Lévy processes and random discrete structures

Lévy processes and random discrete structures

LÉVY2016 SUMMER SCHOOL ON LÉVY PROCESSES

PRELIMINARY VERSION

Igor Kortchemski

The goal of this lecture is to study a family of random trees closely related to stable spectrally positive Lévy processes. Trees appear in many different areas such as computer science (where trees appear in the analysis of random algorithms for instance connected with data allocation), combinatorics (trees are combinatorial objects by essence), mathemat- ical genetics (as phylogenetic trees), in statistical physics (for instance in connection with random maps as we will see below) and in probability theory (where trees describe the genealogical structure of branching processes, fragmentation processes, etc.).

In Section1, we define Bienaymé–Galton–Watson trees and explain how they are coded by random walks. In particular, for a particular family of trees, these random walks are closely related to stable spectrally positive Lévy processes. In Section2, we will use stable spectrally positive Lévy processes to obtain a result concerning Bienaymé–Galton–Watson trees. In Section3, we will use Bienaymé–Galton–Watson trees to obtain a result concerning stable spectrally positive Lévy processes.

1 Bienaymé–Galton–Watson trees and their coding by ran- dom walks

1.1 Trees

In this lectures, by tree, we will always mean plane tree (sometimes also called rooted ordered trees). To define this notion, we follow Neveu’s formalism. LetUbe the set of labels defined by

U=

∞

[

n=0

(N^∗)ⁿ,

where, by convention,(N^∗)⁰={∅}. In other words, an element ofUis a (possible empty) sequence u = u₁· · ·u_j of positive integers. When u = u₁· · ·u_j and v = v₁· · ·v_k are elements ofU, we letuv=u₁· · ·u_jv₁· · ·v_k be the concatenation ofuandv. In particular, u∅ = ∅u = u. Finally, a plane tree is a finite subset of U satisfying the following three conditions:

♠CNRS & CMAP, École polytechnique, Université Paris-Saclay. igor.kortchemski@normalesup.org

(i) ∅ ∈τ,

(ii) ifv∈τandv=ujfor a certainj∈N^∗, thenu∈τ,

(iii) for everyu∈τ, there exists an integerk_u(τ)>0 such that for everyj∈N^∗,uj ∈τif and only if 1 6j6k_u(τ).

∅

1 2 3

11 21

Figure 1: An example of a tree τ, where τ={∅, 1, 11, 2, 21, 3}.

In the sequel, bytreewe will always mean finite plane tree. We will often view the elements ofτas the individuals of a population whoseτis the genealogical tree and∅is the ancestor (the root). In particular, foru∈τ, we say thatku(τ)is the number of children ofu, and writekuwhenτis implicit. The size ofτ, denoted by|τ|, is the number of vertices ofτ. We denote byAthe set of all trees and byAnthe set of all trees of sizen. Finally, for j>1, a forest ofjtrees is an elements ofA^j.

1.2 Bienaymé–Galton–Watson trees

We now define a probability measure onAwhich describes, roughly speaking, the law of a random tree which describes the genealogical tree of a population where individuals have a random number of children, independently, distributed according to a probability measureµ, called the offspring distribution. Such models were considered by Bienaymé [3]

and Galton & Watson [11], who were interested in estimating the probability of extinction of noble names.

We will always make the following assumptions onµ:

(i) µ= (µ(i) :i >0)is a probability distribution on{0, 1, 2, . . .}, (ii) P

k>0kµ(k)61, (iii) µ(0) +µ(1)<1.

Theorem 1.1. Set, for everyτ∈A,

Pµ(τ) =Y

u∈τ

µ(k_u). (1)

ThenPµdefines a probability distribution onA.

Before proving this result, let us mention that in principle we should define theσ-field used forA. Here, sinceAis countable, we simply take the set of all subsets ofAas the σ-field, and we will never mention again measurability issues (one should however be careful when working with infinite trees).

Proof of Theorem1.1. Setc=P

τ∈APµ(τ). Our goal is to show thatc=1.

Step 1.We decompose the set of trees according to the number of children of the root and write

c=X

k>0

X

τ∈A,k∅=k

Pµ(τ) =X

k>0

X

τ₁∈A,...,τk∈A

µ(k)Pµ(τ₁)· · ·Pµ(τ_k) =X

k>0

µ(k)c^k. Step 2. Set, for 0 6 s 6 1, f(s) = P

k>0µ(k)s^k−s. Then f(0) = µ(0) > 0, f(1) = 0,

f⁰(1) = (P

i>0iµ(i)) −1<0 andf⁰⁰ >0 on[0, 1]. Therefore, the only solution off(s) =0 on

[0, 1]iss =1.

Step 3.We check thatc61 by constructing a random variable whose “law” isPµ. To this ender, consider a collection(Ku :u ∈U)of i.i.d. random variables with same lawµ (defined on the same probability space). Then set

T:=

u₁· · ·u_n ∈U:u_i 6K_{u1u2···ui−1} for every 16i6n .

(Intuitively,K_urepresents the number of children ofu∈Uifuis indeed in the tree. ThenT is a random plane tree, but possible infinite. But for a fixed treeτ∈T, we have

P(T=τ) =P(X_u=k_u(τ)for everyu∈τ) =Y

u∈τ

µ(k_u) =Pµ(τ). Therefore

c= X

τ∈A

Pµ(τ) =X

τ∈A

P(T=τ) =P(T∈A)61.

