ON MULTIPLE-PARTICLE CONTINUOUS-TIME RANDOM WALKS

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RANDOM WALKS

PETER BECKER-KERN AND HANS-PETER SCHEFFLER Received 18 August 2003 and in revised form 24 February 2004

Scaling limits of continuous-time random walks are used in physics to model anomalous diﬀusion in which particles spread at a diﬀerent rate than the classical Brownian motion. In this paper, we characterize the scaling limit of the average of multiple particles, independently moving as a continuous-time random walk. The limit is taken by increas- ing the number of particles and scaling from microscopic to macroscopic view. We show that the limit is independent of the order of these limiting procedures and can also be taken simultaneously in both procedures. Whereas the scaling limit of a single-particle movement has quite an obscure behavior, the multiple-particle analogue has much nicer properties.

1. Introduction

Continuous-time random walks (CTRWs) were introduced in [24] to study random walks on a lattice. They are now used in physics to model a wide variety of phenomena connected with anomalous diffusion (see, e.g., [8,9,10,13,23,29,31,34]). An approach different from CTRWs and fractional calculus to anomalous diffusion processes are the so-called random walks in random environments (see, e.g., [7,12,25] and the literature cited therein). However, this paper focuses on the CTRW approach, but it is an interesting open problem to discuss multiple-particle processes for random walks in random environments too. A CTRW is a random walk subordinated to a renewal process. The random walk increments represent the magnitude of particle jumps, and the renewal epochs represent the times of the particle jumps. CTRWs are also calledrenewal reward processes (see, e.g., [33] where applications are given to queuing theory). The usual assumption is that the CTRW is uncoupled, meaning that the random walk is independent of the subordinating renewal process. In this case, if the time between renewals has finite mean, then the renewal process is asymptotically equivalent to a constant multiple of the time variable, and the CTRW behaves like the original random walk for large time [2,15]. In many physical applications, the waiting time between renewals has infinite mean [30]. In [21], we showed that the scaling limit of an uncoupled CTRW with infinite mean waiting

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time is of the formA(E(t)), whereA(t) is the scaling limit of the underlying random walk andE(t) is the hitting time process for aβ-stable subordinator independent ofA(t).

In this paper, we analyze the limiting behavior of the average over multiple infinite mean waiting time CTRWs in the context of operator self-similarity of stochastic processes. As shown in [21], the limiting process{M(t)}t≥0 of a single uncoupled CTRW has quite an obscure behavior, as it is not an operator-stable process nor it has independent increments (i.i.). It follows from [21, Theorem 4.6] that even for a Brownian motion {A(t)}t≥0, the distribution ofM(t) is not even Gaussian.

We consider the average

Z_n(t)=1 n

n k=1

X^(k)(t) (1.1)

of independently moving particles, each moving as a CTRWX^(k)(t)=S^(k)(N_t^(k)), where S^(k)(n) denotes the random walk andN_t^(k)denotes the renewal process. Now there appear three possible ways of central limiting behavior:

(1) first scale the model from microscopic to macroscopic view and then increase the number of particles withn→ ∞;

(2) first increase the number of particles withn→ ∞and then scale the model from microscopic to macroscopic view;

(3) simultaneously increase the number of particles and scale the model, that is, an- alyzeZn(c)(ct), wheren(c)→ ∞asc→ ∞.

It will turn out that in any case of (1), (2), and (3), we get the same limiting process {M(t)˜ }t≥0 which is operator self-similar with exponentF=βEand an operator-stable process in the sense of Maejima [17] with exponentEand with independent but nonsta- tionary increments.

Since we are interested in operator self-similarity, the appropriate mode of convergence is convergence in distribution of all finite-dimensional marginal distributions, denoted by ^{f .d.}⇒. The fact that the limiting behavior of {Zn(t)}t≥0 is the same in (1), (2), and (3) strongly suggests that the limiting process{M˜(t)}t≥0 is a very robust model for anomalous diﬀusions with much nicer properties than the single-particle CTRW limit considered in [14,21]. In [14], the one-dimensional CTRW limit is considered and, by abuse of language, is called a fractional-stable distribution.

