Moment measures of heavy-tailed renewal point processes: asymptotics and applications

DOI:10.1051/ps/2012010 www.esaim-ps.org

MOMENT MEASURES OF HEAVY-TAILED RENEWAL POINT PROCESSES: ASYMPTOTICS AND APPLICATIONS

Cl´ ement Dombry

¹

and Ingemar Kaj

²

Abstract. We study higher-order moment measures of heavy-tailed renewal models, including a re- newal point process with heavy-tailed inter-renewal distribution and its continuous analog, the occupa- tion measure of a heavy-tailed L´evy subordinator. Our results reveal that the asymptotic structure of such moment measures are given by explicit power-law density functions. The same power-law densities appear naturally as cumulant measures of certain Poisson and Gaussian stochastic integrals. This cor- respondence provides new and extended results regarding the asymptotic fluctuations of heavy-tailed sources under aggregation, and clarifies existing links between renewal models and fractional random processes.

Mathematics Subject Classification. 60K05, 60G22, 60F05.

Received February 21, 2012.

1. Introduction and main results

The topic of study in this work is heavy-tailed renewal models, starting from a renewal point process on the line with an inter-renewal time distribution which is heavy-tailed at infinity. The occupation measure of a L´evy subordinator with heavy-tailed L´evy measure provides a continuous analog. Our analysis shows that the discrete and the continuous models are closely related with respect to the structure of their higher-order moment measures. We show that the moment measures in both cases have the same asymptotic structure and that the limit measures are given by specific density functions with a power-law behavior governed by the inter-renewal tail index. As customary, our approach to the discrete renewal model begins with properties of factorial measures.

The power-law density functions which are recognized to determine the asymptotic behavior of moments in the renewal models also appear naturally as cumulant measures of certain Poisson and Gaussian stochastic integrals. This correspondence helps us to unify and extend results on asymptotic fluctuations of heavy-tailed sources under aggregation, and to clarify the role in this context of fractional Poisson motion [2,5] and of fractional Brownian motion as rescaling limit processes.

Hence we believe that the present paper sheds some new light on the rich literature on traffic models and aggregation of heavy-tailed sources. Popular models include the infinite source Poisson model [10,13], the

Keywords and phrases.Heavy-tailed renewal process, moment measures, fractional Brownian motion, fractional Poisson motion.

1 Laboratoire LMA, Universit´e de Poitiers, T´el´eport 2, BP 30179, 86962 Futuroscope-Chasseneuil Cedex, France.

[email protected]

2 Department of Mathematics, Uppsala University, Box 480 SE 75106 Uppsala, Sweden.[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2013

aggregation of ON/OFF sources [5,13,19], renewal processes [7] or renewal/reward processes [12,14,15]. The interested reader should also refer to the monography on heavy-tailed phenomena [17].

We begin this first section of the paper by introducing a renewal point process on the positive half line with heavy-tailed inter-renewal distribution at infinity. Then we introduce the relevant families of power-law density functions, defined on [0,∞)^k for eachk≥1, and recognize the role of these families of functions within two separate frameworks. Firstly they arise as asymptotic limits of the higher-order moment measures for the renewal model. Secondly, the same families of power-law measures express the cumulants for a class of Poisson stochastic integrals with power-law intensity measure. Closely related are Gaussian stochastic integrals defined by a power-law control measure. Based on these links we establish a limit theorem for renewal point processes under aggregation and scaling, and show that the asymptotic fluctuations are given by Poisson or Gaussian stochastic integrals. These are natural extensions of fractional Poisson motion and fractional Brownian motion, which explains and extends earlier results on scaling limits of heavy-tailed renewal counting processes.

All results relating to higher order moment measures of the renewal model have been collected in Section 2. We investigate systematically the asymptotic behavior of factorial moment measures, moment measures and centered moment measures, leading up to identifying the limiting centered moments in terms of power-law density functions. In Section 3 we apply these findings in order to present the proof of our main result Theorem 1.4, which gives the fluctuation limits for renewal systems under simultaneous aggregation and time scaling.

In Section 4 these results are extended to a continuous version of the discrete renewal model based on the occupation measure of a heavy-tailed L´evy subordinator. We establish that the same power-law density measures as for the discrete renewal model provide the higher-order moment asymptotics and the same Poisson and Gaussian integrals yield the scaling limits under aggregation.