By the first two steps, we conclude thatc=1 and this completes the proof.

Remark 1.2. WhenP

i>0iµ(i)>1, let us mention that it is possible to define a probability

measure Pµ on the set of all plane (not necessarily finite) trees in such a way that the formula (1) holds for finite trees. However, since we are only interested in finite trees, we will not enter such considerations.

In the sequel, by Bienaymé–Galton–Watson tree with offspring distributionµ(or simply BGWµtree), we mean a random tree (that is a random variable defined on some probability space taking values inA) whose distribution isPµ. We will alternatively speak of BGW tree when the offspring distribution is implicit.

The most important tool in the study of BGW trees is their coding by random walks, which are usually well understood. The idea of coding BGW trees by functions goes back to Harris [5], and was popularized by Le Gall & Le Jan [10] et Bennies & Kersting [1]. We start by explaining the coding of deterministic trees. We refer to [9] for further applications.

1.3 Coding trees

To code a tree, we first define an order on its vertices. To this end, we use the lexicographic order≺on the setUof labels, for whichv≺wif there existsz ∈Uwithv=z(a₁, . . . ,a_n), w=z(b₁, . . . ,b_m)anda₁ < b₁.

Ifτ ∈A, letu₀,u₁, . . . ,u_|τ|−1be the vertices ofτordered in lexicographic order, an recall thatk_uis the number of children of a vertexu.

Definition 1.3. The Łukasiewicz path W(τ) = (Wn(τ), 0 6 n 6 |τ|) of τ is defined by W0(τ) =0 and, for 06n6|τ|−1:

Wn+1(τ) =Wn(τ) +ku_n(τ) −1.

0 1

2

3 4

5 6 7

8 9 10

11 12

1 2 3 4 5 6 7 8 9 10 11 12 13

−1 0 1 2 3

Figure 2: A tree (with its vertices numbered according to the lexicographic order) and its associated Lukasiewicz path.

See Fig.2for an example. Before proving that the Łukasiewicz path codes bijectively trees, we need to introduce some notation. Forn>1, set

S⁽¹⁾n ={(x₁, . . . ,x_n)∈{−1, 0, 1, . . .}: x₁+· · ·+x_n= −1

etx₁+· · ·+x_j >−1 for every 16j6n−1}.

Ifx= (x₁, . . . ,x_n)∈S^(k)n andy= (y₁, . . . ,y_m)∈S^(km⁰⁾, we writexy= (x₁, . . . ,x_n,y₁, . . . ,y_m) for the concatenation of x andy. In particular, xy ∈ S^(k+k_n+m⁰⁾. If x ∈ S^(k), we may write x=x₁x₂· · ·xk withxi∈S⁽¹⁾for every 16i 6kin a unique way.

Proposition 1.4. For everyn>1, the mappingΦ_ndefined by Φn : An −→ S⁽¹⁾n

τ 7−→ k_ui−1 −1:16i 6n is a bijection.

For τ ∈ A, set Φ(τ) = Φ_|τ|(τ). Proposition 1.4 shows that the Łukasiewicz indeed bijectively codes trees (because the increments of the Łukasiewicz path ofτare the elements ofΦ(τ)) and thatW_|τ|(τ) = −1.

Proof. Fixτ∈An. We first check thatΦn(τ)∈S⁽¹⁾n . For every 1 6j6n, we have Xj

i=1

ku_i−1−1

= Xj

i=1

ku_i−1 −j. (2)

Note that the sum Pj

i=1k_ui−1 counts the number of children of u₀,u₁, . . . ,u_j−1. Ifj < n, the verticesu₁, . . . ,u_j are children ofu₀,u₁, . . . ,u_j−1, so that the quantity (2) is positive. If j=n, the sumP_n

i=1k_ui−1 counts vertices who have a parent, that is everyone except the root, so that this sum isn−1. Therefore,Φn(τ)∈S⁽¹⁾n .

We next show that Φn is bijective by strong induction on n. For n = 1, there is nothing to do. Fix n > 2 and assume thatΦ_j is a bijection fore very j ∈ {1, 2, . . . ,n−1}.

Takex = (a,x₁, . . . ,x_n−1) ∈ S⁽¹⁾n . We have Φ_n(τ) = x if and only if k_∅(τ) = a+1, and (x₁, . . . ,x_n−1)must be the concatenation of the images byΦ of the subtreesτ₁, . . . ,τ_a+1 attached on the children of∅. But(x₁,x₁, . . . ,x_n−1)∈S^(a+1)_n−1 , so(x₁, . . . ,x_n−1) =x₁· · ·x_a+1 can be written as a concatenation of elements ofS⁽¹⁾ in a unique way. Hence

Φ_n(τ) =x ⇐⇒ Φ_|τi|(τ_i) =x_ifor everyi∈{1, 2, . . . ,a+1}

⇐⇒ τ={∅}∪

a+1

[

i=1

iΦ⁻¹_|τ

i|(x_i),

where we have used the induction hypothesis (since|τ_i|<|τ|). This completes the proof.

