Conditional limit theorems for intermediately subcritical branching processes in random environment

www.imstat.org/aihp 2014, Vol. 50, No. 2, 602–627

DOI:10.1214/12-AIHP526

© Association des Publications de l’Institut Henri Poincaré, 2014

Conditional limit theorems for intermediately subcritical branching processes in random environment ¹

V. I. Afanasyev

, Ch. Böinghoff

, G. Kersting

and V. A. Vatutin

aDepartment of Discrete Mathematics, Steklov Mathematical Institute, 8 Gubkin Street, 119 991 Moscow, Russia.

E-mail:viafan@mail.ru;vatutin@mi.ras.ru

bFachbereich Mathematik, Universität Frankfurt, Fach 187, D-60054 Frankfurt am Main, Germany.

E-mail:boeinghoff@math.uni-frankfurt.de;kersting@math.uni-frankfurt.de Received 13 January 2012; revised 18 September 2012; accepted 24 September 2012

Abstract. For a branching process in random environment it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. For the subcritical regime a kind of phase transition appears. In this paper we study the intermediately subcritical case, which constitutes the borderline within this phase transition. We study the asymptotic behavior of the survival probability. Next the size of the population and the shape of the random environment condi- tioned on non-extinction is examined. Finally we show that conditioned on non-extinction periods of small and large population sizes alternate. This kind of ‘bottleneck’ behavior appears under the annealed approach only in the intermediately subcritical case.

Résumé. Nous considérons un processus de branchement dans un environnement aléatoire dont la distribution des enfants des individus varie aléatoirement de façon indépendante d’une génération à l’autre. Dans le régime sous critique, une transition de phase apparaît. Cet article est consacré à l’étude de la région proche de la transition. Nous étudions le comportement asymptotique de la probabilité de survie ainsi que la taille de la population et la forme de l’environnement aléatoire sous la condition de non- extinction. Nous montrons finalement que conditionnée à la non-extinction, la population alterne des périodes de petite et de grande taille. Ce type de comportement apparaît sous la mesure moyennée uniquement dans ce régime sous critique proche de la transition.

MSC:Primary 60J80; secondary 60K37; 60G50; 60F17

Keywords:Branching process; Random environment; Random walk; Change of measure; Survival probability; Functional limit theorem; Tree

1. Introduction and main results

Branching processes in random environment (BPRE), which have been introduced in [7,28], are a discrete time model for the development of a (discrete) population. You can think of a population of plants having a one-year life-cycle.

In each year, the environment varies in a random fashion, independently from one generation to the other. Given the environment, all individuals reproduce independently according to the same mechanism.

More precisely, letΔbe the space of all probability measures onN0. Equipped with the total variation metric,Δ is a Polish space. LetQbe a random variable taking values inΔ. Then an infinite sequenceΠ =(Q1, Q2, . . .)of i.i.d. copies ofQis called arandom environmentandQn is the (random) offspring distribution of an individual at generationn−1. Let us denote byZnthe number of individuals in generationn. ThenZnis the sum ofZn−1inde- pendent random variables, each of which has distributionQn. A sequence ofN0-valued random variablesZ0, Z1, . . .

1This paper is a part of the research project ‘Branching processes and random walks in random environment’ supported by the German Research Foundation (DFG) and the Russian Foundation of Basic Research (RFBF, Grant DFG-RFBR 08-01-91954).

is called abranching process in the random environment Π, ifZ₀ is independent ofΠ and, given Π the process Z=(Z0, Z1, . . .)is a Markov chain with

L

Zn|Zn−1=z, Π=(q1, q2, . . .)

=q_n^∗z (1.1)

for everyn∈N= {1,2, . . .},z∈N0andq1, q2, . . .∈Δ, whereq^∗zis thez-fold convolution of the measureq. ByP, we will denote the corresponding probability measure on the underlying probability space. For convenience, we assume that the process starts with a single founding ancestor, i.e.Z₀=1 and we exclude the trivial caseP(Q(0)=1)=1 throughout this paper. (We shortenQ({y}), q({y})toQ(y), q(y).)

