Aurélien Vasseur

Complements on Sibuya and discrete stable distributions

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Complements on Sibuya and discrete stable distributions Aurélien Vasseur

In [START_REF] Kozubowski | A generalized sibuya distribution[END_REF], Kozubowski and Podgorski state that a random variable X on N has a Sibuya probability distribution if, and only if, for any t ∈ (0, 1],

t • X |t•X =0 D = X, (1)
where • and D = respectively denote the usual thinning operation for discrete random variables and the equality between probability distributions.

In this document, we take advantage of the characterization (1) in order to:

• dene and explore levels of attractiveness for discrete random variables,

• dene and characterize Cox-Sibuya point processes, a consistent adaptation of Sibuya distributions to point processes,

• link these new objects and their properties to discrete stability theory.

For more exhaustive reviews on the present notions, the reader may refer to the works of Steutel and Van Harn [START_REF] Steutel | Discrete Analogues of Self-Decomposability and Stability[END_REF] about discrete stability for random variables, the respective works of Sibuya [START_REF] Sibuya | Generalized hypergeometric, digamma and trigamma distributions[END_REF], Christoph and Schreiber [START_REF] Christoph | Discrete Stable Random Variables[END_REF][START_REF] Christoph | Scaled Sibuya distribution and discrete self-decomposability[END_REF], Devroye [START_REF] Devroye | Random variate generation for the digamma and trigamma distributions[END_REF][START_REF] Devroye | A triptych of discrete distributions related to the stable law[END_REF] and more recently Kozubowski and Podgorski [START_REF] Kozubowski | A generalized sibuya distribution[END_REF] about discrete Sibuya distributions, and the works of Davydov, Molchanov and Zuyev [START_REF] Davydov | Stability for random measures, point processes and discrete semigroups[END_REF] about stability for random measures, in particular point processes. [START_REF] Christoph | Discrete Stable Random Variables[END_REF] Sibuya and discrete stable random variables

Sibuya random variables

Introduced in 1979 by Sibuya [START_REF] Sibuya | Generalized hypergeometric, digamma and trigamma distributions[END_REF], the discrete Sibuya distribution may be dened as follows: a discrete random variable Y on N has a Sibuya distribution with exponent α, if its probability generating function is given, for any s ∈ [0, 1], by

g Y (s) = E[s Y ] = 1 -(1 -s) α ,
that we denote Y ∼ Sib(α).

Such a distribution only exists for α ∈ (0, 1], and a random variable Y ∼ Sib(1) equals 1 almost surely. This set of probability distributions may be extended as follows: for t ∈ [0, 1], a random variable Y on N 0 has a t-scaled Sibuya distribution with exponent α if its probability generating function is given, for any s ∈ [0, 1], by

g Y (s) = E[s Y ] = 1 -t(1 -s) α .
In particular, a 1-scaled Sibuya distribution is a Sibuya distribution with the same exponent α. Furthermore, a random variable with a t-scaled Sibuya distribution with exponent α may be built as a t 1 α -thinning of a random variable with Sibuya distribution. We recall that, for any t ∈ [0, 1], the t-thinning of a discrete random variable X on N is denoted by t • X and, conditionnally to X, has a binomial distribution with parameters X and t. The respective probability generating functions of X and t • X are linked by the following identity: for any

s ∈ [0, 1], g t•X (s) = E[s t•X ] = g X (1 -t(1 -s)).
It is also known that if a random variable Y has a t-scaled Sibuya distribution, then

P(Y = 0) = 1 -t, P(Y = 1) = αt and, for any n ∈ N such that n ≥ 2, P(Y = n) = (1 -α) 1 - α 2 . . . 1 - α n -1 α n t
from which we deduce that, if t ∈ (0, 1] and α ∈ (0, 1), then, for any n ∈ N,

P(Y = n + 1) P(Y = n) = n -α n + 1 .
If α = 1, then Y has a Bernoulli distribution with parameter t and, in particular, if Y has a Sibuya distribution with exponent 1, then Y = 1 almost surely, which corresponds to the only Sibuya distribution with a nite expectation.

If α ∈ (0, 1), it is stated in [START_REF] Kozubowski | A generalized sibuya distribution[END_REF] that a random variable X ∼ Sib(α) has moments of order γ only for γ ∈ (0, α).

A characterization of the Sibuya distribution with exponent α, etablished in [START_REF] Devroye | A triptych of discrete distributions related to the stable law[END_REF] and given by the next theorem, provides an explicit way to simulate some realizations of a random variable with Sibuya distribution. Theorem 1.1. Let α ∈ (0, 1). Let E, G, H be independent random variables such that E has an exponential distribution and G, H have gamma distributions with respective parameters α and 1 -α. Let P be a random variable such that, conditionally to E, G, H, P has a Poisson distribution with parameter EH G . Then, 1 + P has a Sibuya distribution with exponent α.

Concluding this general framework on Sibuya distribution, let focus on two results providing ways to identify a link between Sibuya distributions with different exponents α. The rst result, from [START_REF] Christoph | Scaled Sibuya distribution and discrete self-decomposability[END_REF], is given in the following theorem. Theorem 1.2. Let (X (n) ) n∈N be a sequence of i.i.d. Sibuya random variables with exponent α and Y be a Sibuya random variable with exponent β and independent of (X (n) ) n∈N . Then, the random variable Z dened by Z = Y n=1 X (n) is a Sibuya random variable with exponent αβ.

The second result, from [START_REF] Kozubowski | A generalized sibuya distribution[END_REF], provides a characterization of the family of probability distributions (Sib(α)) α∈(0,1] . Theorem 1.3. Let X be a random variable of N such that P(X = 0) > 0.