Extension to forests. Recall that a forest ofktrees is a sequence ofktrees. Proposition1.4 is easily extended to forests by definingΦ(τ₁, . . . ,τ_k)as the concatenationΦ(τ₁)· · ·Φ(τ_k).

This yields a bijection between the set of all forests withktrees andnvertices (withk6n) andS^(k)_n , where

S^(k)_n :={(x₁, . . . ,x_n)∈{−1, 0, 1, . . .}: x₁+· · ·+x_n = −k

andx₁+· · ·+x_j >−kfor every 16j6n−1}.

Similarly, the Łukasiewicz path of a forest is defined as the concatenation of the jumps of the Łukasiewicz paths of the trees of the forest.

1.4 Coding BGW trees by random walks

We will now identify the law of the Łukasiewicz path of a BGW tree. Consider the random walk (W_n)_n>0 on Z such that W₀ = 0 with jump distribution given by P(W₁=k) =

µ(k+1)for everyk>−1. In other words, forn>1, we may write Wn =X₁+· · ·+Xn,

where the random variables (X_i)_i>1 are independent and identically distributed with P(X₁=k) =µ(k+1)for everyk>−1. This random walk will play a crucial role in the sequel. Finally, forj>1, set

ζ_j =inf{n>1:W_n = −j},

which is the first passage time of the random walk at−j(which could be a priori be infinite

!).

Proposition 1.5. LetTbe a randomBGWµ tree. Then the random vectors (of random length) W0(T),W1(T), . . . ,W_|T|(T)

and (W₀,W₁, . . . ,W_ζ1) have the same distribution.

In particular,|T|andζ₁have the same distribution.

Proof. Fixn>1 and integersx₁, . . . ,x_n>−1. Set

A = P(W1(T) =x₁,W2(T) −W1(T) =x₂, . . . ,Wn(T) −Wn−1(T) =x_n), B = P(W₁=x₁,W₂−W₁=x₂, . . . ,Wn−W_n−1=xn).

We shall show thatA=B.

First of all, if (x₁, . . . ,x_n) 6∈ S⁽¹⁾n , then A = B = 0. Now, if (x₁, . . . ,x_n) ∈ S⁽¹⁾n , by Proposition1.4there exists a treeτwhose Łukasiewicz path is(0,x₁,x₁+x₂, . . .). Then, by (1),

A=P(T=τ) =Y

u∈τ

µ(k_u) = Yn

i=1

µ(x_i+1), et

B = P(W₁=x₁,W₂−W₁=x₂, . . . ,W_n−W_n−1=x_n,ζ₁ =n)

= P(W₁=x₁,W₂−W₁=x₂, . . . ,W_n−W_n−1=x_n)

= Yn

i=1

µ(x_i+1).

For the second equality, we have used the equality of events {W₁ = x₁,W₂−W₁ = x₂, . . . ,Wn−W_n−1 = xn,ζ₁ = n} = {W₁ = x₁,W₂−W₁ = x₂, . . . ,Wn−W_n−1 = xn}, which comes from the fact that(x₁, . . . ,x_n) ∈ S⁽¹⁾n . HenceA = B, and this completes the proof.

Extension to forests. The previous result is immediately adapted to forests:

Corollary 1.6. For everyn > 1, the Łukasiewicz path of a forest ofni.i.d BGWµ trees has the same distribution as(W₀,W₁, . . . ,W_ζn).

Remark 1.7. If µ is an offspring distribution with mean m, we have E[W₁] = m−1.

Indeed,

E[W₁] = X

i>−1

iµ(i+1) =X

i>0

(i−1)µ(i) =m−1.

In particular,(Wn)_n>0is a centered random walk if and only ifm=1 (that is if the offspring distribution is critical).

1.5 A special family of offspring distributions

The connection with Lévy processes arises when considering a special family of offspring distributions. Let us first recall some properties concerning domains of attraction.

Let(X_i)_i>1be i.i.d. real-valued random variables. Assume that there exists a sequence

of positive real numbersan→∞, a sequence of real numbers(bn)and a non-degenerate random variableUsuch that the convergence

X₁+· · ·+X_n

a_n −b_n −→^(d)

n→∞ U

holds in distribution, thenUis anαstable random variable for a certain valueα∈(0, 2].

We say that (the law of)X₁is in the domain of attraction of a stable law of indexα.

In the sequel, we will fixα∈(1, 2)and we will consider offspring distributionsµwhich satisfy:

(i) P

i>0iµ(i) =1 (µis critical),

(ii) µis in the domain of attraction of a stable law of indexα.

It is known that (ii) is equivalent to the fact that µ([n,∞)) = ^L(n)_nα for a slowly varying functionL(that is for every fixedx >0,L(ux)/L(u)→1 asu→∞). A typical example to keep in mind is the case whereµ(n)∼ _n1+α^c for a certain constantc >0.

Recalling the random walk(W_n)_n>0 which was previously defined, W₁ is centered and also is in the domain of attraction of a stable law of indexα. Therefore, there exists a sequencea_n →∞such that

W_n a_n

−→(d)

n→∞ Y^(α), (3)

whereY^(α)is anα-stable spectrally positive random variable normalized so thatEh

e^−λY(α) i

= e^λα for everyλ>0. The fact thatY^(α) is spectrally positive (meaning that its Lévy measure vanishes onR−) follows from the fact that the negative jumps ofW₁are bounded.