1.1. A discrete renewal model

Consider the renewal sequence (S_n)_n≥0generated by a sequence of independent non-negative random variables (X_k)_k≥1, i.e. S_n =S₀+_n

i=1X_i for n≥1. The inter-renewal times (X_k)_k≥1 are identically distributed with probability measureF and independent ofS₀. We writex∈

R

⁺ →F([0, x]) for the distribution function and assume that the inter-renewal distribution has finite meanμ=_∞

0 F([x,∞)) dx >0. The initial distribution, i.e. the distribution ofS₀, is denoted by π. We will use the notation

P

π,

E

π for the probability measure and expectation of the renewal model with initial distributionπ. Two main cases appear naturally: the pure renewal case whereS₀≡0,i.e. π=δ₀, and the stationary renewal case where S₀follows the equilibrium distribution

π_eq(x) = 1 μ

_x

0

F([s,∞)) ds, x≥0. (1.1)

Probability and expectation are denoted by

P

0 and

E

0 in the pure caseπ = δ₀ and by

P

eq and

E

eq in the stationary caseπ=π_eq.

Recall that a function: (0,∞)→(0,∞) is said to be slowly varying at infinity if for allt >0,(tx)/(x)→1, x→ ∞. Our basic assumption is that the inter-renewal distribution F, in addition to finite expected value μ, has a regularly varying tail with exponent 1 +β, β ∈(0,1), i.e. there exists a slowly varying function such that

F([x,∞))∼x^−(1+β)(x) as x→ ∞. (1.2) Thus,X_k, k≥1, have infinite variance. To study the renewal model (S_n) we will use general ideas of renewal point processes and moment measures, see [4] for a full account of such methods.

The renewal point processξ=

n≥0δ_Snon [0,+∞) is locally finite under

P

π. Fork≥1, thek-fold tensorial product ofξwith itself is the random point measure on [0,+∞)^k given by

ξ^⊗k=

n1,...,nk≥0

δ_(Sn

1,...,S_nk).

MOMENT MEASURES OF HEAVY-TAILED RENEWAL POINT PROCESSES: ASYMPTOTICS AND APPLICATIONS

For k≥1, we consider [0,+∞)^k endowed with the Borelσ-algebraBk. The kth moment measure M_k^π of ξ is the measure on [0,+∞)^k defined by

M_k^π(A) =

E

π[ξ^⊗k(A)], A∈ Bk. We also define a centered version of thekth moment measure by

M_k^π(A) =

E

π[(ξ−λ₁/μ)^⊗k(A)], A∈ Bk bounded, whereλ₁ is Lebesgue measure on the real line. In particular, fork= 1,

M₁⁰(A) =

E

0[ξ(A)], M₁^eq(A) = 1

μλ₁(A), M₁⁰(A) =

E

0[ξ(A)]− 1

μλ₁(A), M₁^eq(A) = 0.

For the special case of cylinder sets in [0,+∞)^k we have ξ^⊗k(⊗^k

i=1A_i) = k i=1

ξ(A_i), A₁, . . . A_k∈ B1.

For closed intervals A_i = [0, x_i], this is ξ^⊗k(⊗^k

i=1[0, x_i]) =_k

i=1ξ([0, x_i]), which evaluates tensorial products in terms of the renewal counting process

ξ([0, x]) = Card{n≥0; S_n≤x}, x≥0. (1.3) Similarly, our detailed analysis of higher order moment measures will reveal representations in terms of first order moments, that is the familiar renewal measures

U = ∞ n=1

F^∗n, U +δ₀= ∞ n=0

F^∗n,

whereF^∗n is then-fold convolution ofF andF^∗0=δ₀is Dirac mass at 0. Here,x→U([0, x]) is the associated renewal function.

1.2. A collection of measures

To prepare for our analysis of renewal moment measures we introduce four families of measures defined on [0,+∞)^k, k ≥ 1. They are parameterized by the tail index β, 0 < β <1 introduced in (1.2) and defined by their densities with respect to the Lebesgue measureλ_k, λ_k(dx) = dx, x= (x₁, . . . , x_k). The measures will be denotedP_k^π andP_k^π,k≥1, with the tilde-mark indicating a centered measure and the upper indexπan initial distribution. Whenever it is suitable to distinguish the two choices of initial distributions we use upper index 0 for the caseπ=δ₀ and eq for the equilibrium distributionπ=π_eq in (1.1). Fork= 1, define

dP₁⁰

dλ₁(x) =|x|^−β, dP₁⁰

dλ₁(x) =−|x|^−β and dP₁^eq

dλ₁ (x) = dP₁^eq

dλ₁ (x) = 0.

We use here the convention 0^−β=∞so that the densities may be infinite on a set of zero Lebesgue measure.