As it turns out, the fine structure of the offspring distributions is only of little importance and the asymptotic behavior of the BPRE is mainly determined by the properties of logm(Q), where

m(q)= ∞ y=0

yq(y)

is the mean of the offspring distributionq∈Δ. More precisely, to describe the asymptotic behavior of the BPRE, we have to examine the so-calledassociated random walkS=(Sn)n≥0which is defined byS0=0 and the increments X_n=S_n−S_n−1,n≥1 with

X=logm(Q) resp.X_n=logm(Q_n).

Note that the expectation ofZ_n, conditioned on the environment, can be expressed in terms ofS_n, i.e. using (1.1) and the assumptionZ₀=1 a.s.,

E[Zn|Π] = n k=1

m(Qk)=exp(Sn) P-a.s.

and, averaging over the environment, E[Z_n] =E

m(Q)n

, (1.2)

where we assume that the expectation is finite. Using a first moment estimate yieldsP(Z_n>0|Π )=min0≤k≤nP(Z_k>

0|Π )≤min0≤k≤nE[Zk|Π]and thus P(Z_n>0|Π )≤exp

0min≤k≤nS_k P-a.s. (1.3)

This is a classical upper estimate for the survival probability. IfSis an oscillating random walk, the branching process is calledcritical(see [4]) resp.subcriticalifSdrifts to−∞. From the strong law of large numbers, it results that the conditional non-extinction probability atndecays in this case at an exponential rate for almost every environment as n→ ∞.

As was observed by Afanasyev [1] and later independently by Dekking [18] there are three possibilities for the asymptotic behavior of subcritical branching processes. The asymptotic behavior changes with the sign ofE[Xe^X], which we assume to be finite. A BPRE isweaklysubcritical ifE[Xe^X]>0,intermediatelysubcritical ifE[Xe^X] =0 andstronglysubcritical ifE[Xe^X]<0.

The present article is a part of several publications having started with [3–5], in which we try to develop charac- teristic properties of the different cases. The phase transition already becomes visible when looking at the asymptotic survival probability, i.e. there are positive constants θ1, θ2 such thatP(Zn>0)∼θ1P(min(S1, . . . , Sn)≥0)in the critical case and in the weakly subcritical case (see [3,4]), whereasP(Zn>0)∼θ₂E[m(Q)]ⁿin the strongly subcriti- cal case (see [5]). Characteristics of the different regimes become more evident in the typical magnitude ofZ_k,k≤n, conditioned on{Z_n>0}. In the weakly subcritical caseZ_k is very large, unlesskis close to 0 orn[3]. Contrarily in the strongly subcritical caseZ_k stays small for all 0≤k≤n, see [5]. The phase transition between weak and strong subcriticality can also be observed on the level of large deviations, see [8,13,24,25]. For a more detailed comparative discussion we refer the reader to [12].

Here, our goal is to describe the intermediate case. It is located at the borderline between the weakly and strongly subcritical cases. The passage corresponds to the phase transition in the model, thus a particular rich behavior can be expected for the intermediate case. This is reflected in our results below. In particular we shall observe a kind of bottleneck phenomenon, i.e. that conditioned on survival, there are times when the branching process is small (bottlenecks), yet very large inbetween. This phenomenon does not occur elsewhere under the annealed approach. In the critical regime under the quenched approach, the branching process conditioned on survival also exhibits periods of large population sizes alternating with bottlenecks (see [31,32] and [33]).

Assumption A1. The processZis intermediately subcritical,i.e.

E Xe^X

=0.

The assumption suggests to change fromP to a measure P: For every n∈N and every bounded, measurable functionϕ:Δⁿ×Nⁿ₀⁺¹→R,Pis given by its expectation

E

ϕ(Q1, . . . , Qn, Z0, . . . , Zn)

=γ⁻ⁿE

ϕ(Q1, . . . , Qn, Z0, . . . , Zn)e^Sn−S0 , with

γ=E e^X

.

(We includeS0in the above expression, because later on we shall also consider cases whereS0 =0.) From (1.2) we obtain

E[Zn] =γⁿ. (1.4)

The assumptionE[Xe^X] =0 translates into E[X] =0.

ThusSbecomes a recurrent random walk underP.

As to the regularity of the distribution ofXwe make the following assumptions.

Assumption A2. The distribution ofXhas finite variance with respect toPor(more generally)belongs to the domain of attraction of some stable law with indexα∈(1,2].It is non-lattice.