Then,

∀t ∈ (0, 1], t • X |t•X =0 D = X ⇐⇒ ∃!α ∈ (0, 1] : X ∼ Sib(α)
where D = denotes the equality between probability distributions. Proof. The proof of the theorem is based on the two following lemmas, which are respectively deduced from standard operations on probability generating functions and a classical algebraic result on real functions. Lemma 1.4. Let X be a random variable of N such that P(X = 0) > 0. Then, for any s ∈ [0, 1] and t ∈ (0, 1],

g t•X |t•X =0 (s) = g X (1 -t(1 -s)) -g X (1 -t) 1 -g X (1 -t) = 1 - 1 -g X (1 -t(1 -s)) 1 -g X (1 -t) .
Lemma 1.5. Let f : (0, 1] → (0, +∞) be a continuous function such that, for

any x 1 , x 2 ∈ (0, 1], f (x 1 x 2 ) = f (x 1 )f (x 2 ).
Then, there exists a unique α ∈ R such that, for any x ∈ (0, 1],

f (x) = x α .
(⇐) Since X ∼ Sib(α), we have, for any s ∈ [0, 1],

g X (s) = 1 -(1 -s) α .
By Lemma 1.4, it follows that, for any s ∈ [0, 1] and t ∈ (0, 1],

g t•X |t•X =0 (s) = 1 - 1 -g X (1 -t(1 -s)) 1 -g X (1 -t) = 1 - 1 -(1 -t α (1 -s) α ) 1 -(1 -t α ) = 1 -(1 -s) α = g X (s).
(⇒) For any s ∈ [0, 1] and t ∈ (0, 1],

g t•X |t•X =0 (s) = g X (s) ⇐⇒ g X (1 -t(1 -s)) -g X (1 -t) 1 -g X (1 -t) = g X (s) ⇐⇒ g X (1 -t(1 -s)) = g X (s) + g X (1 -t)(1 -g X (s)) ⇐⇒ 1 -g X (1 -t(1 -s)) = (1 -g X (1 -t))(1 -g X (s)) ⇐⇒ f X (t(1 -s)) = f X (t)f X (1 -s),
where, for any x ∈ [0, 1],

f X (x) = 1 -g X (1 -x).
Since g X is a probability generating function, then g X is continuous and so do f X . Moreover, applying the left hypothesis of the theorem for t = 1, one gets that P(X = 0) = 0, and then one has, for any x ∈ [0, 1),

g X (1 -x) = E[(1 -x) X ] < 1, then, for any x ∈ (0, 1], f X (x) > 0.
Then, by Lemma 1.5, there exists a unique α ∈ R such that, for any x ∈ (0, 1],

f X (x) = x α .
Then, since g X (1) = 1, for any s ∈ [0, 1],

g X (s) = 1 -(1 -s) α .
Since g X is a nonnegative and not identically null functional, it follows that α > 0. If α > 1, then, for any s ∈ (0, 1], g X (s) > s, which is absurd since from the left hypothesis almost surely X ≥ 1. Then α ∈ (0, 1], and then X ∼ Sib(α).

Levels of attractiveness for discrete random variables

The characterization of the Sibuya distribution provided by Theorem 1.3 leads to the following asymptotic result, according to the convergence in distribution, denoted by D -→ and which coincides with the convergence of the probability generating functions.

Theorem 1.6. Let X be a random variable of N 0 such that P(X = 0) > 0 and

Y be a random variable of N. If t • X |t•X =0 D ----→ t→0+ Y, then, ∃!α ∈ (0, 1] : Y ∼ Sib(α).
Proof. For any s ∈ [0, 1] and any t ∈ (0, 1], by Lemma 1.4,

g t•Y |t•Y =0 = g Y (1 -t(1 -s)) -g Y (1 -t) 1 -g Y (1 -t) .
By assumption,

g Y (s) = lim t →0+ g t •X |t •X =0 = lim t →0+ g X (1 -t (1 -s)) -g X (1 -t ) 1 -g X (1 -t ) .
Then,

g Y (1 -t) = lim t →0+ g X (1 -t t) -g X (1 -t ) 1 -g X (1 -t ) and g Y (1 -t(1 -s)) = lim t →0+ g X (1 -t t(1 -s)) -g X (1 -t ) 1 -g X (1 -t ) .
It follows that

g t•Y |t•Y =0 = = lim t →0+ g X (1 -t t(1 -s)) -g X (1 -t ) -g X (1 -t t) -g X (1 -t ) 1 -g X (1 -t ) -g X (1 -t t) -g X (1 -t ) = lim t →0+ g X (1 -t t(1 -s)) -g X (1 -t t) 1 -g X (1 -t t) = lim t→0+ g X (1 -t(1 -s)) -g X (1 -t) 1 -g X (1 -t) = lim t→0+ g t•X |t•X =0 (s) = g Y (s),
which, by Theorem 1.3, concludes the proof. This last result allows to associate to a discrete random variable satisfying assumptions of Theorem 1.6 the corresponding exponent α of the asymptotic Sibuya distribution, from which we can in a natural way introduce the following denition of weak attractiveness. Denition 1.7. A discrete random variable X is said to have a level α of weak

attractiveness if t • X |t•X =0 D ----→ t→0+ Sib(α).
For any α ∈ (0, 1], A * (α) denotes the set of discrete random variables with a level α of weak attractiveness.

This notion of weak attractiveness may be linked to the following alternative denition of attractiveness, whom we show it is stronger than the previous one. Denition 1.8. A discrete random variable X is said to have a level α of (strong) attractiveness if there exists a > 0 such that

lim t→0+ 1 -g X (1 -t) t α = a. (2)
For any α ∈ (0, 1] and a > 0, A(α) denotes the set of discrete random variables with a level α of attractiveness and A(α, a) denotes the set of discrete random variables with a level α of attractiveness and a verifying the equation (2). In particular, for any α ∈ (0, 1],

A(α) = a>0 A(α, a). Theorem 1.9. Let α ∈ (0, 1]. If X ∈ A(α), then X ∈ A * (α).
Proof. For any s ∈ [0, 1] and t ∈ (0, 1], by Lemma 1.4,

g t•X |t•X =0 (s) = 1 - 1 -g X (1 -t(1 -s)) 1 -g X (1 -t) = 1 - 1 -g X (1 -t(1 -s)) t α (1 -s) α (1 -s) α t α 1 -g X (1 -t) .
Applying Denition 1.8, since there exists a > 0 such that

lim t→0+ 1 -g X (1 -t) t α = a, it follows that lim t→0+ g t•X |t•X =0 (s) = 1 -(1 -s) α = g Y (s),
where Y ∼ Sib(α), from which we can conclude.