This paper is organized as follows. InSection 2, we define in detail a multiple-particle CTRW and state our basic assumptions necessary for our main results as well as some basic facts on operator-stable processes. InSection 3, our main results are presented together with its proofs. InSection 4, we compare the models for anomalous diﬀusions emerging from the one-particle versus the multiple-particle CTRW limiting processes.

We conclude this paper by discussing an example of a so-called coupled CTRW, that is, a CTRW where the waiting times and the jumps are dependent.

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2. Basic assumptions and preliminary results

The formulation as well as the proofs of our results rely heavily on the multivariable central limit theory laid out in detail in [22]. In the following, we use the notation as well as some of the results of [22] without further citation. See also [5,21] for more detailed references.

(A) The model. For a linear operatorQonR^d, let exp(Q)=_∞

k=0(k!)⁻¹Q^k denote the exponential and fort >0, definet^Q=exp(Qlogt). LetY,Yi,Y_i^(k),i,k≥1, be i.i.d.R^d- valued random vectors which model the particle jumps. Fort≥0, let

S(t)= [t]

j=1

Yj, S^(k)(t)= [t]

j=1

Y_j^(k) (2.1)

so that S(t)=S^(k)(t)=0 for 0≤t <1 and assume that Y belongs to the strict gener- alized domain of attraction of a full operator-stable random vectorA with exponent E∈GL(R^d), where full means thatAis not concentrated on any proper hyperplane of R^d. In summary, there exists a regularly varying norming functionB:R+→GL(R^d) with exponent−E(denoted byB∈RV(−E)), that is,B(λt)B(t)⁻¹→λ⁻^East→ ∞for anyλ >0, such that for anyt >0,

B(n)S(nt)=⇒A(t) asn−→ ∞, (2.2)

whereA(t)=^d tÊAis the Lévy process generated by the operator-stable random vectorA, which is called an operator Lévy motion. Here⇒denotes convergence in distribution and

=d denotes equality in distribution. Note that we have Re(λ)≥1/2 for any eigenvalueλof the exponentE. Moreover, by independence (see [22, Example 11.2.18]), we obtain

B(n)S^(k)(nt)=⇒^{f .d.}A^(k)(t) asn−→ ∞, (2.3) where{A(t)}t≥0,{A^(k)(t)}t≥0,k≥1, are i.i.d. operator-L´evy motions. Especially, [22, Ex- ample 11.2.18] shows that{A(t)}t≥0 is operator self-similar with exponent E, that is, {A(ct)}t≥0

f .d.

= {c^EA(t)}t≥0for anyc >0, where^{f .d.}=denotes equality of all finite-dimensional marginal distributions.

Further, letJ,Ji,J_i^(k)be i.i.d. random variables withJ≥0 almost surely that model the waiting times between successive jumps of the particles. Fort≥0, let

T(s)= [s]

j=1

J_j, T^(k)(s)= [s]

j=1

J^(k)_j , (2.4)

where again T(t)=T^(k)(t)=0 if 0≤t <1 and assume thatJ belongs to the domain of attraction of some β-stable random variableD with 0< β <1. To summarize this, there exists a regularly varying norming functionb:R+→R+with index−1/β, that is,

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b(λt)b(t)⁻¹→λ⁻^1/βast→ ∞for anyλ >0, such that for anys >0,

b(n)T(ns)=⇒D(s) asn−→ ∞, (2.5)

whereD(s)=^d s^1/βD is aβ-stable subordinator. Note that due to 0< β <1, the random variableJnecessarily has infinite mean. Moreover, by independence we obtain

b(n)T^(k)(ns)=⇒^{f .d.}D^(k)(s) asn−→ ∞, (2.6) where{D(s)}s≥0,{D^(k)(s)}t≥0,k≥1, are i.i.d.β-stable subordinators.