SinceE[X] =0 this means that there is an increasing sequence of positive numbers an=n^1/αn

with a slowly varying sequence1, 2, . . .such that forn→ ∞ P

1 an

S_n∈dx

→s(x)dx

weakly, wheres(x)denotes the density of the limiting stable law. In the case of finite varianceσ²=E[X²]<∞, the slowly varying sequence is constant, i.e._n=σ.

Note that due to the change of measureX⁻always has finite variance and an infinite variance may only arise from X⁺. In case ofα <2 this is the so-called spectrally positive case ([11], Section 8.2.9).

Our last assumption on the environment concerns the standardized truncated second moment ofQ, ζ (a)= 1

m(Q)² ∞ y=a

y²Q(y), a∈N.

Assumption A3. For someε >0and somea∈N E

log⁺ζ (a)α+ε

<∞, wherelog⁺x=log(x∨1).

See [4] for examples where this assumption is fulfilled for anyα∈(1,2]. For binary branching processes in random environment (where individuals have either two children or none)ζ (3)=0, and for cases whereQis a.s. a Poisson distributionζ (2)≤2 or a.s. a geometric distributionζ (2)≤4.

The following theorem has been obtained under quite stronger assumptions in [1,21,30]. Let τ_n=min{k≤n|S_k≤S₀, S₁, . . . , S_n}

be the moment, whenSktakes its minimum withinS0toSnfor the first time.

Theorem 1.1. Under AssumptionsA1toA3,there is a constant0< θ <∞such that asn→ ∞ P(Z_n>0)∼θ γⁿP(τn=n).

In this form the result holds in the strongly subcritical case too [22], however it differs from the corresponding result in the weakly subcritical case [3]. Along the way of proving the subsequent results we also obtain a proof of the above theorem (see the end of Lemma3.3). SinceP(τn=n)∼1/bnwith

b_n=n^1−α−1_n

for some slowly varying sequence(_n)(see Lemma2.2below), it follows P(Zn>0)∼θγⁿ

b_n.

Ifσ²<∞, thenl_nis constant (see [11], Theorems 8.9.12/8.9.13). The next theorem deals with the branching process conditioned on survival at timen.

Theorem 1.2. Under AssumptionsA1toA3the distribution ofZ_nconditioned on the eventZ_n>0converges weakly to a probability distribution onN.Also for everyβ <1the sequenceE[Z^β_n|Z_n>0]is bounded.

Forβ=1 this statement is no longer true, sinceE[Z_n] =γⁿfrom (1.4) and consequentlyE[Z_n|Z_n>0] → ∞for n→ ∞.

The next theorem captures the typical appearance of the random environment, when conditioned on survival. Let Sⁿbe the stochastic process with paths in the Skorohod spaceD[0,1]of càdlàg functions on[0,1]given by

S_tⁿ=Snt, 0≤t≤1.

We agree on the conventionS_nt=S_nt, which we use correspondingly forZ_nt, τ_nt.

Also letL^∗denote a process, which can be understood as a Lévy-process on[0,1]conditioned to attain its mini- mum at timet=1. Formally we will define it in Section2in such a way that(L^∗₁−L^∗_{(1−t )−})_0≤t≤1is a Lévy-meander, as introduced in [16]. IfE[X²]<∞, this is the dual process of a Brownian meander.

Theorem 1.3. Assume AssumptionsA1toA3.Then,asn→ ∞,the distribution ofn−τ_nconditioned on the event Z_n>0converges to a probability distributionponN0and

1

a_nSⁿZ_n>0 d

→L^∗

in the Skorohod spaceD[0,1].Also both quantities are asymptotically independent,namely for every bounded con- tinuousϕ:D[0,1] →Rand everyB⊂N0

E

ϕ 1

a_nSⁿ

;n−τ_n∈BZ_n>0

→E ϕ

L^∗ p(B).

In the strongly subcritical case,n−τ_n also converges to a probability distribution on N0 (consequence of [5], Theorem 1.3). However, this statement is not true for critical (see [4], Theorem 1.4) or weakly subcritical BPREs (see [3], Theorem 1.1 and its proof). The limitL^∗only appears in the intermediate case.