Remark 1.10. The converse of this theorem is not true : consider for example a discrete random variable X whose the probability distribution is given for any n ∈ N such that n ≥ 2 by

P(X = n) = 1 n(n -1)
.

Its probability generating function g X is given for any s ∈ [0, 1] by

g X (s) = +∞ k=2 s k k(k -1)
, which is the double primitive function such that g X (0) = 0 and lim s→1 g X (s) = 1 of the functional given for any s ∈ [0, 1] by

+∞ k=0 s k = 1 1 -s .
We thus obtain that, for any s ∈ [0, 1],

g X (s) = (1 -s) log(1 -s) + s.
On one hand, by Lemma 1.4, for any s ∈ [0, 1] and t ∈ (0, 1],

g t•X |t•X =0 (s) = 1 - 1 -g X (1 -t(1 -s)) 1 -g X (1 -t) ,
and then

g t•X |t•X =0 (s) = 1 - 1 -[t(1 -s) log(t(1 -s)) + 1 -t(1 -s)] 1 -(t log t + 1 -t) = 1 + [t(1 -s) log(t(1 -s)) -t(1 -s)] t -t log t = 1 + [t(1 -s)[log(t(1 -s)) -1] t(1 -log t) = 1 -(1 -s) 1 -log(t(1 -s)) 1 -log t = 1 -(1 -s) 1 - log(1 -s) 1 -log t .
Then, for any s ∈ [0, 1], lim

t→0+ g t•X |t•X =0 (s) = s
and we deduce that t • X |t•X =0 tends to 1 as t tends to 0, which leads to

t • X |t•X =0 D ---→ t→0 Sib (1)
, from which we deduce that X ∈ A * (α).

On the other hand, for any t ∈ (0, 1],

1 -g X (1 -t) t = 1 -[1 -(t log t + 1 -t)] t = 1 -log(t), then lim t→0+ 1 -g X (1 -t) t = +∞,
and then X / ∈ A(1).

According to the previous statement, strong attractiveness is more restrictive than the weak one, but has also good stability properties, as stated in the following theorem.

Theorem 1.11. Let α, β ∈ (0, 1] and a, b ∈ (0, +∞). If X ∈ A(α, a) and

Y ∈ A(β, b) are independent, then X + Y ∈ A(α ∧ β, a + b1 {α=β} ) and X k=1 Y (k) ∈ A(αβ, ab α ) where (Y (k) ) k∈N is a sequence of independant copies of Y . Proof. Since X ∈ A(α, a) and Y ∈ A(β, b), lim t→0+ 1 -g X (1 -t) t α = a.
and

lim t→0+ 1 -g Y (1 -t) t β = b.
Let us show the rst assertion. We can suppose without loss of generality that α ≤ β. For any s ∈ [0, 1], by independence of X and Y ,

g X+Y (s) = g X (s)g Y (s).
Then, for any t ∈ (0, 1],

1 -g X+Y (1 -t) t α = 1 -g X (1 -t)g Y (1 -t) t α = 1 -g X (1 -t) t α + g X (1 -t)(1 -g Y (1 -t)) t α .
Since X ∈ A(α, a), the rst term converges to a as t tends to 0. Since

Y ∈ A(β, b) and α ≤ β, then lim t→0+ 1 -g Y (1 -t) t α = b1 {α=β} and, since g X is continuous in 1, then lim t→0+ g X (1 -t) = g X (1) = 1.
Thus,

lim t→0+ 1 -g X+Y (1 -t) t α = a + b1 {α=β} > 0,
and then X + Y ∈ A(α, a + b1 {α=β} ), which concludes the proof of the rst assertion.

Let us now show the second assertion. For any s ∈ [0, 1], it is known that

g X k=1 Y (k) (s) = g X (g Y (s)). It follows that 1 -g X k=1 Y (k) (1 -t) t αβ = 1 -g X (g Y (1 -t)) t αβ = 1 -g X (1 -(1 -g Y (1 -t))) (1 -g Y (1 -t)) α 1 -g Y (1 -t) t β α .
On one hand, since g Y is continuous in 1, we have

lim t→0+ 1 -g Y (1 -t) = 0,
and then, by hypothesis on X,

lim t→0+ 1 -g X (1 -(1 -g Y (1 -t))) (1 -g Y (1 -t)) α = a.
On the other hand, by hypothesis on Y ,

lim t→0+ 1 -g Y (1 -t) t β = b.
Thus, we deduce that

lim t→0+ 1 -g X k=1 Y (k) (1 -t) t αβ = ab α > 0,
from which we can conclude that X k=1 Y (k) ∈ A(αβ, ab α ). The following statement provide a characterization of the space A(1).

Theorem 1.12. Let X be a random variable in N 0 such that P(X = 0) > 0.

Then,

E[X] < +∞ ⇐⇒ X ∈ A(1) ⇐⇒ X ∈ A(1, E[X]).
Proof. It is known that X has a nite expectation if and only if there exists

a ≥ 0 such that lim t→0+ 1 -g X (1 -t) t = g X (1) = a,
and, in this case, a = E[X]. Moreover, since P(X = 0) > 0, E[X] = 0. Hence, the result.

The previous theorem implies that the sets A(α) with α ∈ (0, 1) contain only probability distributions with innite exceptions and then provide a way to classify explicitly such any distributions. It also shows that most of classical discrete probability distributions belong to the set A(1). For any xed α ∈ (0, 1], we can however always exhibit at least one explicit example of a discrete probability distribution with a level α of strong attractiveness.

Theorem 1.13. Let α ∈ (0, 1]. If X ∼ Sib(α), then, X ∈ A(α, 1). Proof. Since X ∼ Sib(α), for any t ∈ (0, 1], 1 -g X (1 -t) t α = 1 -(1 -(1 -(1 -t)) α ) t α , then 1 -g X (1 -t) t α = 1,
from which we deduce that X ∈ A(α, 1).

