different substrates -dramatically changes the morphology of the ferromagnetic Mnrich inclusions and the magnetotransport properties. We obtained p-type layers with nanocolumns, either parallel or entangled, and n-type layers with spherical clusters. Holes exhibit an anomalous Hall effect, and electrons exhibit a tunneling magnetoresistance, both with a clear dependence on the magnetization of the Mn-rich inclusions; holes exhibit orbital MR, and electrons show only the normal Hall effect, and an additional component of magnetoresistance due to weak localization, all three being independent of the magnetic state of the Mn rich inclusions. Identified mechanisms point to the position of the Fermi level of the Mn-rich material with respect to the valence band of germanium as a crucial parameter in such hybrid layers.

Research on ferromagnetic semiconductors triggered enormous activity due to their potential use in spintronics [1,2]. Up to now, efforts have mainly focused on diluted magnetic semiconductors (DMS) in which magnetic atoms randomly substitute the host matrix atoms [3]. Their magnetic properties can be manipulated by electric fields making them suitable materials for spintronic applications provided that they can be made ferromagnetic above room temperature. However DMS based on II-VI and III-V semiconductors still exhibit very low values of Curie temperature T C .

Many groups have reported T C values well above room temperature, along with remarkable magneto-transport and magneto-optical properties, in semiconductors doped with magnetic transition metals (TM). It is now admitted that these properties may be attributed to TM-rich areas resulting from spinodal decomposition [4]. Such features have been theoretically predicted [5] and reported in (Ge,Mn) [6,7,8,9], and in Cr and Fe-doped GaN [10,11] or ZnTe [12]. In this field of intense materials research, goals are now: (i) controlling spinodal decomposition to reproducibly stabilize high-T C TM-rich areas and tailor desirable magnetic properties, and (ii) enhancing the coupling with carriers to give rise to strong magnetoresistance (MR) or anomalous Hall effect (AHE).

In this paper, we demonstrate the fine control of spinodal decomposition in (Ge,Mn) films grown on GaAs(001) substrates. We focus on (Ge,Mn) because it is compatible with mainstream silicon technology, and spinodal decomposition leads to high T C values in layers grown on Ge substrate. Growing (Ge,Mn) films on GaAs(001) semi-insulating (ρ > 10 7 Ωcm) substrates makes in-plane transport measurements easier, and constitutes a first step towards spin injection from (Ge,Mn) into a GaAs-based spin-LED [13]. Using different sur-face preparations, we clearly identify the role of surface morphology and the role of impurity diffusion from the substrate (either Ga or As atoms), on the nanocolumns growth, on one hand, and on the electrical properties, on the other hand. We thus address the major issue of the influence of co-doping (either n-type or p-type) on spinodal decomposition in group IV magnetic semiconductors, demonstrating a major influence on the shape of the Mn-rich precipitates. We also provide new hints to control and optimize magneto-transport propertie of the (Ge,Mn) films. We show that AHE and MR are not optimized simultaneously, and we propose a general picture based on the electrical doping of the matrix and on the position of Fermi level in the precipitates with respect to the valence band of Ge.

(Ge,Mn) films were grown by low temperature Molecular Beam Epitaxy (MBE), using growth conditions as described in Ref. [6], with the substrate temperature T g = 100 • C and deposition rate ∼ 0.2 Å.s -1 . We have used two different methods to prepare the GaAs surface. In the first one, the native oxide was thermally desorbed from an epiready substrate, by raising the substrate temperature up to almost 600 • C. The (Ge,Mn) layers was grown directly on the resulting Ga-rich GaAs surface, which was rough as observed by RHEED. Such samples will be labelled Ga-(Ge,Mn). In the second case, As-(Ge,Mn) samples, a thin undoped GaAs buffer layer was grown first in a separate III-V system, protected with an amorphous As capping, and transferred in air to the IV-IV MBE machine. Desorbing the As capping layer at 200 • C results in a very flat, (2×4) reconstructed, Asrich surface. Samples grown on Ge substrates, labelled Ge-(Ge,Mn), constitute our reference samples.