The last theorem characterizes the typical behavior ofZ, conditioned on survival. For a partial result see Theorem 1 in [2]. Recall thatτ_nt is the moment whenS₀, . . . , S_nttakes its minimum.

Theorem 1.4. Let0< t1<· · ·< tr<1.Fori=1, . . . , r let μ(i)=min

j≤i: inf

t≤t_jL^∗_t =inf

t≤t_iL^∗_t

.

Then under AssumptionsA1toA3there are i.i.d.random variablesV₁, . . . , V_r with values inNand independent of L^∗such that

(Zτ_nt1, . . . , Zτ_ntr)|Zn>0 d

→(Vμ(1), . . . , Vμ(r))

asn→ ∞.Also there are i.i.d.strictly positive random variablesW1, . . . , Wr independent ofL^∗such that Znt₁

e^{Snt1−Sτnt1}

, . . . , Znt_r

e^{Sntr−Sτntr}

Z_n>0 d

→(W_μ(1), . . . , W_μ(r)) asn→ ∞.

Note that this theorem cannot be generalized to a functional limit result in Skorohod space. The limiting process would consist of paths which are constant within excursions between descending ladder points of the processL and change independently from one excursion to the next. However, such a process is not càdlàg at these ladder points.

The content of Theorems1.2to1.4may be understood as follows: As in Theorem1.2one expects that the popu- lation size is small at timen, conditioned on{Z_n>0}. This requires thatS_nis not much larger than min(S0, . . . , S_n), because otherwise the population would grow again at the end. Indeed this is confirmed by Theorem1.3. Similarly one expects that the population size is small in decreasing ladder points beforen, which is stated in the first part of Theo- rem1.4. On the other hand within upward excursions between such points of minimum, the population development is unaffected of the condition{Z_n>0}.

More precisely forr=1 andt₁=tTheorem1.4says the following: At timeτ_nt the population consists only of few individuals, whereas at timentit is large, namely of order e^Snt−Sτnt-many individuals, which for everyε >0 is bigger than e^δan with probability 1−ε, ifδ >0 is small enough. Thus the minimum of the random walk at timeτ_nt acts as a bottleneck for the population, whereas afterwards the increasing random walk generates an environment, which is favorable for growth.

Moreover: In case ofr=2 eitherτ_nt1< τ_nt2 orτ_nt1 =τ_nt2, which for the limiting processL^∗meansμ(2)=2 or μ(2)=1. The theorem says that in the first situation of two bottlenecks the population sizesZ_τnt

1 andZ_τnt

2 are asymp- totically independent, as well as the sizesZ_nt1 andZ_nt2. In the second situation of one common bottleneck certainly Z_τnt

1 andZ_τnt

2 are equal. Interestingly this is asymptotically true as well forZ_nt1/e^{Snt1−Sτnt1} andZ_nt2/e^{Snt2−Sτnt2}. Here a law of large numbers is at work, in a similar fashion as for supercritical Galton–Watson processes.

Theorems1.3and1.4may lead to the conjecture that(_a¹

nlogZ_nt)_0≤t≤1converges to a Lévy-process, conditioned to take its minimum at the end and reflected at zero. For the finite dimensional distributions this follows from the theorems together with path properties of Lévy-processes. For linear fractional offspring distributions this was already obtained in [2], Theorem 2. The proof of tightness is somewhat involved and given in a separate paper (see [14]).

The proofs rest largely on the fact that the eventZn>0 asymptotically entails thatτntakes a value close ton, as stated in Theorem1.3. Thus it is our strategy to replace the conditioning eventZn>0 by eventsτn=n−m, which are

easier to handle. Here we can build on some random walk theory. For the proof of the last theorem we also make use of constructions oftrees with stemgoing back to Lyons, Perez and Pemantle [26] and Geiger [20] for Galton–Watson processes. They establish a connection between branching processes conditioned to survive and branching processes with immigration.

The paper is organized as follows: In Section2we compile and prove several results on random walks. In Section3 the proofs of the first three theorems are given. Section4deals with trees with stem and Section5contains the proof of our last theorem.

2. Results on random walks

In this section we assemble several auxiliary results on the random walkS. We allow for an arbitrary initial value S₀=x. Then we writePx(·)andEx[·]for the corresponding probabilities and expectations. ThusP=P0.