1.3

Stability for discrete random variables Discrete stable distributions are introduced in [START_REF] Steutel | Discrete Analogues of Self-Decomposability and Stability[END_REF], where is given a more exhaustive framework on these probability distributions. We only recall a few statements which will be useful in the following results. For α ∈ (0, 1], a random variable X on N 0 has a discrete stable distribution with exponent α if, for any t ∈ [0, 1],

t 1 α • X (1) + (1 -t) 1 α D = X,
where X (1) and X (2) are two independant copies of X. The probability generating function of such a X may then be written, for any s ∈ [0, 1], by:

g X (s) = e -a(1-s) α
where a is a nonnegative real number, and then the notation

X ∼ DSt(α, a)
identies the probability distribution of X. If α = 1, X has a Poisson distribution with parameter a. More generally, a discrete stable probability distribution with parameter a ∈ [0, +∞) and exponent α ∈ (0, 1] admits the following representation: if X ∼ DSt(α, a), then

X D = Z k=1 Y (k) , (3)
where Z is a Poisson random variable with parameter a and (Y (k) ) k∈N is a sequence of independent copies of a Sibuya random variable Y with exponent α.

Since the Poisson random variable Z may be seen as a particular case of a discrete stable random variable (with exponent 1), our following result provides an extension of the previous representation. Theorem 1.14. Let α, β, γ ∈ (0, 1] such that α = βγ, a ∈ [0, +∞) and independent random variables Z ∼ DSt(γ, a) and Y ∼ Sib(β). Then,

Z k=1 Y (k) ∼ DSt(α, a), where (Y (k) ) k∈N is a sequence of independant copies of Y . Proof. For any s ∈ [0, 1], denoting X = Z k=1 Y (k) , g X (s) = g Z (g Y (s)),
and the respective expressions of the probability generating functions provide

g X (s) = e -a(1-(1-(1-s) β )) γ = e -a(1-s) βγ ,
hence, the result.

The representation of discrete stable distributions given by (3) provides a way to connect discrete stability and attractiveness, as stated in the two followings results.

Theorem 1.15. Let α ∈ (0, 1] and a > 0. If X ∼ DSt(α, a), then

X ∈ A(α, a).
Proof. This result is a consequence of the fact that the distribution of X admits the characterization given by (3). It is then sucient to combine Theorem 1.13 with the second assertion of Theorem 1.11. Theorem 1.16. Let α ∈ (0, 1] and a > 0. If X ∈ A(α, a), then

1 n 1 α • n k=1 X (k) D -----→ n→+∞ DSt(α, a).
Proof. For any s ∈ [0, 1] and n ∈ N, combining standard operations on probability generating functions, one has

g 1 n 1 α • n k=1 X (k) (s) = g X 1 - 1 n 1 α (1 -s) n . It follows that g 1 n 1 α • n k=1 X (k) (s) = exp n log g X 1 - 1 n 1 α (1 -s) = exp n log h n (s) 1 n (1 -s) α + 1
where

h n (s) = g X 1 - 1 n 1 α (1 -s) -1 1 n (1 -s) α
, and then

g 1 n 1 α • n k=1 X (k) (s) = exp log h n (s) 1 n (1 -s) α + 1 h n (s)(1 -s) α 1 n h n (s)(1 -s) α . Since X ∈ A(α, a), for any s ∈ [0, 1], lim n→+∞ h n (s) = a, then lim n→+∞ h n (s)(1 -s) α n = 0,
and then, since

lim x→0+ log(1 + x) x = 1, it follows that lim n→+∞ log h n (s) 1 n (1 -s) α + 1 h n (s)(1 -s) α 1 n = 1,
from which we can deduce that, for any s ∈ [0, 1], The following notions concern point process theory and come essentially from [START_REF] Daley | An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods[END_REF]. We consider a locally compact metric space X endowed with its Borel tribe X , a (not necessarily diuse) Radon measure on X. The family of relatively compact Borel sets is denoted by X 0 . The space of measures on X will be denoted M, the space of probability measures on X denoted M 1 , the space of sub-probability measures on X denoted M * 1 , the space of probability measures on M denoted M 1 , and B + (X) denotes the set of bounded measurable functions from X to [0, +∞) with compact support. If µ ∈ M and f is a function on X integrable with respect to µ, then f, µ designs the integral of f with respect to µ.

lim n→+∞ g 1 n 1 α • n k=1 X (k) (s) = e -a(
A counting measure ξ on X is a measure on X such that, for any A ∈ X 0 , ξ(A) ∈ N 0 . A conguration (respectively nite conguration) on X is a locally nite (respectively nite) counting measure on X.

A point process Φ on X is a random conguration on X. Its intensity measure is the measure M on (X, X ) dened, for any A ∈ X , by

M (A) = E[Φ(A)]. Its probability generating functional G Φ is dened on the space T = {u : X → (0, 1] : 1 -u ∈ B + (X)} ∪ {0} X
and given for any function u ∈ T by:

G Φ (u) = E exp ˆX log u(x)Φ(dx) .
If β is a function from X to [0, 1], the point process on X denoted β • Φ and built by keeping with probability β(x) and deleting with probability 1 -β(x) each point x of Φ independently is called the β-thinning of Φ. Its probability generating functional is given for any u ∈ T by:

G β•Φ (u) = G Φ (1 -β(1 -u)).
Let M be a Radon measure on X. The Poisson point process Φ with intensity measure M is dened as the unique point process on X with intensity measure M such that, for any disjoint relatively compact subsets Λ 1 , Λ 2 , the random variables Φ(Λ 1 ) and Φ(Λ 2 ) are independent.

A Poisson point process on X with nite intensity measure M may be dened as a nite point process Φ on X such that its total number of points N has a Poisson distribution with parameter M (X) and, conditionally to N , Φ is a binomial point process on X with parameter N and supported by M (•) M (X) . Let Φ be a Poisson point process on X with intensity measure M . Then, its probability generating functional is given for any u ∈ T by:

G Φ (u) = exp -ˆX(1 -u(x))M (dx)
from which we can easily deduce that, for any t ∈ [0, 1],

t • Φ (1) + (1 -t) • Φ (2) D = Φ,
where Φ (1) and Φ (2) are independent copies of the point process Φ.