Now fork≥1 andt≥0, letN_t^(k)=max{n≥0 :T^(k)(n)≤t}denote the renewal process of the cumulative waiting times and letE^(k)(t)=inf{s≥0 :D^(k)(s)> t}be the cor- responding hitting time process by (2.6). LetNt be i.i.d. asN_t^(k),k≥1, and letE(t) be i.i.d. asE^(k)(t),k≥1. Note that{E(t)}t≥0has nondecreasing sampling paths and by [21, Corollary 3.2] has moments of all orders. Moreover, for some regularly varying function b˜with indexβandc·b(˜b(c))→1 asc→ ∞, we have by [21, Theorem 3.6] that

1

b(c)˜ N_ct^(k)=⇒^{f .d.}E^(k)(t) asc−→ ∞ (2.7)

for anyk≥1.

Now fork≥1 andt≥0, let

X^(k)(t)=S^(k)N_t^(k), X(t)=SNt

(2.8) be CTRWs, each describing the movement of a single-particle. Then{X^(k)(t)}t≥0are i.i.d.

as{X(t)}t≥0. In the following, we assume that{Y,Yi,Y_i^(k),J,Ji,J_i^(k):i,k≥1}are independent so that each CTRW{X^(k)(t)_}_t_≥0 isuncoupled. Let ˜B(c)₌B(˜b(c)). It follows from [21, Theorem 4.2] that for anyk≥1,

B(c)X˜ ^(k)(ct)=⇒^{f .d.}M^(k)(t)=A^(k)E^(k)(t) asc−→ ∞, (2.9) where {M^(k)(t)_}_t_≥0 are i.i.d. as {M(t)_}_t_≥0= {A(E(t))_}_t_≥0, each describing the macroscopic movement of a particle. It is shown in [21, Section 4] that{M^(k)(t)}t≥0is operator self-similar with exponentF=βE, that is,

M^(k)(ct)_t_≥₀ ^{f .d.}=

c^FM^(k)(t)_t_≥₀ (2.10) for anyk≥1 and anyc >0, but neither is operator-stable nor has stationary or independent increments.

(B) Operator-stable processes. In this section, we briefly recall the definition of an operator-stable process and analyze in detail the example of an operator L´evy motion, which is crucial for our main results. We follow the basic definition of [17]. Assume that A is some strictly operator-stable random vector with exponentEand distributionν, that is,νis infinitely divisible andν^t=(t^Eν) for any t >0, whereν^t denotes thet-fold

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convolution power ofν. It follows from [22, Theorem 7.2.1] that the real parts of the eigenvalues ofEare greater than or equal to 1/2.

Definition 2.1. AnR^d-valued stochastic process{Z(t)}t≥0is called anoperator-stable pro- cesswith exponentE, if for any 0< t1<···< tmandm≥1, the random vector (Z(t1),. . ., Z(tm)) is operator-stable with exponentE(m)=diag(E,. . .,E) on (R^d)^m.

This definition generalizes the well-known notion of a Gaussian or symmetricα-stable process. See [27] for details on those processes.

The following example is crucial for our main results: let{A(t)}t≥0be an operator Lévy motion with exponentEand without normal component, that is, the distributionνof A=A(1) has the Lévy representation [a, 0,ϕ], wherea∈R^dandϕdenotes the Lévy measure ofν(see [22, Theorem 3.1.11]). It then follows from the operator stability ofνthat A(t) has distributionν^tand hence by [22, Definition 3.1.23] has Lévy measuret·ϕ. Then by [17], we have that{A(t)}t≥0is an operator-stable process in the sense of Definition2.1.