Fork≥2, letx₍₁₎≤. . .≤x_(k)be the order statistic of the vectorx= (x₁, . . . , x_k)∈[0,+∞)^k and put dP_k⁰

dλ_k(x) =|x₍₁₎|^−β+ k i=2

|x_(i)−x_(i−1)|^−β, dP_k⁰

dλ_k(x) =|x_(k)−x₍₁₎|^−β− |x_(k)|^−β (1.4)

and

dP^eq_k dλ_k (x) =

k i=2

|x_(i)−x_(i−1)|^−β, dP^eq_k

dλ_k (x) =|x_(k)−x₍₁₎|^−β. (1.5) These measures will play a key role in this work as they provide non-obvious links between the renewal models on one hand and Poisson integrals on the other. It is worth noting that they enjoy interesting scaling properties:

for alla >0 andA∈ B_k,

P_k^π(aA) =a^k−βP_k^π(A), P_k^π(aA) =a^k−βP_k^π(A),

valid for either initial condition δ₀ or π_eq. Also, in the stationary case, the limit measures are invariant by translation: for allh≥0 andA∈ Bk

P_k^eq(A+h) =P_k^eq(A), P_k^eq(A+h) =P_k^eq(A).

In the pure case, for all boundedA∈ B_k,

h→+∞lim P_k⁰(A+h) =P_k^eq(A), lim

h→+∞P_k⁰(A+h) =P_k^eq(A).

1.3. Scaling limits for moment measures

We are interested in the asymptotic behavior of the moment measures under rescaling of the space [0,+∞).

Fora >0 andk≥1, let s_a=s^k_a denote the component wise defined scaling mapx→x/afrom [0,+∞)^k onto itself. We define the rescaled point processξs⁻¹_a by

ξs⁻¹_a (A) =ξ(aA), A∈ Bk, and noticeξs⁻¹_a =

n≥0δ_Sn/a. Thekth moment measureM_k^πs⁻¹_a and thekth centered moment measureM_k^πs⁻¹_a ofξs⁻¹_a , are given by

M_k^πs⁻¹_a (A) =M_k^π(aA), M_k^πs⁻¹_a (A) =M_k^π(aA), A∈ Bk.

The asymptotic behavior of the rescaled moment measures asa→ ∞will be given in terms of weak convergence as follows. If mis a signed Radon measure and f a bounded and compactly supported function both defined on [0,∞)⁺ we writem[f] =

f dm, for the corresponding integral. Let Fk=

f : [0,+∞)^k→

R

; compact support , ∂^kf

∂x₁. . . ∂x_k exists and is continuous

, k≥1. (1.6) The technical reason for this choice of test functions is the use of an integration by parts formula (see Lem.2.1 below) and the fact thatf^⊗k ∈ Fk whenf ∈ F1. If (m_n)_n≥1 andm are signed Radon measures on [0,+∞)^k, we say that (m_n)_n≥1 Fk-converges tom, denotedm_n−→^Fk m, if and only if

m_n[f]→m[f], n→ ∞, f ∈ Fk. (1.7)

Our main result for the discrete renewal model is the following limit theorem, which identifies the asymptotic form ofk-moment measures expressed in terms of the power law measuresP_k^π andP_k^π.

Theorem 1.1. Suppose that assumption (1.2)is satisfied. Forπ=δ₀ orπ=π_eq andk≥1, M_k^πs⁻¹_a −(a/μ)^kλ_k

a^k−β(a)

Fk

−→ 1

βμ^k+1P_k^π and M_k^πs⁻¹_a a^k−β(a)

Fk

−→ (−1)^k

βμ^k+1P_k^π asa→+∞.

MOMENT MEASURES OF HEAVY-TAILED RENEWAL POINT PROCESSES: ASYMPTOTICS AND APPLICATIONS

1.4. A class of Poisson and Gaussian integrals

We relate the measuresP_k^eqandP_k⁰ introduced in (1.4) to the cumulants of Poisson and Gaussian integrals.

Recall that we have fixed a parameterβ ∈(0,1) and letN(dx,du) be a Poisson random measure on

R

×[0,+∞) with intensity measure

n(dx,du) = dx(β+ 1)u^−β−2du

and let N(dx,du) = N(dx,du)−n(dx,du) denote the corresponding compensated Poisson measure. Once again, we use the convention 0^−β−2=∞and the intensity measure density may be infinite on a set with zero Lebesgue measure. Similarly, letN₊(dx,du) be the Poisson random measure on [0,+∞)×[0,+∞) with intensity n₊(dx,du) = 1_x≥0n(dx,du) and putN₊=N₊−n₊.

It is convenient for our purpose to view each Poisson point (x, u) as an interval [x, x+u] on the real line and the Poisson measureN(dx,du) as a collection of overlapping sessions formed by these intervals. With this in mind we will restrict attention to integrands obtained as the weighted occupation times

g(x, u) =

[0,+∞)1_[x,x+u](y)f(y) dy.

Forf : [0,+∞)→

R

^andk≥1, we define the tensor productf^⊗k : [0,+∞)^k→

R

^by

f^⊗k(x₁, . . . , x_k) = k i=1

f(x_i).