Stability for point processes

It is shown in [START_REF] Davydov | Stability for random measures, point processes and discrete semigroups[END_REF] that Poisson point processes may be included in a larger class of point processes, which is described as follows. A point process Φ is said to be discrete α-stable if, for any t ∈ [0, 1] and any independent copies Φ (1) and Φ (2) of the point process Φ,

t 1 α • Φ (1) + (1 -t) 1 α • Φ (2) D = Φ,
which it is equivalent to say that its probability generating functional G Φ veries, for any u ∈ T ,

G Φ (1 -t 1 α (1 -u))G Φ (1 -(1 -t) 1 α (1 -u)) = G Φ (u). (4)
More explicitly, a nite point process Φ is discrete α-stable if and only if its probability generating functional is given for any function u ∈ T by:

G Φ (u) = exp - ˆM1 1 -u, µ α σ(dµ) , (5)
for some locally nite spectral measure σ on M 1 such that, for any B ∈ X 0 , ˆM1 µ(B) α σ(dµ) < +∞.

Since the probability distribution of such a point process Φ is then only depending on the exponent α and the spectral measure σ, we may from now denote Φ ∼ DSt(α, σ).

Spectral measures of sums and thinnings of discrete stable point processes, are given explicitly as follows: if

Φ (1) , . . . , Φ (n) are independent copies of Φ ∼ DSt(α, σ) and t ∈ [0, 1], then n k=1 Φ (k) ∼ DSt(α, nσ), t • Φ ∼ DSt(α, t α σ), 1 n 1 α • n k=1 Φ (k) D = Φ.
The representation provided by (3) for discrete stable random variables may be adapted for discrete stable point processes, but we need to give rst the denition of a Sibuya point process, introduced in [START_REF] Davydov | Stability for random measures, point processes and discrete semigroups[END_REF] for this purpose. Denition 2.1. Let µ be a probability distribution on X. A point process Υ on X is called a Sibuya point process with exponent α and parameter measure µ, if its probability generating functional is given for any function u ∈ T by:

G Υ (u) = 1 -1 -u, µ α , denoted Υ ∼ Sib(α, µ).
In order to get a realization of a Sibuya point process, the number Y of points is drawn according to a Sibuya distribution with exponent α, and the Y points are placed independently according to the parameter measure µ on X. In particular, a Sibuya point process on with exponent 1 and parameter measure µ ∈ M 1 has exactly one point, which is placed on X according to µ.

Let us note that this denition may be a bit extended. Indeed, if t ∈ [0, 1] and Υ ∼ Sib(α, µ), then the t-thinning of Υ has a probability generating functional dened for any function u ∈ T by:

G t•Υ (u) = 1 -1 -u, tµ α ,
and then we can enlarge the previous denition for the case where the parameter measure µ is such that µ(X) ≤ 1 (denoted µ ∈ M * 1 ).

Denition 2.2. Let µ ∈ M * 1 . A point process Υ on X is called the Sibuya point process with exponent α and parameter measure µ (also denoted Υ ∼ Sib(α, µ)) if its probability generating functional is given for any function u ∈ T by:

G Υ (u) = 1 -1 -u, µ α .
This extension of the denition of a Sibuya point process to sub-probability measures allows to state the two following results, using probability generating functional. On one hand, if Υ µ is a Sibuya point process with exponent α and parameter measure µ, then t • Υ µ is a Sibuya point process with exponent α and parameter measure tµ. One the other hand, if Z is a random variable with Bernoulli distribution with parameter t α , then the point process Φ t dened by Φ t = 1 {Z=1} Υ µ is a Sibuya point process with exponent α and parameter measure tµ.

An analog of Theorem 1.2 for Sibuya point processes is now stated in which follows.

Theorem 2.3. Let (Υ (n) ) n∈N be a sequence of i.i.d. Sibuya random variables with exponent α and parameter measure µ ∈ M * 1 , and Y be a Sibuya random variable with exponent β and independent of (Υ (n) ) n∈N . Then, the point process Ψ dened by

Ψ = Y n=1 Υ (n)
is a Sibuya point process with exponent αβ and parameter measure µ.

Proof. For any function

u ∈ T , G Ψ (u) = E E x∈ Y n=1 Υ (n) u(x) Y = E[(1 -1 -u, µ α ) Y ]. Then, since E[s Y ] = 1 -(1 -s) β , G Ψ (u) = 1 -(1 -(1 -1 -u, µ α )) β = 1 -1 -u, µ αβ .
This proof is thus complete.

We may now recall the cluster representation of a nite discrete stable point process. This result is stated in [START_REF] Davydov | Stability for random measures, point processes and discrete semigroups[END_REF] also for point processes which are not necessarily nite, but our investigations only focus on the nite case.

Theorem 2.4. A nite point process Φ ∼ DSt(α, σ) can be represented as a cluster process with a Poisson center process on M 1 driven by intensity measure σ and daughter processes being Sibuya point processes Υ µ with exponent α and parameter measure µ ∈ M 1 . Put another way, the point process Φ may be represented as:

Φ = µ∈ζ Υ µ ,
where ζ is a Poisson point process on M 1 with intensity measure σ. Its probability generating functional is given for any function u ∈ T by:

G Φ (u) = exp ˆM1 (G Υµ (u) -1)σ(dµ) .
Noting that the Poisson point process ζ in the previous theorem is a particular case of a discrete stable point process (with exponent 1), we provide both a generalization of the previous result and an analog of Theorem 1.14 for point processes in the following theorem. Theorem 2.5. Let α, β, γ ∈ (0, 1] such that α = βγ. Let σ be a locally nite measure on M 1 such that, for any B ∈ X 0 , ˆM1 µ(B) α σ(dµ) < +∞.

Let ι : µ ∈ M 1 → δ µ ∈ M 1 and Φ be a discrete stable point process on M 1 with exponent γ and spectral measure σ = σ • ι -1 . Then, the point process

Φ = µ∈Φ Υ β
µ is a discrete stable point process on X with exponent α and spectral measure σ.

Proof. For any function

u :∈ T , since Φ = µ∈Φ Υ β µ , G Φ (u) = G Φ (G Υ β µ (u))
, and the respective expressions of the probability generating functionals provide

G Φ (u) = exp - ˆM 1 1 -u, µ β , µ γ σ (dµ ) . Then, since σ = σ • ι -1 , G Φ (u) = exp - ˆM1 1 -u, µ β , δ ν γ σ(dν)
and nally

G Φ (u) = exp - ˆM1 1 -u, ν βγ σ(dν) = exp - ˆM1 1 -u, ν α σ(dν) ,
hence, the result.