We now describe the L´evy measure of its finite-dimensional marginal distributions. Let 0=t0< t1<···< tm be arbitrary. Then the random vectorsA(ti)−A(ti−1),i=1,. . .,m, are independent andA(t_i)−A(t_i₋1)=^d A(t_i−t_i₋1) has L´evy measure (t_i−t_i₋1)·ϕ. It fol- lows from the independence of theR^d-valued components that the (R^d)^m-valued random vector

ξm= At1

,At2

−At1

,. . .,Atm

−Atm−1

(2.11)

is operator-stable on (R^d)^mwith exponentE_(m)and has L´evy measure Φ˜t1,...,tm=

m i=1

t_i−t_i₋1

·ϕ_i, (2.12)

with

ϕ_i₌ε0⊗ ··· ⊗ε0⊗ϕ_⊗ε0⊗ ··· ⊗ε0 (2.13) fori=1,. . .,m, whereϕappears in the ith component of the product measure andε0

denotes Dirac measure at the origin. Now let Ψm:R^d^m−→

R^d^m, Ψm

x1,. . .,xm

=

x1,x1+x2,. . .,x1+···+xm

. (2.14)

ThenΨmis linear and invertible and we haveΨm◦tÊ^(m)=tÊ^(m)◦Ψm. Hence (A(t1),A(t2), . . .,A(tm))=Ψm(ξm) is operator-stable on (R^d)^m with exponentE(m)and has Lévy measure

Φt1,...,tm=ΨmΦ˜t1,...,tm

= m i=1

ti−ti−1

·Ψm ϕi

. (2.15)

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3. Main results

In this section, we present our main results together with their proofs. We show that the multiple-particle average CTRW process{Z_n(t)}t≥0 defined in (1.1) will, properly normalized, converge in the =⇒^{f .d.}-sense in all three cases (1), (2), and (3) to the same limiting process{M˜(t)}t≥0. It will turn out that this limiting process is an operator-stable process with exponentEwhich is operator self-similar with exponentF=βE, whereβ andEare as inSection 2. Let f d−ᏸdenote the convergence in distribution of finite- dimensional marginals of the process.

Theorem3.1 (case (1)).

f d−ᏸ−lim

n→∞lim

c→∞Bb(c)n˜ nZ_n(ct)=M(t),˜ (3.1) where{M(t)˜ }t≥0is operator self-similar with exponentF=βE. Moreover,{M(t)˜ }t≥0is an operator-stable process with exponentE. In caseA=A(1)in (2.3) has no normal component, for0< t1<···< t_m, the random vector( ˜M(t1),. . ., ˜M(t_m))has L´evy measure

Ξt1,...,tm=Φ_C

βt^β1,...,Cβt^βm, (3.2)

where Φs1,...,sm is as in (2.15) and C_β =E(E(1)); E denoting expectation. Especially, {M(t)˜ }t≥0has independent increments andM˜(t)−M(s)˜ =^d (Cβ(t^β−s^β))^EAfor0≤s < t.

Theorem3.2 (case (2)).

f d−ᏸ−lim

c→∞lim

n→∞Bb(c)n˜ nZn(ct)=M(t),˜ (3.3) where the limiting process{M˜(t)_}_t_≥0is as inTheorem 3.1.

Theorem3.3 (case (3)). Letn(c)→ ∞asc→ ∞. Then f d−ᏸ−lim

c→∞Bb(c)n(c)˜ n(c)Zn(c)(ct)=M(t),˜ (3.4) where the limiting process{M˜(t)}t≥0is as inTheorem 3.1.

We now give the proofs of our main results together with a technical lemma necessary for the proofs. We start withTheorem 3.1.

Proof ofTheorem 3.1. Fix any 0=t0< t1<···< tm. Let for allimean fori=1,. . .,m. It follows from [21, Theorem 4.2] (see (2.9)) that for allk≥1, we have

B(c)X˜ ^(k)cti

:∀i=⇒

M^(k)ti

:∀i asc−→ ∞. (3.5)

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SinceB∈RV(−E), we knowB(˜b(c)n)B(˜b(c))⁻¹→n⁻^Easc→ ∞. Hence, by independence ink, we conclude that

Bb(c)n˜ nZn

cti

:∀i= n k=1

Bb(c)n˜ Bb(c)˜ ⁻¹B(c)X˜ ^(k)cti

:∀i

=⇒

n k=1

n⁻^EM^(k)ti

:∀i asc−→ ∞.