Let us mention that ifµ([n,∞)) = ^L(n)_nα , the sequencea_nis chosen such that nL(a_n)

n^α −→

n→∞

1

|Γ(1−α)|,

so that a_n is equal to n^1/α multiplied by another slowly varying function. Finally, we mention that the Lévy measureΠofY^(α) is

Π(dr) = α(α−1) Γ(2−α) · 1

r^1+α1r>0dr.

2 Maximum out-degree of Bienaymé–Galton–Watson trees

In this section, we fix 1 < α < 2 and consider a critical offspring distribution µ which belongs to the domain of attraction of a stable law of index α. For a tree τ ∈ A, we let M(τ) =maxu∈τk_u(τ)be the maximum number of children (or the maximum out-degree) of a vertex inτ.

The goal of this Section is to establish the following result, due to Bertoin [2], and we present the proof given in [2].

Theorem 2.1(Bertoin). LetTbe aBGWµtree. Then

P(M(T)> x) _x→∼_∞ β x,

whereβ > 0only depends onαand is the unique positive solution of the equation X∞

n=0

(−1)ⁿ βⁿ

(n−α)n! =0.

The fact that this equation has a unique solution follows for instance from the fact that f(x) =P_∞

n=0(−1)ⁿ_(n−α)n!^xn is continuous onR+, increasing withf(0)<0 andf(1)>1.

The first idea it to reduce the problem to a forest of BGWµ trees.

Lemma 2.2. Forn>1, denote byM_nthe maximum out-degree in a forest ofni.i.d.BGWµtrees.

Assume that

P(Mn 6n) −→

n→∞ e^−β. Then

P(M₁ > x) _x→∼_∞ β x. Proof. We first check that

P(M₁> n) _n→∼_∞ β

n (4)

whenn→∞along integer valuers. By independence,P(M_n 6n) = (1−P(M₁> n))ⁿ. Hence, by assumption,

nln(1−P(M₁ > n)) _n→∼_∞ −β.

SinceP(M₁ > n)→ 0 asn→∞, we have ln(1−P(M₁> n)) ∼−P(M₁ > n)asn→∞, and we readily get (4).

To finish the proof, we use a monotonicity argument by writing, forx >0, P(M₁>[x] +1)>P(M₁ > x)>P(M₁>[x]),

where[x]denotes the integer part ofx. By (4), P(M₁ >[x] +1) _x→∞∼ β

[x] +1 _x→∞∼ β

x, P(M₁>[x]) _x→∞∼ β

[x] _x→∞∼ β x. This shows that

P(M₁> x) _x→∞∼ β x and completes the proof.

2.1 Link with Lévy processes

A first step towards the proof of Theorem2.1is a connection with Lévy processes. To this end, as in Section1.4, we introduce the random walk(W_n)_n>0such thatW₀=0 and with jump distributionP(W₁ =i) =µ(i+1)fori >−1. As seen was seen in Section1.5, there exists a sequencean →∞such that

W_n a_n

−→(d)

n→∞ Y^(α),

whereY^(α)is anα-stable spectrally positive random variable normalized so thatEh

e^−λY(α)i

= e^λα for everyλ>0.

We will need a functional extension of this result (known as Donsker’s invariance principle in the case of brownian motion). Before, let us recall several facts concerning the Skorokhod topology. Denote byD(R+,R)the space of all real-valued càdlàg functions defined onR+, equipped with the SkorokhodJ₁metric so thatD(R+,R)is a Polish metric space (meaning that it is complete and separable). Recall (see e.g. [6, Theorem 1.14 in Chapter VI]) that iff_n,f∈D(R+,R)the convergencef_n →finD(R+,R)holds if and only if there exist time changesλ_nsuch that:

– λ_n(0) =0,λ_nis strictly increasing and continuous,λ_n(∞) =∞, – sup

t>0

|λ_n(t) −t| −→

n→∞ 0,

– for every integerN>1, sup

t6N

|f_n(λ_n(t)) −f(t)| −→

n→∞ 0.

Then the convergence

W_[nt]

a_n :t >0

(d) n→∞−→ Y

holds in distribution inD(R+,R)asn→∞(see Theorem 16.14 in [7]). Alternatively, there exists a sequenceae_n →∞such that the convergence

W_[

aent]

n :t >0

(d)

n→∞−→ Y (5)

holds in distribution inD(R+,R)asn→∞.

We now introduce some notation. Let(Y_t)_t>0be an α-stable spectrally positive Lévy process withY₀=0 normalised so thatE

e^−λY1

=e^λα for everyλ>0. Fors,t >0 we set

∆Y_s =Y_s−Y_s−, ∆^∗_t =sup

s<t

∆Y_s, with the convention∆Y₀=0 and∆^∗₀=0. Finally, set

τ₁ =inf{t >0:Y_t<−1}.

We are now ready to state and prove the connection with Lévy processes.

Proposition 2.3. Then the convergence M_n

n

−→(d) n→∞ ∆^∗_τ

1

holds in distribution.

Proof. Since ∆Y_τ1 = 0 almost surely, by continuity properties of the J₁ metric (see [6, Proposition 2.12 in Chapter VI]), the convergence (8) implies the convergence in distribution of the càdlàg functions stopped at the first time they hit−1. In other words,

W_[

aent]

n :06t 6 ζ_n aen

(d)

n→−→∞ (Y_t:06t6τ₁), (6) where we recall thatζ_n =inf{k>0:W_k = −n}.