2.1. Some asymptotic results Let us introduce forn≥1

L_n=min(S1, . . . , S_n), M_n=max(S1, . . . , S_n) and as above forn≥0

τ_n=min

k≤n: S_k=min(0, Ln) .

There is a connection betweenM_nandτ_n, set up by the dual random walk Sˆ_k=S_n−S_n−k, 0≤k≤n.

Namely{τ_n=n} = { ˆM_n<0}withMˆ_n=max(Sˆ₁, . . . ,Sˆ_n)and consequently P(τn=n)=P(Mn<0).

In particularP(τn=n)is decreasing.

Next define the renewal functionsu:R→Randv:R→Rby u(x)=1+^∞

k=1

P(−S_k≤x, M_k<0), x≥0,

v(x)=1+^∞

k=1

P(−S_k> x, L_k>0), x <0, v(0)=E

v(X);X <0 ,

and 0 elsewhere. In particularu(0)=1. It is well-known that 0< v(0)≤1, for details we refer to [15], Appendix B, and [32]. (Our functionv(x)coincides with the functionv(x)in [3] up to a constant.) Alsou(x)andv(−x)are of order O(x)forx→ ∞.

Lemma 2.1. Under AssumptionA2,for everyr >0there exists aκ >0such that E

e^−rSn;L_n≥0

∼κn⁻¹a_n⁻¹ asn→ ∞.

For the proof we refer to Proposition 2.1 in [3].

Lemma 2.2. Under AssumptionA2there are real numbers

b_n=n^1−α−1_n, n≥1

with a sequence(_n)slowly varying at infinity such that for everyx≥0 P(Mn< x)∼v(−x)b⁻_n¹

asn→ ∞.Also there is a constantc >0such that for allx≥0 P(Mn< x)=P₋x(M_n<0)≤cv(−x)b⁻_n¹.

Proof. The corresponding statements for P(Mn ≤x) are well-known. Indeed the first one is contained in Theo- rem 8.9.12 in [11], whereρnow is equal to 1−α⁻¹, since we are in the spectrally positive case (note that the proof therein works for allx≥0 and not only, as stated, for the continuity points ofv).

Forx >0 this proof completely translates toP(Mn< x). Therefrom the casex=0 can be treated as follows:

P(Mn<0)=E

PX₁(Mn−1<0);X1<0

=b_n−1E

PX₁(Mn−1<0)

bn−1 ;X₁<0

.

As for everyx >0,PX1(M_n<0)≤PX1(M_n< x), applying the bound forPX1(M_n< x)yields thatb⁻_n−¹₁PX1(M_n<0) is bounded by an integrable function. Thus from dominated convergence and fromb_n∼b_n−1we get

P(Mn<0)∼b_nE

v(X₁);X₁<0 .

Now from Eq. (2.1) below the conditional expectation in the right-hand side is equal tov(0), as defined above, which gives the claim.

The second statement is obtained just as in Lemma 2.1 in [4].

2.2. The probability measuresP⁺andP⁻

The fundamental properties ofu, vare the identities E

u(x+X);X+x≥0

=u(x), x≥0,

(2.1) E

v(x+X);X+x <0

=v(x), x≤0,

which hold for every oscillating random walk (see e.g. [32]). It follows that uandvgive rise to further probability measuresP⁺andP⁻. The construction procedure is standard and explained forP⁺ in detail in [4,10]. We shortly summarize it below.

Consider the filtrationF=(Fn)_n≥0, whereFn=σ (Q₁, . . . , Q_n, Z₀, . . . , Z_n). ThusS is adapted toFandX_n+1 (as well asQ_n+1) is independent otFnfor alln≥0. Then for every bounded,Fn-measurable random variableR_n

E⁺_x[R_n] = 1 u(x)Ex

R_nu(S_n);L_n≥0

, x≥0, E⁻_x[Rn] = 1

v(x)Ex

Rnv(Sn);Mn<0

, x≤0.

These are Doob’s transforms from the theory of Markov chains. Shortly speakingP⁺_x andP⁻_x correspond to condi- tioning the random walkSnot to enter(−∞,0)and[0,∞)respectively.