Proposition 1.2. For any measurable and bounded function f : [0,+∞) →

R

with compact support, the Poisson integrals

J_β^eq[f] =

R×[0,+∞)

[0,+∞)1_[x,x+u](y)f(y) dyN(dx,du), J_β⁰[f] =

[0,+∞)×[0,+∞)

[0,+∞)

1_[x,x+u](y)f(y) dyN₊(dx,du), are well defined and have finite cumulants of any order given by

C_k(J_β^eq[f]) = 1

βP_k^eq[f^⊗k], C_k(J_β⁰[f]) = 1

βP_k⁰[f^⊗k], k≥2, (1.8) andC₁(J_β^eq[f]) =C₁(J_β⁰[f]) = 0. Furthermore, the distributions of J_β^eq[f]andJ_β⁰[f]are uniquely determined by their sequence of cumulants.

We now define the Gaussian counterpart of these Poisson integrals, seee.g.[18] for background and details.

Let W(dx,du) be a Gaussian random measure on

R

^×[0,+∞) with control measure n(dx,du). Similarly, let W₊(dx,du) be a Gaussian random measure on [0,+∞)×[0,+∞) with control measuren₊(dx,du).

Proposition 1.3. For any measurable and bounded function f : [0,+∞) →

R

with compact support, the Gaussian integrals

G^eq_β [f] =

R×[0,+∞)

[0,+∞)1_[x,x+u](y)f(y) dy W(dx,du), G⁰_β[f] =

[0,+∞)×[0,+∞)

[0,+∞)1_[x,x+u](y)f(y) dy W₊(dx,du), are well defined and have finite cumulants of any order given by

C₂(G^eq_β [f]) = 1

βP₂^eq[f^⊗2], C₂(G⁰_β[f]) = 1

βP₂⁰[f^⊗2] (1.9)

and fork= 1 ork≥3,

C_k(G^eq_β[f]) = 0, C_k(G⁰_β[f]) = 0.

1.5. Scaling limits for superposition of sources

Letξⁱ,i≥1 be i.i.d. copies of the random measureξ. We will consider ξ_a,m= 1

b(a, m) m i=1

(ξⁱs⁻¹_a −M₁^πs⁻¹_a ),

obtained by aggregation of m centered copies ofξ under spatial scaling by s_a, with a suitable normalization sequence b(a, m), and study the asymptotic behavior of ξ_a,m as a → ∞ and m → ∞. To deal with the simultaneous limit, we considera=a_m→+∞as m→+∞.

The two limit regimes of interest to us are known in some applications as regimes of intermediate connection rate and of fast connection rate, respectively, and are hence labeled in the following (ICR) and (FCR). The intermediate scaling condition is given by

m(a)

a^β →μ c^β, 0< c <∞, (ICR)

whereas the fast scaling condition entails

m(a)

a^β →+∞. (FCR)

We consider the random fields{ξ_a,m[f] :f ∈ F₁}and prove convergence of the finite dimensional distributions, which we denote by−→. For^{f idi} f ∈ F1, we define f_c byf_c(x) =cf(cx).

Theorem 1.4.

(1) Consider the rescaling assumption (ICR)with normalization b(a, m) =a.

In the pure caseπ=δ₀,

ξ_a,m[f]−→ −^{f idi} 1

μJ_β⁰[f_c], f ∈ F1; in the stationary caseπ=π_eq,

ξ_a,m[f]−→ −^{f idi} 1

μJ_β^eq[f_c], f ∈ F1.

(2) Under the scaling assumption (FCR)with normalization b(a, m) = (ma^2−β(a))^1/2, in the pure caseπ=δ₀

ξ_a,m[f]−→^{f idi}G⁰_β[f], f ∈ F₁; in the stationary caseπ=π_eq,

ξ_a,m[f]−→^{f idi}G^eq_β [f], f ∈ F₁.

Remark 1.5. The above theorem is related to a limit theorem for renewal counting processes which is the main result in [7]. Indeed, considering the parameterized class of functionsf_t= 1_[0,t],t≥0, we haveξs⁻¹_a [f_t] = ξ([0, at]). Applied to the equilibrium renewal model for which

E

eqξⁱ([0, t])) =t/μ, this yields

ξ_a,m[f] = 1 b(a, m)

m i=1

(ξⁱ([0, at])−at/μ).