Magnetization was measured using a Superconducting QUantum Interference Device (SQUID). Magnetotrans-port properties (magnetoresistance and Hall effect) were investigated using Hall bars defined by optical lithography, aligned along a < 110 > direction, of width 20 µm, with voltage probes separated by 140 µm. A typical morphology is that of long Mn-rich nanocolumns, growing normal to the substrate surface. On a Ge substrate [6] or on a Ge buffer layer grown on GaAs(001) (Fig. 1a), these nanocolumns are well aligned along the [001] growth direction. On a Ge buffer layer grown on a Ge(001) substrate with {113} facets obtained by anisotropic chemical etching in an H 2 O 2 aqueous solution, Fig. 1b, they grow perpendicular to the facets. Finally, in Ga-(Ge,Mn) films, they are bent according to the initial surface roughness (Fig. 1c), and this results in a highly disordered pattern. This general picture fully agrees with 2D spinodal decomposition, driven by surface diffusion and aggregation of Mn atoms, with nucleation of Mn-rich areas taking place during the first stage of the growth [5]. As a consequence of this mechanism, the columns are always perpendicular to the growing surface. By contrast, in Fig. 1d, the As-(Ge,Mn) layers feature randomly distributed Mn-rich precipitates. In addition, a few Ge 3 Mn 5 clusters already start to form, as evidenced by their typical Moiré contrast (see inset of Fig. 2b). The same random distribution of nanoclusters is observed when increasing the nominal Mn content from 2% to 6% and 10% at T g =100 • C, but for an increase of the average precipitate density and length along the growth direction. This suggests that the decomposition is of 3D character and mostly driven by nucleation.

Secondary Ion Mass Spectroscopy (SIMS, not shown) performed on the As-(Ge,Mn) samples evidences an Asrich topmost layer, extending over 3±1 nm below the sample surface and containing up to 6×10 19 As cm -3 , i.e., an integrated amount of almost 1 ML As. This is a consequence of the segregation of As atoms, initially present at the GaAs surface, during the growth of Ge, with a well-known surfactant effect [14]. Accordingly, as described below, the As-(Ge,Mn) films appear as ndoped: As atoms are shallow donors in Ge, and in this topmost layer they compensate p-type doping by substitutional Mn.

The presence of As near the surface of the growing layer offers a possible explanation for this change of character of the spinodal decomposition, from 2D to 3D. According to Ref. [15], Mn atoms are incorporated into germanium in a subsurface interstitial position, and further diffuse within the growth plane: this offers a mechanism for 2D spinodal decomposition [5]. Codoping with As changes the charge state of Mn atoms, thus reducing Coulomb repulsion and enhancing the effect of attractive Mn-Mn pair interaction, making the nucleation of Mn-rich precipitates easier [4]. In addition to that mechanism, the presence of donors like As is expected to displace the equilibrium between interstitial Mn (another donor) and substitutional Mn (an acceptor), enhancing the amount of substitutional Mn (which form nucleation centers for further Mn aggregation [16]), and reducing the amount of interstitial Mn (thus decreasing the incorporation into already existing clusters). These different mechanisms induced by the presence of As conspire to favor a growth process dominated by nucleation, contributing to make the spinodal decomposition 3D.