The following lemma is taken from [4,10].

Lemma 2.3. Assume Assumption A2 and let U₁, U₂, . . . be a sequence of uniformly bounded random variables, adapted to the filtrationF.IfU_n→U_∞P⁺-a.s.for some limiting random variableU_∞,then asn→ ∞

E[Un|Ln≥0] →E⁺[U_∞].

Similarly,ifU_n→U_∞P⁻-a.s.,then asn→ ∞ E[U_n|M_n<0] →E⁻[U_∞].

The first part coincides with Lemma 2.5 from [4]. The proof of the second part follows exactly the same lines using Lemma2.2.

2.3. Two functional limit results

Because of AssumptionA2there exists a Lévy-processL=(L_t)_t≥0such that the processesSⁿ=(_a¹

nS_nt)_0≤t≤1con- verge in distribution toLin the Skorohod spaceD[0,1]. LetL⁻=(L⁻_t )_0≤t≤1denote the corresponding non-positive Lévy-meander. This is the process(L_t)_0≤t≤1, conditioned on the event sup_t≤1L_t≤0 (see [9] and [16]).

Lemma 2.4. Under AssumptionsA1andA2for everyx≥0andn→ ∞ 1

an

SⁿM_n<−x d

→L⁻

in the Skorohod spaceD[0,1].

The proof follows exactly the same arguments as the proof of Lemma 2.3 in [4], i.e. using the suitably adapted decomposition (2.10) therein and [19].

FromL⁻we obtain the processL^∗as follows. LetΛ:D[0,1] →D[0,1]be the mappingg→ ˆggiven by ˆ

g(t)=g(1)−g(s−), 0≤t≤1, withs=1−t

and g(0−)=0. Λ is a continuous mapping and Λ⁻¹=Λ. Note that Λ maps the subset D⁻= {g∈D[0,1]:

sup_t≥εg(t) <0 for allε >0}onto the setD^∗= {g∈D[0,1]: infs≤1−εg(s) > g(1)for allε >0}.

Now let L^∗=Λ

L⁻ .

SinceL⁻∈D⁻a.s. it follows thatL^∗∈D^∗a.s. This means thatL^∗takes its infimum at the end a.s.L^∗may be viewed as the process(L_t)_0≤t≤1, conditioned to attain its infimum att=1. This becomes clear from the following result.

Lemma 2.5. Under AssumptionsA1andA2,forn→ ∞ 1

a_nSⁿτ_n=n d

→L^∗

inD[0,1].

Proof. In Lemma2.4, we may replaceSⁿby the processTⁿgiven byT_tⁿ=S_tⁿ_+1/n fort≤1−_n¹ andT_tⁿ=S₁ⁿfor 1−¹_n< t≤1. It follows that

1 a_nΛ

TⁿM_n<0 d

→L^∗.

NowΛ(Tⁿ)is obtained fromSⁿ, if we just interchange the jumps inSⁿ fromX₁, . . . , X_ntoX_n. . . , X₁. This corre- sponds to proceeding to the dual random walk, and it follows

1

a_nSⁿτ_n=n d

→L^∗.

This is the claim.

We end this section by some remarks on the distribution ofL^∗₁. First L₁ has a stable distribution, thus it has a density with respect to Lebesgue measure and is unbounded from below. Since we are in the spectrally positive case,Lhas no negative jumps a.s. Therefore we may use fluctuation theory for the processL^↓, which is the Lévy- process, conditioned to take values in(−∞,0], see [9], Section VII.3. From Corollary 16 therein it follows thatL^↓₁ has a density and is unbounded from below, too. As stated in [16] (see also [15]) the distributions ofL^↓₁ andL^∗₁are mutually absolutely continuous, therefore also the distributionνofL^∗₁has a density and is not concentrated on some compact interval.

2.4. Further limit results LetQ_j=Q1forj ≤0.

Lemma 2.6. Under AssumptionsA1andA2form≥0, k≥1,forn→ ∞the distribution of

(Qτ_n+1, . . . , Qτ_n+k), (Qτ_n, . . . , Qτ_n−k+1),(S_τn, S_n−m) a_n

converges weakly to a probability measureμ⁺_k ⊗μ⁻_k ⊗μ,whereμ⁺_k,μ⁻_k are the distributions of(Q1, . . . , Qk)under the probability measuresP⁺,P⁻andμis a non-degenerate probability measure onR²,i.e.the measure is not a Dirac measure.