MOMENT MEASURES OF HEAVY-TAILED RENEWAL POINT PROCESSES: ASYMPTOTICS AND APPLICATIONS

While 1_[0,t]∈ F/ ₁ the Poisson integralsJ_β^eq[1_[0,t]] andJ_β⁰[1_[0,t]] are still well-defined. PutJ_β^eq(t) =J_β^eq[1_[0,t]] and G^eq_β(t) =G^eq_β[1_[0,t]], t≥0, that is

J_β^eq(t) =

R×[0,+∞)

_t

0 1_[x,x+u](y) dyN(dx,du), and

G^eq_β (t) =

R×[0,+∞)

_t

0 1_[x,x+u](y) dyW(dx,du), t≥0.

According to [7], under assumption (ICR), 1

a m i=1

(ξⁱ([0, at])−at/μ)−→ −^{f idi} 1

μcJ_β^eq(t/c), whereas under assumption (FCR),

1 (m(a)a^2−β)^1/2

m i=1

(ξⁱ([0, at])−at/μ)−→^{f idi}G^eq_β (t), t≥0.

By construction,G^eq_β(t),t≥0, is a Gaussian mean zero process with stationary increments and variance given

by

R×[0,+∞)

_t

0 1_[x,x+u](y) dy 2

n(dx,du) = 1

β(1−β)t^2−β. The normalized processesB_H(t) =

β(1−β)G^eq_β(t) andP_H(t) =

β(1−β)J_β^eq(t) have stationary increments and covariance function

Cov(B_H(t), B_H(s)) = Cov(P_H(t), P_H(s)) = 1

2(t^2H+s^2H− |t−s|^2H).

The relation

E

|B_H(t)−B_H(s)|²=

E

|P_H(t)−P_H(s)|²=|t−s|^2−β

with β <1 together with the Kolmogorov-Centsov criterion implies that there exists a modification of these processes with continuous sample paths. Thus, B_H is a fractional Brownian motion with Hurst index H = 1−β/2 ∈ (1/2,1). By analogy, the mean zero non-Gaussian processP_H has been called fractional Poisson motion, cf. [2,5]. By a refined analysis of higher-order moments it can be shown moreover that P_H and B_H share the same H¨older-regularity: in both cases the paths areγ-H¨older continuous of any orderγ < H, [6].

2. Asymptotics of moment measures

This section is devoted to the proof of Theorem1.1. We begin with some preliminaries.

2.1. Preliminaries

We give a simple criterion for the convergence −→^Fk of signed Radon measures defined in (1.7). For x = (x₁, . . . , x_k)∈[0,+∞)^k, we denote [0, x] = [0, x₁]×. . .×[0, x_k].

Lemma 2.1. Let (m_n)_n≥1 and m be signed Radon measures. Suppose that the following conditions holds: for allx∈[0,+∞)^k,

(i) m_n([0, x])→m([0, x])as n→ ∞;

(ii) sup_u∈[0,x]sup_n≥n0|mn([0, u])|<∞for somen₀≥1.

Then, m_n−→^Fk m asn→ ∞.

Proof. The proof is based on the following standard integration by part formula: for any signed Radon measure mandf ∈ F_k with support included in [0, x],

m[f] =

[0,+∞)^kf dm= (−1)^k

[0,x]

∂^kf

∂x₁. . . ∂x_k(u)m([0, u])du.

Using this, we have, asn→ ∞,

m_n[f] = (−1)^k

[0,x]

∂^kf

∂x₁. . . ∂x_k(u)m_n([0, u])du

−→(−1)^k

[0,x]

∂^kf

∂x₁. . . ∂x_k(u)m([0, u])du=m[f].

The convergence is ensured by Lebesgue’s dominated convergence theorem: condition (i) entails the pointwise

convergence and (ii) the domination condition.

The following simple properties will be useful. A mappingh: [0,+∞)^r→[0,+∞)^lis said to be proper if any compactK⊂[0,+∞)^l has a compact preimageh⁻¹(K)⊂[0,+∞)^r.

Lemma 2.2. Suppose that (m_n)_n≥1 and m are signed Radon measure on [0,+∞)^r such that m_n −→^Fr m as n→ ∞. Then

(1) for all smooth and proper functionsh: [0,+∞)^r→[0,+∞)^l,m_nh⁻¹−→^Fl mh⁻¹ asn→ ∞; (2) for alll≥1,m_n⊗λ_l^F−→^r+lm⊗λ_l asn→ ∞;

Proof. For the first point, it is enough to remark that if f ∈ Fl andhis smooth and proper, thenf h∈ Fr so that

(m_nh⁻¹)[f] =m_n[f h]−→m[f h] = (mh⁻¹)[f] asn→ ∞. For the second point, letf ∈ F_r+land define the function ˜f by

f˜(x₁, . . . , x_r) =

[0,+∞)^lf(x₁, . . . , x_r, x_r+1, . . . , x_r+l) dx_r+1. . .dx_r+l. It is easy to show that ˜f ∈ Frso that

(m_n⊗λ_l)[f] =m_n[ ˜f]−→m[ ˜f] = (m⊗λ_l)[f] as n→ ∞.