(3.6)

Now for any Borel setsU_i⊂R^d,i=1,. . .,m, we have by independence of{A^(k)(t)}t≥0and {E^(k)(t)}t≥0for anykthat

P n k=1

M^(k)ti

:∀i∈U1× ··· ×Um

=P n k=1

A^(k)E^(k)t1

∈U1,. . ., n k=1

A^(k)E^(k)t_m∈U_m

=

R^m+

···

R^m+

P n k=1

A^(k)x⁽¹⁾_k ∈U1,. . ., n k=1

A^(k)x^(m)_k ∈Um

×dP_(E⁽¹⁾_(t₁_),...,E⁽¹⁾_(t_m))

x₁⁽¹⁾,. . .,x^(m)₁ ···dP_(E(n)(t1),...,E⁽ⁿ⁾(tm))

x⁽¹⁾_n ,. . .,x^(m)_n

=

R^m+

···

R^m+

PAx⁽¹⁾1 +···+x⁽¹⁾_n ∈U1,. . .,Ax^(m)1 +···+x^(m)_n ∈Um

×dP_(E⁽¹⁾_(t₁_),...,E⁽¹⁾_(t_m))

x₁⁽¹⁾,. . .,x^(m)₁ ···dP_(E⁽ⁿ⁾_(t₁_),...,E⁽ⁿ⁾_(t_m))

x⁽¹⁾_n ,. . .,x^(m)_n

=P A _n

k=1

E^(k)t1

∈U1,. . .,A _n

k=1

E^(k)tm

∈Um

.

(3.7) In order to justify the formula above, we have to show that

P n k=1

A^(k)x⁽¹⁾_k ∈U1,. . ., n k=1

A^(k)x_k^(m)∈Um

=PAx⁽¹⁾₁ +···+x⁽¹⁾_n ∈U1,. . .,Ax^(m)₁ +···+x^(m)_n ∈Um

.

(3.8)

Form=1, we have by independence that P

n k=1

A^(k)x⁽¹⁾_k ∈U1

=PAx⁽¹⁾₁ +···+x⁽¹⁾_n ∈U1

. (3.9)

Note that since the sample paths of {E(t)}t≥0 are nondecreasing, we necessarily have x_i⁽¹⁾≤ ··· ≤x^(m)_i for alli=1,. . .,n. By induction and using the fact that{A^(k)(t)}t≥0has

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stationary and independent increments, we therefore get that

P n k=1

A^(k)x⁽¹⁾_k ∈U1,. . ., n k=1

A^(k)x^(m)_k ∈Um

=P n k=1

A^(k)x⁽¹⁾_k ∈U1,. . ., n k=1

A^(k)x^(m_k ⁻¹⁾∈U_m₋1, n

k=1

A^(k)x_k^(m⁻¹⁾+A^(k)x^(m)_k −A^(k)x_k^(m⁻¹⁾∈Um

=

R^d···

R^dP n k=1

A^(k)x⁽¹⁾_k _∈U1,. . ., n k=1

A^(k)x^(m_k ⁻¹⁾_∈U_m₋1, n

k=1

A^(k)x_k^(m⁻¹⁾+yk

∈Um

×dP_A(x₁^(m)₋_x^(m₁⁻¹⁾₎y1

···dP_A(x^(m)_n ₋_x_n^(m⁻¹⁾₎yn

=

R^d···

R^dPAx₁⁽¹⁾+···+x_n⁽¹⁾∈U1,. . .,Ax₁^(m⁻¹⁾+···+x_n^(m⁻¹⁾∈U_m₋1, Ax₁^(m⁻¹⁾+···+x_n^(m⁻¹⁾+y1+···+y_n∈U_m

×dP_A(x^(m)

1 −x^(m1⁻¹⁾)

y1

···dP_A(x^(m)_n ₋_x_n^(m⁻¹⁾₎yn

=P Ax⁽¹⁾1 +···+x⁽¹⁾_n ∈U1,. . .,Ax^(m1 ⁻¹⁾+···+x^(m_n ⁻¹⁾∈Um−1, A

x₁^(m⁻¹⁾+···+x_n^(m⁻¹⁾+ n k=1

x_k^(m)−x^(m_k ⁻¹⁾∈Um

=PAx1⁽¹⁾+···+x_n⁽¹⁾∈U1,. . .,Ax1^(m)+···+x^(m)_n ∈Um

,

(3.10) proving (3.8).