Proof. Let forr≥0

Q⁺(r)=(Q_r+1, . . . , Q_r+k), Q⁻(r)=(Q_r, . . . , Q_r−k+1).

Letφ₁, φ₂:Δ^k→Rbe bounded functions andφ₃, φ₄:R→Rbe bounded continuous functions. A decomposition with respect toτ_nyields

E

φ₁

Q⁻(τ_n) φ₂

Q⁺(τ_n) φ₃

S_τn an

φ₄

S_n−m−S_τn an

= n r=0

E

φ₁ Q⁻(r)

φ₂ Q⁺(r)

φ₃ S_r

a_n

φ₄

S_n−m−S_r a_n

;τ_n=r

. (2.2)

LettingLr,n=min(Sr+1, . . . , Sn)−Sr and using duality we get forr > k E

φ₁

Q⁻(r) φ₂

Q⁺(r) φ₃

S_r a_n

φ₄

S_n−m−S_r a_n

;τ_n=r

=E

φ₁ Q⁻(r)

φ₃ Sr

a_n

φ₂ Q⁺(r)

φ₄

Sn−m−Sr

a_n

;τ_r=r, L_r,n≥0

=E

φ1

Q⁻(r) φ3

S_r a_n

;τr=r

E

φ2

Q⁺(0) φ4

S_n−m−r a_n

;Ln−r≥0

=E

φ1

Q⁺(0) φ3

S_r a_n

;M_r<0

E

φ2

Q⁺(0) φ4

S_n−m−r a_n

;L_n−r≥0

.

Moreover forr > k

E[φ₁(Q⁺(0))φ3(S_r/a_n);M_r<0]

P(Mr<0)

=E

φ1

Q⁺(0) ES_k

φ3

S_r−k a_n

Mr−k<0

PSk(M_r−k<0)

P(Mr<0) ;Mk<0

.

Therefore by Lemmas2.2,2.4and dominated convergence, ifrn∼t nfor some 0< t <1, thenar_n/an∼t^1/αand E[φ1(Q⁺(0))φ3(Sr_n/an);Mr_n<0]

P(Mr_n<0)

→E φ₁

Q⁺(0)

v(S_k);M_k<0 E

φ₃

t^1/αL⁻₁

=E⁻ φ1

Q⁺(0) E

φ3

t^1/αL⁻₁ .

In much the same way, lettingL⁺be the positive Lévy meander and using Lemma 2.3 from [4], it follows that E[φ₂(Q⁺(0))φ4(S_n−m−rn/a_n);L_n−rn≥0]

P(Ln−rn≥0)

→E⁺ φ2

Q⁺(0) E

φ4

(1−t )^1/αL⁺₁ .

SinceP(Mr_n<0)P(Ln−r_n≥0)=P(τn=r_n), we obtain forr_n∼t nand 0< t <1 E

φ₁

Q⁻(r_n) φ₂

Q⁺(r_n) φ₃

S_rn a_n

φ₄

S_n−m−S_rn a_n

τ_n=r_n

→E⁻

φ₁(Q₁, . . . , Q_k) E⁺

φ₂(Q₁, . . . , Q_k) E

φ₃

t^1/αL⁻₁ E

φ₄

(1−t )^1/αL⁺₁ .

Now in view of AssumptionA2, the generalized arcsine law (see [11]) is valid forτ_n, i.e.τ_n/nis convergent in distribution to a Beta-distribution with a density, which we denote byg(t)dt. Therefore it follows from (2.2) that

E

φ₁

Q⁻(τ_n) φ₂

Q⁺(τ_n) φ₃

Sτ_n

a_n

φ₄

Sn−m−Sτ_n

a_n

→E⁻

φ1(Q1, . . . , Qk) E⁺

φ2(Q1, . . . , Qk)

× 1

0

E φ₃

t^1/αL⁻₁ E

φ₄

(1−t )^1/αL⁺₁ g(t)dt.

This gives the claim.