By Corollary1.6,(W₀, . . . ,W_ζn)has the same distribution as the Łukasiewicz path of a forest ofni.i.d. BGWµ trees. But by construction of the Łukasiewicz path, the maximum outdegree of the forest, is the maximum jump of its Łukasiewicz path plus one. Therefore,

Mn−1

n has the same distribution as the maximum jump of _W

[ante ]

n :06t 6 ^ζn

aen

. The desired result follows by the continuity of the largest jump of càdlàg functions for theJ₁ metric on compact time sets (see [6, Proposition 2.4 in Chapter VI]).

The goal is now to show thatP ∆^∗_τ

1 6x

=e^−βx for everyx >0.

2.2 A fluctuation identity for spectrally positive Lévy processes

Here we calculate the Laplace transform of hitting times for general spectrally positive Lévy processes, a result which will be needed in the calculation ofP ∆^∗_τ

1 6x

.

Let(eY_t,t >0)be a Lévy process with Lévy measureΠ. It is known (see e.g. [8, Theorem 3.6]) that for everyλ∈R,

∀t >0, Eh e^λeYti

<∞ ⇐⇒

Z

|x|>1e^λx Π(dx)<∞

Now assume thatYeis spectrally positive, that isΠ=0 onR−and thatΠ6=0 onR+. As a consequence,Eh

e^−λYeti

<∞for everyt,λ>0, and we may define the Laplace exponent Ψby

Eh e^−λeYt

i

=e^tΨ(λ) forλ,t>0. In particular,

Eh

e^{−λeYt−tΨ(λ)} i

=1. (7)

The Laplace exponentΨ is continuous onR+, is strictly convex and satisfiesΨ(0) = 0, Ψ(∞) =∞andΨ⁰(0) = −Eh

Ye₁i

(see [8, Exercise 3.5]).

Finally, for everyq>0 set

Φ(q) =sup{λ>0:Ψ(λ) =q} and for everyx >0 set

˜

τ_x=inf

t >0:Ye_t <−x

.

Note that forq>0, the equationΨ(λ) = qhas only solution, except whenq=0 and Eh

Ye₁i

>0.

Theorem 2.4. For everyq>0, E

e^−qτ˜x1τ˜x<∞

=e^−xΦ(q). In particular,P(τ˜x <∞) =1if and only ifEh

Ye₁ i

60.

Proof. We follow the proof of [8, Theorem 3.12]. Denote by Ft the P-completedσ-field generated byσ(Ye_s,s6t).

We start with the case q > 0. For t > 0, using the fact that Ye_τ˜x = −x on the event {τ˜_x <∞}sinceYeis spectrally positive, write

Eh

e^{−Φ(q)eYt−qt} Fτ˜x

i

= e^{−Φ(q)eYt−qt}1t6τ˜x+Eh

e^{−Φ(q)(eYt−eYτx˜ +eYτx˜ )−q(t−τ˜x+τ˜x)}1t>˜τx

Fτ˜x

i

= e^{−Φ(q)eYt−qt}1t6τ˜x+e^{xΦ(q)−qτ˜x}Eh

e^{−Φ(q)(Yet−eYτx˜ )−q(t−τ˜x)}1t>τ˜x

Fτ˜x

i

= e^{−Φ(q)eYt−qt}1t6τ˜x+e^{xΦ(q)−qτ˜x}1t>τ˜x

= e^{−Φ(q)eYt∧τx˜ −qt∧τ˜x}

where we have combined the strong Markov property with (7) for the penultimate equality.

Taking expectations and using (7) again, we get that:

Eh

e^{−Φ(q)Yet∧τx˜ −qt∧τ˜x}i

=1.

Then take t → ∞, and since e^{−Φ(q)eYt∧τx˜ −qt∧τ˜x} 6 e^xΦ(q), we get the desired result by dominated convergence.

The caseq=0 is settled by takingq↓0 in the identity which has just been established.

2.3 End of the proof

Theorem2.1will readily follow from Lemma2.2and Proposition2.3once the following result is established.

Proposition 2.5. For everyx >0, we have P ∆^∗_τ

1 6x

= β x.

Proof. Denote byΠ(dr) = _r1+α^c 1r>0drthe Lévy measure ofY (we will not need the explicit value of the constantc).

The idea is to consider the processY obtained by suppressing jumps greater than or equal toxby writing

Y_t= Y_t−X

s6t

∆_s1∆s>x

!

| {z }

Yet

+X

s6t

∆_s1∆s>x

| {z }

Zt

.

By the Lévy-Itô decomposition, the processesYeandZare independent andZis a compound poisson process.