Cox-Sibuya point processes

The denition of Sibuya point processes (Denition 2.1) proposed by Davydov et al. [START_REF] Davydov | Stability for random measures, point processes and discrete semigroups[END_REF] appears for several reasons as the most natural way to extend Sibuya probability distributions from discrete random variables to point processes: rstly, the similarity between the expressions of their respective probability generating functions; secondly, the fact that a Sibuya point process is the simplest point process such that the number of points has a Sibuya distribution (these points are then independent and identically distributed); thirdly, the compatibility with the extension of the compound Poisson representation of a discrete stable distribution given by (3) to the Poisson cluster representation of a discrete stable point process (Theorem 2.3). An alternative method is to consider the characterization of a Sibuya distribution given by Theorem 1.3 and relative to thinning, which is easily transposed from discrete random variables to point processes, as follows.

Denition 2.6. Let Φ be a point process on X such that P(Φ = 0) > 0. We say that Φ is a Cox-Sibuya point process if, for any t ∈ (0, 1],

t • Φ |t•Φ =0 = Φ.
In order to identify more explicitly such any point processes, we state rst the two following lemmas. Lemma 2.7. A Cox-Sibuya point process Φ on X is almost surely nite.

Proof. Supposing that P(Φ(X) = +∞) > 0, we can dene the point process Ψ as Ψ = Φ |Φ(X)=+∞ .

If a conguration φ of X is such that φ(X) = +∞, then, for any t ∈ (0, 1],

P(t • φ = 0) = 0.
It follows that, almost surely,

(t • Φ |t•Φ =0 ) |Φ(X)=+∞ = t • Φ |t•Φ =0,Φ(X)=+∞ = t • Φ |Φ(X)=+∞ ,
and we deduce that, almost surely,

t • Φ |Φ(X)=+∞ = Φ |Φ(X)=+∞ = Ψ.
Moreover, one hand hand, for any u ∈ T and t ∈ (0, 1], On the other hand,

G t•Ψ (u) = G Ψ (1 -t(1 -u)) = E x∈Φ (1 -t(1 -u))(x)
G t•Ψ (u) = G Ψ (1 -t(1 -u)) ----→ t→0+ 1,
then, Ψ = 0 almost surely, which is absurd (by construction of Ψ) and concludes the proof. Lemma 2.8. Let Φ be a point process on X such that P(Φ = 0) > 0. Then, for any u ∈ T and t ∈ (0, 1],

G t•Φ |t•Φ =0 (u) = G Φ (1 -t(1 -u)) -G Φ (1 -t) 1 -G Φ (1 -t) = 1 - 1 -G Φ (1 -t(1 -u)) 1 -GΦ(1 -u) .
where F Φ (•) = 1 -G Φ (1 -•).

Let now (Φ (k) ) k∈N be a sequence of independent copies of Φ. Let Z be a Poisson random variable with parameter 1. We dene the point process Ψ as

Ψ = Z k=1 Φ (k) .
Let show that Ψ is a discrete α-stable, that is to show, for any u ∈ T and t ∈ [0, 1], that

G Ψ (1 -t 1 α (1 -u))G Ψ (1 -(1 -t) 1 α (1 -u)) = G Ψ (u).
Noting that, for any u ∈ T ,

G Ψ (u) = g Z (G Φ (u)) = exp{-(1 -G Φ (u))},
it follows that, for any u ∈ T and t ∈ [0, 1],

G Ψ (1 -t 1 α (1 -u))G Ψ (1 -(1 -t) 1 α (1 -u)) = = exp{-(1 -G Φ (1 -t 1 α (1 -u)))} exp{-(1 -G Φ (1 -(1 -t) 1 α (1 -u)))} = exp{-F Φ (t 1 α (1 -u))} exp{-F Φ ((1 -t) 1 α (1 -u))}.
Hence, according to [START_REF] Devroye | Random variate generation for the digamma and trigamma distributions[END_REF], Moreover, since G Φ (0) = P(Φ = 0) = 0, it follows that σ 1 (M 1 ) = 1. The random probability measure M on M 1 is then chosen such that its probability distribution is σ 1 , from which we can conclude.

G Ψ (1 -t 1 α (1 -u))G Ψ (1 -(1 -t)

1 {Φ

 1 )1 {Φ(X)=+∞} P(Φ(X) = +∞) = G t•Φ |Φ(X)=+∞ (u) = G Ψ (u).

1 α ( 1

 11 -u)) = = exp{-tF Φ ((1 -u))} exp{-(1 -t)F Φ ((1 -u))} = exp{-F Φ (1 -u)} = exp{-(1 -G Φ (u))} = G Ψ (u),then, according to the characterization provided by (4), Ψ is a discrete α-stable point process. Thus, since Ψ is almost surely nite (as a consequence of Lemma 2.7), by equality (5), there exists a measure σ 1 on M 1 such that, for any u ∈ T ,G Ψ (u) = exp{-ˆM1 1 -u, µ α σ 1 (dµ)}, then exp{-(1 -G Φ (u))} = exp{-ˆM1 1 -u, µ α σ 1 (dµ)},and thenG Φ (u) = 1 -ˆM1 1 -u, µ α σ 1 (dµ).

Proof. For any u ∈ T and t ∈ (0, 1], by combining basic operations on probability generating functionals,

We can now state the following characterization of a Cox-Sibuya point process.

Theorem 2.9. Let Φ be a point process on X such that P(Φ = 0) > 0. Then, Φ is a Cox-Sibuya point process if, and only if, there exist α ∈ (0, 1] and a random element M of M 1 with probability law σ 1 such that conditionally to M ,

In this case, its probability generating functional is given, for any u ∈ T , by

Proof. (⇐) Since, conditionally to M , Φ ∼ Sib(α, M ), we have, for any u ∈ T ,

By Lemma 2.8, it follows that, for any u ∈ T and t ∈ (0, 1],

.

from which we can conclude.