2.2. Asymptotics of factorial moment measures

Before considering moment and centered moment measures, we need to consider factorial moment measures that have a simpler structure. Thekth factorial moment measureM_[k]^π,k≥1, is defined by

M_[k]^π(A) =

E

π

δ_{(Sn1,...,Snk)}(A)

, A∈ Bk, (2.1)

where the sum runs over the set ofk-tuple’s (n₁, . . . , n_k) of pairwise distinct non-negative integers, see [4].

To clarify the structure of factorial moment measures for the renewal point process ξ, we introduce the following additional notation. Letθ_k: [0,+∞)^k→[0,+∞)^k be the injective mapping

θ_k(x₁, . . . , x_k) = (x₁, x₁+x₂, . . . , x₁+. . .+x_k),

and letΣ_kdenote the set of permutations of{1, . . . , k}. A permutationσoperates on [0,+∞)^k by permutation of the coordinates:

σ(x₁, . . . , x_k) = (x_σ(1), . . . , x_σ(k)), (x₁, . . . , x_k)∈[0,+∞)^k.

MOMENT MEASURES OF HEAVY-TAILED RENEWAL POINT PROCESSES: ASYMPTOTICS AND APPLICATIONS

Lemma 2.3. For any k≥1, thekth factorial moment measureM_[k]^π satisfies M_[k]^π =

σ∈Σk

(π+π∗U)⊗U^⊗(k−1)

θ_k⁻¹σ⁻¹,

with U =_∞

n=1F^∗n.

Proof. By reindexing the terms in the sum (2.1), M_[k]^π(A) =

E

π

σ∈Σk

0≤n1<...<nk

δ_{(Snσ(1),...,Snσ(k))}(A)

=

E

π

σ∈Σk

0≤n1<...<nk

δ_(Sn

1,...,S_nk)(σ⁻¹(A))

=

σ∈Σk

0≤n1<...<nk

P

π[(S_n1, . . . , S_nk)∈σ⁻¹(A)].

Then, for allσ∈Σ_k, by changing the summation indices according to i₁=n₁≥0,i₂=n₂−n₁≥1, . . . , i_k= n_k−n_k−1≥1,

0≤n1<...<nk

P

π[(S_n1, . . . , S_nk)∈σ⁻¹(A)] =

i1≥0

i2>0,...,ik>0

P

π[(S_n1, S_n2−S_n1, . . . , S_nk−S_nk−1)∈θ⁻¹_k σ⁻¹(A)].

Since the law of (S_n1, S_n2−S_n1, . . . , S_nk−S_nk−1) is equal to (π∗F^∗i1)⊗F^∗i2 ⊗. . .⊗F^∗ik withF^∗0=δ₀, we get

0≤n1<...<nk

P

π[(S_n1, . . . , S_nk)∈σ⁻¹(A)] =

i1≥0

i2>0,...,ik>0

[(π∗F^∗i1)⊗F^∗i2⊗. . .⊗F^∗ik](θ⁻¹_k σ⁻¹A)

= [(π+π∗U)⊗U^⊗(k−1)](θ⁻¹_k σ⁻¹A)

and this proves the desired result.

We are now in position to consider scaled factorial moment measures. At this stage we recall the basic assumption (1.2) that the inter-renewal distribution has a regularly varying decay with tail parameter 1 +β as well as the notion ofFk-convergence introduced in (1.7).

Lemma 2.4.

In the pure case when π=δ₀, we have asa→ ∞, M_[k]⁰ s⁻¹_a −(a/μ)^kλ_k

a^k−β(a)

Fk

−→ 1

βμ^k+1P_k⁰, k≥1.

In the stationary case whenπ=π_eq, asa→ ∞, M_[k]^eqs⁻¹_a −(a/μ)^kλ_k

a^k−β(a)

Fk

−→ 1

βμ^k+1P_k^eq, k≥1.

Proof. Since the measureλ_kθ⁻¹_k is absolutely continuous with respect to λ_k with density 1_{{y1≤...≤yk}}, we have

σ∈Σk

λ_kθ⁻¹_k σ⁻¹=λ_k.

By Lemma2.3,

M_[k]^π s⁻¹_a −(a/μ)^kλ_k

a^k−β(a) = 1

a^k−β(a)(M_[k]^π −μ^−kλ_k)s⁻¹_a

= 1

a^k−β(a)

σ∈Σk

(π+π∗U)⊗U^⊗k−1−μ^−kλ_k

θ⁻¹_k σ⁻¹s⁻¹_a . Using the commutation relationss_aσ=σs_a ands_aθ_k =θ_ks_a, this implies

M_[k]^πs⁻¹_a −(a/μ)^kλ_k

a^k−β(a) =

σ∈Σk

((π+π∗U)⊗U^⊗(k−1))s⁻¹_a −(a/μ)^kλ_k a^k−β(a)

θ_k⁻¹σ⁻¹. (2.2) The further analysis of (2.2) will be carried out separately for the pure and the stationary cases.