We start with calculating the Laplace exponentΨeofY. First note that fore λ>0, Eh

e^−λY1 i

=exp(λ^α) =exp Z_∞

0

e^−λu−1+λu

Π(du)

and

Eh e^−λZ1i

=exp Z_∞

x

e^−λu−1

Π(du)

. SinceE

e^−Y1

=Eh

e^−eY1e^−Z1 i

=Eh e^−eY1

iE e^−Z1

by independence, we get that Ψ(λ) =e

Z_∞

0

e^−λu−1+λu

Π(du) − Z_∞

x

e^−λu−1

Π(du)

= c Z_x

0

e^−λu−1+λu 1

u^1+α du−cλ Z_∞

x

u u^1+α du

= c Z_x

0

e^−λu−1+λu 1

u^1+α du− cλ (α−1)u^α−1

Now, to finish the proof, introduce

˜

τ₁=inf

t >0:Yet<−1

, J=inf{t >0:∆Yt > x}=inf{t >0:∆Zt> x} and note that there is the equality of events

∆^∗_τ

1 6x ={τ˜₁< J}.

But ˜τ₁is a measurable function ofYeandJis a measurable function ofZ, so ˜τ₁ andJare independent. In addition,Jis distributed according to an exponential random variable of parameter

Π([x,∞)) = Z_∞

x

Π(du) = c αx^α. Therefore

P ∆^∗_τ

1 6x

=P(τ˜₁ < J) =Eh

e^{−αxαc τ˜1}i . SinceEh

Ye₁i

6E[Y₁] =0, by Theorem2.4, Eh

e^{−αxαc τ˜1} i

=e^−p(x) wherep(x)is the positive solution of

c Z_x

0

e^−p(x)u−1+p(x)u 1

u^1+α du− cp(x)

(α−1)u^α−1 = c αx^α. The change of variableu=xvin the integral gives

Z₁

0

e^−xp(x)v−1+xp(x)v 1

u^1+α du−xp(x) α−1 = 1

α. Settingβ=xp(x), we see thatβsatisfies the question

Z₁

0

e^−βv−1+βv 1

u^1+α du− β

α−1 = 1 α. Expandinge^−βvthen readily gives the desired result.

3 An identity for stable spectrally positive Lévy processes

3.1 The cyclic lemma

For 1 6k6n, set

S^(k)n :={(x₁, . . . ,x_n)∈{−1, 0, 1, . . .}:x₁+· · ·+x_n = −k},

and

S^(k)n ={(x₁, . . . ,x_n)∈S^(k)_n :x₁+· · ·+x_j >−kpour tout 16j6n−1}.

In the following, we identify an element ofZ/nZ with its unique representative in {0, 1, . . . ,n−1}. Forx= (x₁, . . . ,x_n)∈S^(k)n andi∈Z/nZ, we set

x⁽ⁱ⁾ = (x_i+1, . . . ,x_i+n),

where the addition of indices is considered modulon. We say thatx⁽ⁱ⁾ is obtained fromx by a cyclic permutation. Note thatS^(k)n is stable by cyclic permutations.

Definition 3.1. Forx∈S^(k)n , set I_x =

i ∈Z/nZ:x⁽ⁱ⁾ ∈S^(k)n

. See Fig.3for an example.

1 2 3 4 5 6 7 8 9 10 11 12

−4

−3

−2

−1 0 1

1 2 3 4 5 6 7 8 9 10 11 12

−4

−3

−2

−1 0 1 2

Figure 3: For x = (x₁,x₂, . . .), we represent x₁+· · ·+x_i as a function of i. On the left, we takex = (1,−1,−1,−1,−1, 2,−1,−1,−1, 0, 2,−1)∈S⁽³⁾₁₃, where I_x = {4, 5, 9}. On the right, we take x⁽⁵⁾, which is indeed an element ofS⁽³⁾₁₃.

Note that ifx∈S^(k)n andi ∈Z/nZ, then Card(Ix) =Card(I_x(i)).

Theorem 3.2. (Cyclic Lemma) For everyx∈S^(k)n , we haveCard(Ix) =k.

Therefore, ifx∈ ∪_n>kS^(k)n , the setI_xdepends onx, but its cardinal does not depend on x!

Proof. We start with an intermediate result: we check that Card(I_x)does not change if one concatenates

(a,−1, . . . ,| {z −1}

atimes

)

to the left ofx, for an integera>1. To this end, fixx= (x₁, . . . ,xn)∈S^(k)n and set ex= (a,−1, . . . ,| {z −1}

atimes

,x₁, . . . ,x_n).

First, it is clear that 0 ∈I

exif and only if 0∈I_x. Then, if 0< j 6n−1, we have ex^(j+a+1) = (x_j+1, . . . ,x_n,a,−1, . . . ,−1,x₁, . . . ,x_j).

It readily follows thatj∈I_xif and only ifj+a+1∈I

ex. Next, we check that if 0< i6a+1, theni 6∈I

ex. Indeed, if 0< i6a+1, then ex⁽ⁱ⁾ = (−| 1, . . . ,{z −1}

a−i+1 times

,x₁,x₂, . . . ,x_n,a,−1, . . . ,−1). The sum of the elements ofex⁽ⁱ⁾ up to elementx_nis

x₁+· · ·+x_n− (a−i+1) = −k− (a−i+1)6−k. Henceex⁽ⁱ⁾ 6∈I

ex. This shows our intermediate result.