A complete study of the magnetic properties will be published elsewhere [17]. All samples exhibit two [case of Ga-(Ge,Mn), Fig. 2a] or three [case of As-(Ge,Mn), Fig. 2b] magnetic phases, as evidenced from the temperature dependence of the saturation magnetization M s (inset of Fig. 2) and the remanent magnetization M r , and the ZFC-FC curves. They exhibit: (i) a strong paramagnetic signal with a 1/T temperature dependence at low temperature, attributed to Mn atoms diluted in the Ge matrix, and well fitted using a 3/2-Brillouin function [18]; (ii) a contribution attributed to the superparamagnetic Mn-rich nanocolumns or precipitates, with finite T C and blocking temperature T B ; (iii) in As-(Ge,Mn) only, a contribution from Ge 3 Mn 5 clusters with a broad range of blocking temperatures. A detailed analysis [17] of the Ga-(Ge,Mn) sample in Fig. 2a shows that 40±6% of the magnetic moments are in nanocolumns, with T C ≈ 150 K and T B = 15±5 K, 1.0±0.1 µ B /Mn, and an average magnetic moment of a nanocolumn 520±50 µ B . For the As-(Ge,Mn) sample in Fig. 2b, we found that 52±3% of the magnetic moments are in the matrix, and 22±2% in the Mn-rich precipitates with T C ≈ 50 K, T B =15±5 K, 1.2±0.2 µ B /Mn, and ≈100±20 µ B per precipitate. From the high field slope of the Hall effect, Ga-Ge 0.9 Mn 0.1 films are p-type, with a hole density up to 3×10 19 cm -3 at 300 K, comparable to what we gave as a lower bound (due to a strong AHE) of the hole density in Ge-(Ge,Mn). The resistivity (Fig. 3a) is of the insulating type (it was metallic in Ge-(Ge,Mn)). Mn was reported as a double acceptor in germanium, with acceptor levels 160 meV and 370 meV above the valence band edge [19] respectively. For such a deep acceptor, the Mott critical density is expected to be well in the 10 20 cm -3 range, and for a measured density one order of magnitude lower we should observe a strongly activated conductivity [20]: we actually observe (Fig. 3a) a weak temperature dependence. We conclude that the Hall effect is not due solely to the holes from the Mn acceptors in the germanium matrix. This is possible if the Fermi level of the Mn-rich material forming the precipitates lies below the top of the valence band in germanium, so that no Schottky barrier, and even an accumulation layer, is formed around each precipitate. That induces the build-up of an electric field pattern around each nanocolumn, which drags holes towards the nanocolumn.

Then the magnetotransport properties can be understood as follows: (i) as the nanocolumns configuration is well below the percolation threshhold, holes have to propagate through the germanium matrix; that makes the basis of the conductivity; (ii) however, the electric field pattern drags the holes through the nanocolumns, where the conductivity is higher; applying a magnetic field suppresses this effect, creating the geometrically enhanced orbital MR, or Extraordinary MR (EMR) [21] which we observed to be strong in Ge-(Ge,Mn) [6]; (iii) finally, the absence of Schottky barrier enhances the interaction of holes with Mn atoms in the nanocolumns, thus allowing a spin polarization and a strong AHE to appear [6].

In the Ga-(Ge,Mn) samples, we still observe the EMR (Fig. 3b), with the same temperature dependence as in Ref. [6], but much weaker (although much higher than classical Lorentz MR). This is readily explained by considering the dependence of EMR on the carrier mobility (it scales as µ 2 ): as seen in Fig. 3a, the zero-field resistivity ρ 0 in Fig. 3a is of the order of 10 -2 Ω cm, so that the mobility is lower than in Ge-(Ge,Mn) by more than one order of magnitude, possibly due to the higher disorder and induced defects. Finally, at very low temperature, negative MR is observed, which may be due to spin disorder scattering [22] on Mn atoms diluted in the Ge matrix (as typical in metallic DMS), or to GMR on the Mn-rich nanocolumns. In both cases, this effect is expected to be very weak; in addition the spin diffusion length of holes is very short due to spin-orbit coupling, making GMR on Mn-rich nanocolumns unlikely.

AHE in Ga-(Ge,Mn) nicely matches the magnetization of the nanocolumns (Fig. 3c-d). Again, the effect is weaker than in Ge-(Ge,Mn), by almost two orders of magnitude. This is expected from the lower mobility: it was pointed out in Ref. [23] that scattering on impurities such as Ga atoms (SIMS measurements have indeed shown that Ga out-diffused from the GaAs substrate) partly suppresses the effect of skew scattering.

As-(Ge,Mn) films exhibit metallic n-type conductivity. This is clearly due to the presence of As donors in the topmost layer. Hence magnetotransport essentially measures the properties of this 3 nm thick layer. We observe no AHE, as expected since spin-orbit scattering is small for electrons in the conduction band of germanium, and also because the same assumptions as above (Fermi level of precipitates lying below the top of the valence band) creates a high Schottky barrier for electrons.

MR is highly anisotropic (Fig. 4a): we show now that this is due to 2D weak localization in the As-doped layer, which vanishes when the field is applied in-plane (θ = 0). The isotropic part (Fig. 4c-d) will be analyzed later on.