We first consider the pure case. For k= 1, we use the classical result for the renewal function U([0, x]) =

E

0ξ([0, x]) with heavy-tailed inter-renewal distribution that satisfies assumption (1.2) (see Thm. 4 in [20]):

U([0, x])−x

μ ∼ (x)x^1−β

μ²β(1−β) as x→ ∞. (2.3)

Fork= 1 we haveM_[1]⁰ =δ₀+U and, forx >0,P₁⁰([0, x]) =_1−β¹ x^1−β. This entails condition (i) of Lemma2.1:

forx >0 and asa→ ∞, M_[1]⁰s⁻¹_a −a/μλ₁

a^1−β(a)

([0, x]) = U([0, ax]) + 1−ax/μ

a^1−β(a) −→ x^1−β

μ²β(1−β) = 1

βμ²P₁⁰([0, x]).

Condition (ii) of Lemma2.1is the property that there existsa₀>0 such that sup

u∈[0,x]sup

a≥a0

U([0, au])−au/μ

a^1−β(a) <∞. (2.4)

The proof of (2.4) relies on the theory of regularly varying functions and is more technical; it is postponed to Appendix A.

Turning to the casek≥2 and considering (2.2) withπ=δ₀, we will prove theFk-convergence of measures (δ₀+U)⊗U^⊗(k−1)

s⁻¹_a −(a/μ)^kλ_k a^k−β(a)

Fk

−→ 1

βμ^k+1Q⁰_k, as a→ ∞, (2.5) where (dQ⁰_k/dλ_k)(x) =_k

i=1x^−β_i . To this end, we note that forx= (x₁, . . . , x_k)∈[0,+∞)^k, Q⁰_k([0, x]) = 1

1−β _k

i=1

x_j k i=1

x^−β_i

and, using (2.3),

((δ₀+U)⊗U^⊗(k−1))([0, ax]) = (1 +U([0, ax₁]) k i=2

U([0, ax_i]) (2.6)

= k i=1

ax_i/μ+(ax_i)(ax_i)^1−β

μ²β(1−β) +o((a)a^1−β)

= (a/μ)^k _k

i=1

x_i 1 + (a)a^−β μβ(1−β)

k i=1

x^−β_i +o((a)a^−β)

= a^k

μ^kλ_k([0, x]) +a^k−β(a)

βμ^k+1 Q⁰_k([0, x]) +o((a)a^k−β),

MOMENT MEASURES OF HEAVY-TAILED RENEWAL POINT PROCESSES: ASYMPTOTICS AND APPLICATIONS

which proves condition (i) of Lemma2.1. Equations (2.4) and (2.6) together imply condition (ii),i.e., sup

u∈[0,x] sup

a≥a0

((δ₀+U)⊗U^⊗(k−1))([0, ax])−(a/μ)^kλ_k([0, x])

a^k−β(a) <∞

and hence the Fk-convergence (2.5) follows from Lemma 2.1. The mapping σθ_k is smooth and proper and so (2.2) and (2.5) together with Lemma2.2entail

M_[k]⁰ s⁻¹_a −(a/μ)^kλ_k a^k−β(a)

Fk

−→ 1 βμ^k+1

σ∈Σk

Q⁰_kθ⁻¹_k σ⁻¹.

It remains to verify that the measure

σ∈ΣkQ⁰_kθ⁻¹_k σ⁻¹equalsP_k⁰. For this we observe thatQ⁰_kθ⁻¹_k is absolutely continuous with respect toλ_k, with density

d(Q⁰_kθ⁻¹_k ) dλ_k (y) =

|y1|^−β+ k i=2

|yi−y_i−1|^−β

1_{{y1≤...≤yk}}.

Hence the measure

σ∈ΣkQ⁰_kθ⁻¹_k σ⁻¹ has the density function

σ∈Σk

|y_σ(1)|^−β+ k i=2

|y_σ(i)−y_σ(i−1)|^−β

1_{{yσ(1)≤...≤yσ(k)}}=|y₍₁₎|^−β+ k i=2

|y_(i)−y_(i−1)|^−β= dP_k⁰ dλ_k(y).

Taking into account the property (2.4) of regularly varying functions in Appendix A, this completes the proof of the lemma for the pure case.

The proof in the stationary case is quite similar and is only given in brief. Fork= 1, M_[1]^eq=π_eq+π_eq∗U =λ₁/μ,

hence

M_[1]^eq−λ₁/μ=P₁^eq= 0.