Therefore, by (3.6) and (3.7), we have shown that Bb(c)n˜ nZn(t)_t_≥₀=⇒^{f .d.} n⁻^EA

_n

k=1

E^(k)(t)

t≥0

(3.11)

asc→ ∞. Since by [21, Corollary 3.2], we haveE(E(t)^γ)=Cβt^βγfor anyγ >0, the weak law of large numbers implies

1 n

n k=1

E^(k)t_i−→EEt_i asn−→ ∞ (3.12)

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in probability for 1≤i≤m. In view of (3.7), we therefore get for any continuity sets Ui⊂R^d,i=1,. . .,m, using that{A(ct)}t≥0

f .d.

= {c^EA(t)}t≥0, P

n k=1

n⁻^EM^(k)t_i:∀i∈U1× ··· ×U_m

=P n⁻^EA _n

k=1

E^(k)t1

∈U1,. . .,n⁻^EA _n

k=1

E^(k)tm

∈Um

=P A 1

n n k=1

E^(k)t1

∈U1,. . .,A 1

n n k=1

E^(k)t_m

∈U_m

−→PAEEt1

∈U1,. . .,AEE(tm

∈Um

(3.13)

asn→ ∞, using the continuity in distribution of (y1,. . .,y_m)→(A(y1),. . .,A(y_m)) and (3.12). Hence (3.1) follows, where

M˜(t)_t_≥₀ ^{f .d.}=

AEE(t)_t_≥₀=

AC_βt^β_t_≥₀. (3.14) Hence {M˜(t)}t≥0 has independent increments with ˜M(t)−M˜(s)=^d (C_β(t^β−s^β))^EA.

Moreover,

c^FM(t)˜ _t_≥₀^{f .d.}=

c^β^EAC_βt^β

t≥0 f .d.

=

AC_βc^βt^β_t_≥₀ ^{f .d.}= M(ct)˜ _t_≥₀, (3.15) proving that{M(t)˜ }t≥0 is operator self-similar with exponentF=βE. Finally, if Ahas no normal component and Lévy measureϕ, by (2.15) the operator-stable random vector (A(s1),. . .,A(s_m)) for 0< s1<···< s_mhas Lévy measureΦs1,...,sm. Hence the Lévy measure of the operator-stable random vector ( ˜M(t1),. . ., ˜M(tm)) with exponentE(m) has Lévy measureΞt1,...,tmas in (3.2). This concludes the proof.

For the proof of Theorems3.2and3.3, we need the following lemma, which might be of independent interest.

Lemma3.4. For anyγ >0, we have E 1

b(c)˜ Nct

γ

−→EE(t)^γ asc−→ ∞. (3.16) Proof. Note that

E 1 b(c)˜ N_ct

γ

=γ _∞

0 x^γ⁻¹P 1

b(c)˜ N_ct≥x

dx (3.17)

and that by [21, Theorem 3.6], we have P

1

˜b(c)N_ct≥x

−→PE(t)≥x asc−→ ∞. (3.18)

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We now show that for some smallδ >0, there exists ac0>0 such that P

1

b(c)˜ Nct≥x

≤e^te⁽⁻^1+δ)x (3.19)

for allc≥c0and allx >0. Then by dominated convergence, we obtain γ

_∞

0 x^γ⁻¹P 1

b(c)˜ N_ct≥x

−→γ _∞

0 x^γ⁻¹PE(t)≥xdx=EE(t)^γ (3.20) asc→ ∞.