Let us now establish the Cyclic Lemma by strong induction onn. Forn =k, there is nothing to do, as the only element ofS^(k)n isx= (−1,−1, . . . ,−1). Then consider an integer n > k such that the Cyclic Lemma holds for elements ofS^(k)_j with j= k, . . . ,n−1. Take x= (x₁, . . . ,xn)∈S^(k)n . Since Card(Ix)does not change under cyclic permutations ofxand since there existsi ∈{1, 2, . . . ,n}such thatx_i >0 (becausen > k), without loss of generality we may assume thatx₁>0. Denote by 1=i₁ < i₂<· · ·< i_mthe indicesisuch thatx_i >0 and seti_m+1=n+1 by convention. Then

−k= Xn

i=1

x_i = Xm

j=1

x_ij− (i_j+1−i_j−1)

sincei_j+1−i_j−1 count the number of consecutive−1 that immediately followsx_ij. Since this sum is negative, there existsj∈{1, 2, . . . ,m}such thatx_ij 6i_j+1−i_j−1. Theforex_ij is immediately followed by at leastx_ij consective times−1. Then letexbe the vector obtained fromxby supressingx_ijimmediately followed byx_ijtimes−1, so that Card(I

ex) =Card(I_x) by the intermediate result. Hence Card(I_x) =kby induction hypothesis.

Remark 3.3. Fix x = (x₁, . . . ,x_n) ∈ S^(k)_n . Set m = min{x₁+· · ·+x_i : 1 6 i 6 n}, and ζ_i(x) =min{j>1:x₁+· · ·+x_j =m+i−1}for 16i6k. Then

I_x ={ζ₁(x), . . . ,ζ_k(x)}.

Indeed, this follows from the fact that this property is invariant under insertion of(a,−1, . . . ,−1) for an integera>1 (where−1 is writtenatimes).

3.2 Applications to random walks

In this section, we fix a random walk (W_n = X₁+· · ·+X_n)_n>0 onZ such thatW₀ = 0, P(W₁ >−1) =1 andP(W₁=0)<1. We set, fork>1,

ζ_k =inf{i >0:W_i = −k}.

Definition 3.4. A functionF:Zⁿ→Ris said to be invariant under cyclic permutations if

∀x∈Zⁿ, ∀i∈Z/nZ, F(x) =F(x⁽ⁱ⁾).

Let us give several example of functions invariant by cyclic permutations. If x = (x₁, . . . ,x_n), one may takeF(x) =max(x₁, . . . ,x_n),F(x) =min(x₁, . . . ,x_n),F(x) =x₁x₂· · ·x_n, F(x) =x₁+· · ·+x_n, or more generallyF(x) =x^λ₁+· · ·+x^λ_navecλ >0. IfA⊂Z,

F(x) = Xn

i=1

1x_i∈A,

which counts the number of elements inA, is also invariant under cyclic permutations.

IfFis invariant under cyclic permutations andg:R→Ris a function, theng◦Fis also invariant under cyclic permutations. Finally,F(x₁,x₂,x₃) = (x₂−x₁)³+ (x₃−x₂)³+ (x₁− x₃)³is invariant under cyclic permutations but not invariant under all permutations.

Proposition 3.5. LetF : Zⁿ → Rbe a function invariant under cyclic permutations. Then for every integersk6nthe following assertions hold.

(i) E

F(X₁, . . . ,X_n)1ζ_k=n

= _n^kE[F(X₁, . . . ,X_n)1Wn=−k], (ii) P(ζ_k =n) = ^k_nP(W_n = −k).

The assertion (ii) is known as Kemperman’s formula.

Proof. The second assertion follows from the first one simply by takingF≡ 1. For (i), to simplify notation, setX_n = (X₁, . . . ,X_n). Note that the following equalities of events hold

{W_n = −k}=

Xn ∈S^(k)n

and {ζ_k =n}=

Xn∈S^(k)n

. In particular,

E

F(X₁, . . . ,X_n)1ζk=n

=Eh

F(Xn)1_X

n∈S^(k)n

i . Then write

Eh

F(Xn)1

Xn∈S^(k)n

i

= 1 n

Xn

i=1

Eh

F(X⁽ⁱ⁾n )1

X⁽ⁱ⁾n ∈S^(k)n

i

(sinceX⁽ⁱ⁾n andXnhave the same law)

= 1 n

Xn

i=1

Eh

F(Xn)1_X⁽ⁱ⁾

n ∈S^(k)n

i

(invariance ofFby cyclic permutations)

= 1 nE

"

F(X_n)1_X

n∈S^(k)n

Xn

i=1

1_X⁽ⁱ⁾

n ∈S^(k)n

!#

= k nEh

F(X₁, . . . ,X_n)1_X

n∈S^(k)n

i ,

where the last equality is a consequence of the equality of the random variables 1_X

n∈S^(k)n

Xn

i=1

1_X⁽ⁱ⁾

n ∈S^(k)n

!

=k1_X

n∈S^(k)n

by the Cyclic Lemma. We conclude that Eh

F(Xn)1_X

n∈S^(k)n

i

= k

nE[F(X₁, . . . ,X_n)1Wn=−k], which is the desired result.

We now present two applications of this result.

Proposition 3.6. Assume thatE[W₁]60. Then, for everyk>1, X∞

n=1

1

nP(W_n = −k) = 1 k.

Proof. We first check thatP(ζ_n <∞) =1 for everyn>1. By the strong Markov property, it is enough to show thatP(ζ₁ <∞) =1.