(⇒) Let Φ be a Cox-Sibuya point process on X. Since, by Lemma 2.7, Φ is almost surely nite, it makes sense to determine the probability distribution of Φ(X). Firstly, P(Φ = 0) = 0, then P(Φ(X) = 0) > 0. By Lemma 1.4, for any s ∈ [0, 1] and t ∈ (0, 1],

We deduce, by Lemma 2.8, that

and then, since Φ is a Cox-Sibuya point process,

Hence, Φ(X) is a Cox-Sibuya discrete random variable, and then, by Theorem 1.3, there exists α ∈ (0, 1] such that Φ(X) ∼ Sib(α).

It follows that, for any u ∈ T and t ∈ (0, 1], applying Lemma 2.8,

Then, since Φ is a Cox-Sibuya point process,

and then

and nally

Denition 2.10. Let Φ be a Cox-Sibuya point process with coecient α and random measure M given by Theorem 2.9. The coecient α and the probability distribution σ 1 of M are respectively called the exponent and the spectral law of Φ, that we denote Φ ∼ CSib(α, σ 1 ).

Remark 2.11. The terminology "Cox-Sibuya" refers to Cox point processes [START_REF] Cox | Point processes[END_REF].

Such a point process is dened as a Poisson point process conditionally to its random intensity measure. The previous theorem states that a Cox-Sibuya point process may be seen in a similar way as a Sibuya point process conditionally to its random parameter measure.

An equivalent of Theorem 1.6 for point processes is provided in the next theorem.

Theorem 2.12. Let Φ be a point process on X such that P(Φ = 0) > 0 and Ψ be a point process on X. If

then Ψ is a Cox-Sibuya point process. Proof. For any u ∈ T and any t ∈ (0, 1], by Lemma 2.8,

By assumption,

Then,

It follows that

which, by Theorem 2.9, concludes the proof.

Adapting the Poisson cluster representation given by Theorem 2.3 with Cox-Sibuya point processes, a new characterization of discrete stable point processes is stated in the following theorem. Theorem 2.13. Let Ψ be a nite point process on X. Then, the two following assertions are equivalent:

• there exist a Poisson random variable Z with parameter c ∈ [0, +∞) and a point process Φ ∼ CSib(α, σ 1 ) such that

where (Φ (k) ) k∈N is a sequence of independent copies of Φ.

In this case, σ = cσ 1 and c = σ(M 1 ).

Proof. A nite point process Ψ is a discrete α-stable point process, if and only if, by formula (5), its probability generating functional is given for any function u ∈ T by:

for a nite measure σ on M 1 . Using Denition 2.1 and standard operations on probability generating functionals, it is equivalent to say that, for any u ∈ T ,

where Z has a Poisson distribution with parameter c = σ(M 1 ) and (Φ (k) ) k∈N is a sequence of independent copies of a Cox-Sibuya point process Φ with exponent α and spectral law σ 1 = σ σ(M1) .


S R Wang

C Raynaud
email: raynaud@cegely.insa-lyon.fr

D Planson

M Lazar

J P Chante

OBIC measurements on 1.3 kV 6H-SiC bipolar diodes protected by Junction Termination Extension

Keywords: 6H-SiC, bipolar diodes, JTE, OBIC, breakdown voltage

Silicon carbide has suitable properties for high power electronic applications. We present here results on 6H-SiC bipolar diodes protected by Junction Termination Extension. We have achieved breakdown voltage of ~1200 V, which is close to the theoretical value, and forward current density of 120 A cm -2 at 5 V. Optical Beam Induced Current measurements and simulations have shown that JTE are very efficient and that their parameters are close to the optimum. Nevertheless, a little increase of their doping level should increase the breakdown voltage up to ~1400 V.

Introduction

Due to its good electrical properties, mainly a high critical electric field and a large bandgap, silicon carbide has demonstrated potentialities for high power devices. We have designed and realised bipolar diodes to sustain a reverse voltage of about 1.3 kV. To avoid surface premature breakdown and thus benefit the advantages of SiC, we have protected our diodes with an aluminium-implanted ring, surrounding the emitter. This ring is named Junction Termination Extension (JTE). We have investigated the performances of these diodes in terms of Current-Voltage (IV) measurements, and analysed the effect of the JTE using Optical Beam Induced Current (OBIC) measurements.

Samples details

6H-SiC n-type epitaxial layers with values of thickness of 10 µm, and of doping level ND = 1.6´10 16 cm -3 , on n + -type 6H-SiC substrates, have been purchased from Cree Research for this study. The p + emitter has been realised by multiple aluminium implantations at 300°C with energies varying between 25 and 300 keV, with a total dose of 1.75´10 15 cm -2 . Lateral protections have been realised by RT Al implantations with a total dose of 1.3´10 13 cm -2 . This produces a p-type region doped at 3´10 17 cm -3 on a depth of ~0.5 µm. Then samples are annealed in a rf-induction heating furnace at 1700°C during 30 min. [1]. Diodes have been metallized with Ni in backside and Al-Ti in the front side. Different kinds of diodes are available, with varying emitter diameters and JTE lengths. The surface is not passivated.

Current-voltage measurements

IV measurements have been performed with a Keithley 237 Source Measurement Unit. Reverse biases are delivered by a 12.5 kV-100 mA FUG supplier. In forward biases, ~50% of diodes (protected or non protected) show current densities > 150 A cm -2 at 5 V (Figure 1), with a best result of 220 A cm -2 . In reverse biases, about 40% diodes protected with JTE have been able to block up to 1100 V (Figure 2) with a best result at 1270 V, which is very closed to the theoretical value given by numerical simulations (see Table I), using the ionisation coefficients determined by Baliga [START_REF] Raghunathan | IEEE Int. Symp. On Power Semiconducor and Devices and IC's[END_REF]. The lowest significant current is 10 -11 A. The last data points which are drawn are considered as the breakdown voltage (next point is more than 0.1 µA). The majority of nonprotected diodes have shown breakdown voltages Vbr at about 500 V. In the following, we will use the Baliga coefficients, due to their good agreement with our measurements on both protected and non-protected diodes.