Next let 0=t₀< t₁<· · ·< t_r< t_r+1=1 and for 1≤i≤r

σ_i,n=min{k: nt_i−1≤k≤nt_i, S_k≤S_j for allnt_i−1≤j≤nt_i} (2.3) be the first moment, whenS_ktakes its minimum betweennt_i−1andnt_i.

Lemma 2.7. Letm≥0andk, r≥1.Then under AssumptionsA1,A2,given the eventτ_n−m=n−m,the random elements inΔ^2k

Q⁽ⁱ⁾=

(Q_σi,n+1, . . . , Q_σi,n+k), (Q_σi,n, . . . , Q_σi,n−k+1)

, i=1, . . . , r,

are forn→ ∞asymptotically independent with asymptotic distributionμ⁺_k ⊗μ⁻_k.Also,givenτ_n−m=n−m,they are asymptotically independent from the random vector

1 an

(S_σ1,n, S_nt1, . . . , S_σr,n, S_ntr).

Proof. Recall from above that, givenτ_n=n, the distribution of _a¹

nS_nis weakly convergent to a probability measure νonR⁻, the distribution ofL^∗₁, which possesses a density and is not concentrated on a compact interval.

Let

σ_r+1,n=min{k: nt_r≤k≤n−m, S_k≤S_jfor allnt_r ≤j≤n−m} and fori≤r

U_i= 1 an

(S_σi,n−S_nti−1), V_i= 1 an

(S_nti−S_nti−1), V_r+1= 1 an

(S_n−m−S_ntr)

and Wi =(Ui, Vi). Since (an)is regularly varying, from the last lemma and from our assumptions on indepen- dence it follows that the random variables Q⁽¹⁾, . . . , Q^(r), W1, . . . , Wr, Vr+1 are asymptotically independent. The eventσr+1,n=n−mis independent ofQ⁽¹⁾, . . . , Q^(r), W1, . . . , Wr and thus only affectsVr+1. Thus from Lemma2.6

Q⁽¹⁾, . . . , Q^(r), W₁, . . . , W_r, V_r+1|σ_r+1,n=n−m d

→

μ⁺_k ⊗μ⁻_k⊗r

μ₁⊗ · · · ⊗μ_r⊗ν,

where the probability measuresμ_ialso depend ont_i−t_i−1. If a Borel setA⊂R^2r+1satisfiesμ₁⊗· · ·⊗μ_r⊗ν(A) >0 andμ₁⊗ · · · ⊗μ_r⊗ν(∂A)=0, it follows

Q⁽¹⁾, . . . , Q^(r)|(W1, . . . , Wr, Vr+1)∈A, σr+1,n=n−m d

→

μ⁺_k ⊗μ⁻_k⊗r

.

We apply this result toAof the formA=B∩C, where the Borel setBsatisfies the same conditions asA, and

C=

(u₁, v₁, . . . , u_r, v_r, v_r+1): u_j>

r+1

i=j

v_i forj≤r

.

Sinceνis not concentrated on a compact set,μ₁⊗ · · · ⊗μ_r⊗ν(C) >0, and becauseνhas a density,μ₁⊗ · · · ⊗μ_r⊗ ν(∂C)=0. As

{τ_n−m=n−m} = {S_σj,n> S_n−mforj≤r, σ_r+1,n=n−m}

=

(W₁, . . . , W_r, V_r+1)∈C, σ_r+1,n=n−m we obtain

Q⁽¹⁾, . . . , Q^(r)|(W₁, . . . , W_r, V_r+1)∈B, τ_n−m=n−m d

→

μ⁺_k ⊗μ⁻_k⊗r

.

The choiceB=R^2r+1gives the asymptotic distribution of(Q⁽¹⁾, . . . , Q^(r)). Since(Sσ_1,n, Snt₁, . . . , Sσ_r,n, Snt_r)is ob- tained from(W1, . . . , Wr, Vr+1)by linear combinations, also the asymptotic independence follows.

3. Proofs of Theorems1.1to1.3

Define η_k=^∞

y=0

y(y−1)Qk(y) _∞

y=0

yQ_k(y) 2

, k≥1.

Lemma 3.1. Assume AssumptionsA1toA3.Then for allx≥0 ∞

k=0

ηk+1e^−Sk<∞ P⁺_x-a.s.