To this end, define the offspring distributionµbyµ(i) =P(W₁=i−1)fori>0 and let Tbe a BGWµ random tree. By Corollary1.6, forn>1, we haveP(ζ₁=n) =P(|T|=n). SinceE[W₁]60, we haveP

i>0iµ(i)61, so that by Theorem1.1we have

1=X

n>1

P(|T|=n) =X

n>1

P(ζ₁=n) =P(ζ₁<∞). Next, by Proposition3.5(ii),

X∞ n=1

k

nP(W_n = −k) = X∞ n=1

P(ζ_k =n) =P(ζ_k <∞) =1, and this completes the proof.

Proposition3.6gives the following interesting deterministic identity:

Corollary 3.7. For every06λ61, X

n>1

(λn)ⁿ⁻¹e^−λn

n =1.

Proof. In Proposition3.6, we take k = 1 and W₁ = Poisson(λ) −1. In particular,W_n+n is distributed according to a Poisson random variable of parameter λn. Therefore, by Proposition3.6,

1= X∞ n=1

1

nP(W_n = −1) = X∞ n=1

1

nP(W_n+n=n−1) = X∞ n=1

1

n·(λn)ⁿ⁻¹e^−λn (n−1)! , and the desired result immediately follows.

Proposition 3.8(Chen, Curien & Maillard [4]). Assume thatE[W₁]60. Letf:Z→R+be a function. Then, for everyk>2,

E



 1 ζ_k−1

ζ_k

X

i=1

f(X_i)



=E

f(X₁) k k+X₁

.

Proof. Sincek>2, we haveζ_k >2. Now, forn>2, we have by Proposition3.5(i):

E

"X_n

i=1

f(X_i)1ζ_k=n

#

= k n E

"X_n

i=1

f(X_i)1Wn=−k

#

= kE[f(X₁)1Wn=−k]

= kE E

f(X₁)1Wn=−k

X₁

= kE

f(X₁)E

1Wn=−k

X₁

= kE[f(X₁)P(W_n−1= −k−X₁|X₁)]. Therefore

E



 1 ζ_k−1

ζ_k

X

i=1

f(X_i)



 = X

n>2

1 n−1E

" _n X

i=1

f(X_i)1ζk=n

#

= E



kf(X₁)X

n>2

1

n−1P(W_n−1= −k−X₁)





= E

f(X₁) k k+X₁

where we have used Proposition3.6for the last equality. This completes the proof.

3.3 Application to stable spectrally positive Lévy processes

We will now pass the identity of Proposition 3.8 through the scaling limit to obtain an identity concerning stable spectrally positive Lévy processes.

As in Section2.1, we let (Y_t)_t>0 be anα-stable spectrally positive Lévy process with Y₀ =0 normalised so thatE

e^−λY1

=e^λα for everyλ>0. Fors >0 we set∆Y_s =Y_s−Y_s−

wih the convention∆Y₀=0. Finally, set

τ₁ =inf{t >0:Y_t<−1}. Recall that the Lévy measure ofYis

Π(dr) = α(α−1) Γ(2−α) · 1

r^1+α1r>0dr.

Proposition 3.9(Chen, Curien & Maillard [4]). For every measurable functionf:R+ →R+

withf(0) =0, we have

E



 1 τ₁

X

s6τ₁

f(∆s)



= Z_∞

0

f(x)

1+x Π(dx).

Proof. By linearity and standard approximation techniques, it is enough to establish the identity withfof the formf(x) =1x>awith fixeda >0.

Choose(W_n=X₁+· · ·+X_n)such thatW₀=0,P(W₁>−1) =1,E[W₁ =0]and P(W₁>n) _n→∞∼ c

n^1+α withc= ^α(α−1)_Γ(2−α). Then, as in Section2.1, the convergence

W_[n^α_t]

n :t>0

(d)

n→−→∞ Y (8)

holds in distribution inD(R+,R)asn→∞. In addition, by (6), we have W_[n^α_t]

n :06t6 ζ_n n^α

(d)

n→−→∞ (Y_t:06t6τ₁). Therefore

1

ζn−1 n^α

ζn

X

i=1

f X_i

n

(d) n→∞−→

1 τ₁

X

s6τ₁

f(∆_s). (9)

We now check that the expectations converge as well. First, by Proposition3.8, E

"

1

ζn−1 n^α

ζn

X

i=1

f X_i

n #

=n^1+αE

f X₁

n

1 n+X₁

.

Recall that [x]denotes the integer part of x. To estimate the right-hand side, using the explicit expression off, write

n^αE

f X₁

n

n n+X₁

= X∞ k=an

P(X₁=k) n^α+1 n+k

= Z_∞

[an]

dsP(X₁ = [s]) n^α+1 n+ [s]

= Z_∞

[an]/n

duP(X₁ = [un]) n^α+1 1+^[un]_n . Thefore, by dominated convergence,

n^αE

f X₁

n

k k+X₁

n→−→∞

Z_∞

a

1

1+u· c

u^1+αdu= Z_∞

0

f(u)

1+uΠ(du).

An extension of Proposition3.8and similar arguments show that E



 1

ζn−1 n^α

ζn

X

i=1

f X_i

n

!²



is bounded (and actually converges) asn→∞(we leave the details to the reader). There- fore, by uniform integrability, the convergence (9) also holds in expectation, and the desired result follows.

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