Konstantinov coef. Ruff coef. Baliga coef. Theoretical 1D Vbr 1240 [V] 920 [V] 1540 [V] 2D Vbr : with JTE 950 [V] 790 [V] 1100 [V] 2D Vbr : without JTE 380 [V] 310 [V] 460 [V]
Table I : Numerical simulations of breakdown voltages of our bipolar diodes. Theses values have been calculated with Medici TM software for diodes protected or not protected by JTE, using 1-D (one Dimensional) and 2-D models, and using the ionisation coefficients published by Konstantinov [3], Ruff [4], and Baliga [START_REF] Raghunathan | IEEE Int. Symp. On Power Semiconducor and Devices and IC's[END_REF]. NB: In the JTE, all dopants are ionised due to the presence of the electric field. So Vbr should not depend on temperature due to the ionisation energies of acceptors levels, when JTE are fully depleted before the breakdown occurs (i.e. for low JTE doses).

The large breakdown voltage difference between protected and non-protected diodes, obtained in experimental measurements, is significant of the beneficial role of JTE, when dopants are correctly activated (i.e. incorporation of Al in substitutional SiC lattice sites) by the post implantation annealing. Moreover, we have noticed that testing the diodes in air, in silicone oil or in SF6 does not increase significantly Vbr. We have therefore used Optical Beam Induced Current (OBIC) measurements in order to study the distribution of equipotential lines at the surface of the samples.

OBIC measurements

Details about this technique have been described elsewhere [2]: a UV beam produced by an Ar ++ laser (300 nm < l < 363,8 nm) is focused at the surface of the samples, modulated at f = 70 Hz by a mechanical chopper, and therefore generates free carrier. When they are generated in a space charge region, electrons and holes are separated by the electric field and a current is generated and detected by a lock in amplifier (SR 830). By translating the beam, we can measure the generated current along the diameters of the diodes. In this study, we have used a wavelength of 351.1 nm, which corresponds to a penetration depth of ~ 9.4 µm in 6H-SiC.

Figure 3 shows such a measurement performed on a diode without JTE. The structure has been drawn on the figure for easier comparison. It is clear that on non-protected diodes, the OBIC signal increases very fast at the edge of the emitter, which is due to the high electric field present in this corner region. This leads to a severe reduction in breakdown voltages of these diodes (~500 V). OBIC signal obtained on a protected diode is represented in Figure 4. The difference with Figure 3 is evident: the current peak is much wider. This is relevant to the spreading out of the equipotential lines. The peak width agrees well with the JTE length (120 µm) and it means that JTE are depleted even at very low voltage. On diodes with shorter JTE (50 µm), the spreading out of the OBIC signal is also in agreement with the JTE length. We can also notice that at 600 V, a peak also appears at the emitter edge of protected diodes. This peak remains at the edge at the emitter edge even at 900V with non-equilibration by the JTE contribution. Normally an optimal doping of the JTE involves two peaks with the same range of magnitude at the edge of the emitter and the JTE. This indicates that the electric field increases much more near the emitter edge. This could mean that the doping level of JTE is close to the optimised value but a little lower. In order to confirm these assumptions on the distribution of the equipotential, OBIC signal has been simulated with Medici TM .

OBIC simulations

Figure 5 shows both theoretical (computed using Baliga coefficients for the ionisation rate) and experimental OBIC signal obtained on a diode without JTE, when 0 V is applied on the anode. From the slope in the semi-logarithmic scale, we have deduced the value of the minority carrier lifetime t of 450 ns. In the following, the theoretical curves have been calculated using this value.

The simulated behaviour of the OBIC signal is in agreement with experimental results. For diodes without JTE, the OBIC peak which arises close to the emitter edge increases strongly at low voltage (~ 400 V in Figure 6). For JTE-protected diodes, a peak close to the emitter occurs at much higher voltage (~800 V), due to the lateral extension of the space charge region, as shown on Figure 7. The OBIC signal is rather constant on the whole length of the JTE. The lack of a peak on the OBIC signal at the edge of the JTE even for a polarisation of 1100 V in reverse bias confirms an insufficient protection from the JTE. As a result, to reach the theoretical one Dimensional Vbr value (Table 1) a little increase of the JTE doping level is necessary. This deduction confirms also a good activation of the dopants implanted to create the JTE and the significant role of the post-implantation annealing in the performance of these devices in reverse bias.

Figure 6 : Theoretical OBIC intensity computed for the diode without JTE, using a lifetime of 450 ns.

Figure 7 : Theoretical OBIC intensity computed for the protected diode, using a lifetime of 450 ns.

Conclusions

We have fabricated bipolar diodes with effective protection against premature breakdown. The observed breakdown voltage is in good agreement with the calculated one. This is due to the beneficial role of Junction Termination Extension, which are close to the optimum. OBIC measurements and simulations have shown that JTE are depleted and come into effect even at low voltages, and that for voltages near Vbr, the electric field increases at the periphery of the emitter. This is a proof that nevertheless the JTE doping level is a little too low. As Vbr is close to the desired value (for which the diodes have been designed), we can also conclude that during the postimplantation annealing, almost all the dopants have been activated in the JTE.

Figure 1 :

 1 Figure 1 : IV characteristics of some diodes with JTE, in forward bias, measured with a SMU K237.

Figure 2 :

 2 Figure 2 : IV characteristics of 3 diodes (emitter diameter AE = 160 µm) with JTE, in reverse bias, by step of 10 V.

Figure 3 :

 3 Figure 3 : OBIC measurements performed on a p + n diode without JTE, for increasing reverse voltage (emitter diameter AE = 160 µm).

Figure 4 :

 4 Figure 4 : OBIC measurements performed on a p+n diode protected by JTE, for increasing reverse voltage (emitter diameter AE = 250 µm).

Figure 5 :

 5 Figure 5 : Comparison of theoretical curves with experimental data obtained at 0 V on the diode without JTE (cf. Figure 3), using minority carrier lifetime values of 50 ns and 450 ns. The theoretical curves have been plotted only at the edge of the emitter.

  Photocurrent (pA)

Acknowledgements

The authors gratefully acknowledge A. Poggi, R. Nipoti and G. Cardinali (from IMM Sezione di Bologna, Italy) for metal contact realization. The authors also thank CIME (Grenoble) team for technological means availability. They thank also K. Isoird for fruitful discussions on OBIC measurements. Schneider Electric group is thanked a lot as well for interest and support. The authors are very grateful to Le Grand Lyon for its financial support to international researcher (post-doctorate grant).