Fork≥2, equation (2.2) yields M_[k]^eqs⁻¹_a −(a/μ)^kλ_k

a^k−β(a) =

σ∈Σk

(μ⁻¹λ₁⊗U^⊗(k−1))s⁻¹_a −(a/μ)^kλ_k a^k−β(a)

θ⁻¹_k σ⁻¹.

Similarly as in (2.5), we then prove

(μ⁻¹λ₁⊗U^⊗(k−1))s⁻¹_a −(a/μ)^kλ_k a^k−β(a)

Fk

−→ 1 βμ^k+1Q^eq_k with (dQ^eq_k /dλ_k)(x) =_k

i=2x^−β_i . The desired result now follows from the fact that

σ∈Σk

Q^eq_k θ⁻¹_k σ⁻¹=P_k^eq.

2.3. Asymptotics of moment measures

Moment measures can be expressed in terms of factorial moment measures. Trivially, fork= 1,M₁^π=M_[1]^π. Fork= 2 and A₁, A₂∈ B₁:

M₂^π(A₁×A₂) =

E

π

⎡

⎣

n1,n2≥0

δ_(Sn

1,Sn2)(A₁×A₂)

⎤

⎦

=

E

π

⎡

⎣

n1 =n2

δ_(Sn

1,Sn2)(A₁×A₂)

⎤

⎦+

E

π

⎡

⎣

n≥0

δ_(Sn,Sn)(A₁×A₂)

⎤

⎦

=M_[2]^π(A₁×A₂) +M_[1]^π(A₁∩A₂).

Definei₁₂: [0,+∞)→[0,+∞)² given byi₁₂(x) = (x, x). We havei⁻¹₁₂(A₁×A₂) =A₁∩A₂so that M₂^π=M_[2]^π +M_[1]^πi⁻¹_1,2.

Whenk= 3, a similar computation yields

M₃^π(A₁×A₂×A₃) =M_[3]^π(A₁×A₂×A₃) +M_[2]^π((A₁∩A₂)×A₃) +M_[2]^π((A₁∩A₃)×A₂) +M_[2]^π((A₂∩A₃)×A₁) +M_[1]^π(A₁∩A₂∩A₃).

Define the mappings i_12,3(x, y) = (x, x, y), i_13,2(x, y) = (x, y, x), i_23,1(x, y) = (y, x, x) and i₁₂₃(x) = (x, x, x).

Then,

M₃^π =M_[3]^π +M_[2]^πi⁻¹_12,3+M_[2]^πi⁻¹_13,2+M_[2]^πi⁻¹_23,1+M_[1]^πi⁻¹₁₂₃.

In the general casek≥1, the combinatorial relation between moment measures and factorial moment measures can be found in [4], Exercise 5.4.5, page 143. For 1≤j≤k, letPjkbe the set of all partitions of{1, . . . , k}intoj nonempty subsets. An elementT ={S1(T), . . . , S_j(T)} ∈ Pjk formed byjdisjoint subsets labelled in arbitrary order is called aj-partition. The cardinality of the subsetS_i(T) is denoted by|Si(T)|so that_j

i=1|Si(T)|=k.

For anyj-partitionT ∈ Pjk, the injection i_T : [0,+∞)^j→[0,+∞)^k is defined byi_T(x₁, . . . , x_j) = (y₁, . . . , y_k) wherey_p=x_i ifp∈S_i(T). Then

M_k^π= k j=1

T ∈Pjk

M_[j]^πi⁻¹_T . (2.7)

Proof of Theorem 1.1.Moment measures.

To prove Theorem1.1as regards properties of the moment measuresM_k⁰andM_k^eq, we will show that factorial moment measures and moment measures share the same asymptotic behavior. More precisely, using (2.7),

M_k^πs⁻¹_a −(a/μ)^kλ_k

a^k−β(a) =M_[k]^π s⁻¹_a −(a/μ)^kλ_k a^k−β(a) +

k−1

j=1

T ∈Pjk

M_[j]^πi⁻¹_T s⁻¹_a a^k−β(a) ·

We have shown in Lemma2.4that the factorial moments in the first term on the right hand sideFk-converge to the desired limit measures. To complete the proof it suffices to verify that the second summation term vanishes in the limit:

k−1

j=1

T ∈Pjk

M_[j]^πi⁻¹_T s⁻¹_a a^k−β(a)

Fk

−→0. (2.8)

We will establish (2.8) for the pure case π =δ₀. A straightforward adaptation yields the stationary case. By Lemma 2.4, for 1≤j≤k−1,

M_[j]⁰s⁻¹_a (a/μ)^j

Fj

−→λ_j