It remains to show (3.19). Sinceb(˜b(c))∼1/c asc→ ∞, we havec⁻¹T(˜b(c))⇒Das c→ ∞. In view of [28, Example 24.12], for a suitable choice of the norming functionb in (2.6), we get for the Laplace transformE(e⁻^sD)=exp(−s^β) fors≥0. Hence by the continuity theorem for Laplace transforms, we have for anys≥0 that

Ee⁻^sc⁻¹^T(˜^b(c))−→Ee⁻^sD=e⁻^s^β asc−→ ∞. (3.21) Therefore, fors=1, we obtain

Ee⁻^c⁻¹^J^[˜^b(c)]−→e⁻¹ asc−→ ∞, (3.22)

so for any 0< δ <1, there exists ac0>0 such that

Ee⁻^c⁻¹^J^b(c)^˜ ≤e⁻^1+δ ∀c≥c0. (3.23)

Since{T(n)≤t} = {Nt≥n}, using Markov’s inequality, we have P

1

b(c)˜ Nct≥x

=PNct≥˜b(c)x

=Pc⁻¹Tb(c)x˜ ≤t

=Pexp−c⁻¹Tb(c)x˜ ≥e⁻^t

≤e^tEexp−c⁻¹T˜b(c)x

=e^tEe⁻^c⁻¹^J^b(c)x^˜

≤e^te⁽⁻^1+δ)x

(3.24)

for allc≥c0and allx >0, proving (3.19). This concludes the proof.

Proof ofTheorem 3.2. Again, fix any 0< t1<···< tmand note that by (2.3) and independence, we have for allk≥1 that

B(n)S^(k)nti

:∀i=⇒

A^(k)ti

:∀i asn−→ ∞. (3.25)

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Similar to the proof of (3.7) and (3.8), we have for any Borel setsU_i⊂R^dthat

P n k=1

X^(k)t_i:∀i∈U1× ··· ×U_m

=P n k=1

S^(k)N_t^(k)₁ ∈U1,. . ., n k=1

S^(k)N_t^(k)_m ∈Um

=P S _n

k=1

N_t^(k)₁

∈U1,. . .,S _n

k=1

N_t^(k)_m

∈U_m

.

(3.26)

Using (3.25), we therefore get for any continuity setsUi⊂R^dthat PBb(c)n˜ nZn

cti

:∀i∈U1× ··· ×Um

=P Bb(c)n˜ S _n

k=1

N_ct^(k)₁

∈U1,. . .,Bb(c)n˜ S _n

k=1

N_ct^(k)_m

∈Um

=

R^m+

PBb(c)n˜ Snx1

∈U1,. . .,Bb(c)n˜ Snx_m∈U_m

×dP_((1/n)ⁿ

k=1Nct1^(k),...,(1/n)ⁿ_k₌1N_ctm^(k))

x1,. . .,xm

.

(3.27)

Note that by [4, (4.3)], together with the regular variation ofB, that is,B(˜b(c)n)B(n)⁻¹→ b(c)˜ ⁻^E, we have

PBb(c)n˜ Snx_i∈U_i:∀i−→Pb(c)˜ ⁻^EAx_i∈U_i:∀i (3.28) asn→ ∞uniformly on compact subsets of{0≤x1≤ ··· ≤x_m}. Moreover, by the weak law of large numbers, we haven⁻¹ⁿ_k₌₁N_ct^(k)_i →E(Ncti) asn→ ∞in probability and hence in distribution. Then by [4, Proposition 4.1], we conclude

PB˜b(c)nnZ_nct_i:∀i∈U1× ··· ×U_m−→Pb(c)˜ ⁻^EAEN_ct_i∈U_i:∀i (3.29) asn→ ∞.

In view of the operator self-similarity of {A(t)}t≥0 with exponentE together with Lemma 3.4, we finally obtain

Pb(c)˜ ⁻^EAENcti

∈Ui:∀i=P

A

E 1 b(c)˜ Ncti

∈Ui:∀i

−→PAEEt_i∈U_i:∀i

(3.30)

asc→ ∞which concludes the proof.