The MR of non-interacting electrons in the 2D weak localization regime is ∆ρ/ρ ≈ -∆σ/σ = -A.f 2 [4eµ 0 Hsin(θ)L φ / ] [START_REF] Altshuler | Electron -Electron Interactions in Disordered Systems[END_REF], where A = e 2 /2π 2 σ 2D (0), L φ is the phase relaxation length, and the function f 2 (x) is defined in [START_REF] Altshuler | Electron -Electron Interactions in Disordered Systems[END_REF]. Here we neglect the effect of spin-orbit coupling and the anisotropy of L φ [17]. Fits in Fig. 4a were obtained with only two adjustable parameters: A = 4%, close to the value A = (6 ± 2)% calculated using the experimental value of the 2D conductivity at H = 0, and L φ = 11.5 nm, large enough with respect to the thickness of the conducting layer to justify the use of the 2D regime of weak localization. Moreover, fits of the temperature dependence of the anisotropic MR (Fig. 4b) were obtained by simply writing L φ = Dτ φ , using the temperature dependence of the diffusion coefficient D for an n-doped degenerate semiconductor [START_REF] Sze | Physics of Semiconductors Devices[END_REF], and a temperature dependence of the phase relaxation time τ φ ∝ T -α with α ≈1.7, similar to that obtained in [START_REF] Polyanskaya | [END_REF] for Ge:Sb (α=1.5) and in [27] for Si-MOSFETs (α=1.6), and currently attributed to both electron-electron and electron-phonon collisions.

Turning back to the isotropic MR, Fig. 4c-d, it contains a negative contribution, which features two maxima at the coercive field of Mn-rich precipitates, and vanishes above 50 K as does their magnetization. Hence, we tentatively ascribe it to tunneling MR (TMR) through the precipitates and the Schottky barriers formed around them. By analogy with spin injection from a ferromagnetic metal to a semiconductor [28], efficient spin injection from the precipitate to the matrix requires an interface resistance provided by the Schottky barrier. This barrier must be high enough to prevent full spin relaxation inside the precipitate but reasonably transparent to allow tunnel MR to occur.

To summarize, we have shown that surface morphol-ogy and co-doping have major influence on spinodal decomposition in (Ge,Mn) films grown on GaAs(001) substrates. For films grown on Ga-rich rough surfaces, we recovered 2D spinodal decomposition with bent nanocolumns. Electrical properties are similar to what we obtained on Ge(001) substrates except that the presence of defects in the films leads to weaker positive MR and AHE. For films grown on As-rich flat surfaces, 3D spinodal decomposition is observed due to As co-doping and magnetotransport is dominated by TMR and weak localization, AHE is negligible. These results are consistent with the assumption that the Fermi level of Mn-rich precipitates lies in the valence band of the Ge matrix.

FIG. 1 :

 1 FIG. 1: TEM cross section of 80 nm thick layers : (a) Ge0.94Mn0.06 grown at 100 • C after depositing a 40 nm thick Ge buffer on GaAs(001) and (b) at 100 • C on the facetted Ge surface. (c) Ga-Ge0.9Mn0.1 and (d) As-Ge0.98Mn0.02 films grown at 100 • C.

FIG. 2 :

 2 FIG. 2: Temperature dependence of magnetization in (a) Ga-Ge0.9Mn0.1 and (b) As-Ge0.94Mn0.06: ZFC-FC curves at 0.015 T and magnetic remanence Mr after maximum field cooling at 5 T. Inset: in (a), saturation magnetization at 2 T, and in (b), TEM image of a Ge3Mn5 cluster.

FIG. 3 :

 3 FIG. 3: Magnetotransport in Ga-Ge0.9Mn0.1, magnetic field applied normal to the plane: temperature dependence of the zero-field resistivity (a), MR up to 8 T at different temperatures (b), and AHE (red) compared to magnetization (blue) at 2-3 K (c) and 4-5 K (d). Inset: AHE at 10, 100 and 150 K.

FIG. 4 :

 4 FIG. 4: Magnetotransport in As-Ge0.94Mn0.06. (a) field dependance and (b) temperature dependence of the MR anisotropy, for different orientations of the field; symbols are experimental data, solid line calculated. (c,d) MR and magnetization as a function of the field applied in-plane.

We thank A. Arnoult (LAAS, Toulouse) for providing As-capped GaAs substrates, T. Dietl and J. Pernot for fruitful discussions. This work was granted by the Agence Nationale pour la Recherche (project GeMO) and the Nanoscience Foundation in Grenoble (project IMAGE).