PERSISTENCE PROBABILITIES FOR STATIONARY INCREMENT PROCESSES

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PERSISTENCE PROBABILITIES FOR STATIONARY INCREMENT PROCESSES

Frank Aurzada, Nadine Guillotin-Plantard, Françoise Pene

To cite this version:

Frank Aurzada, Nadine Guillotin-Plantard, Françoise Pene. PERSISTENCE PROBABILITIES FOR STATIONARY INCREMENT PROCESSES. Stochastic Processes and their Applications, Elsevier, 2018, 128 (5), pp.1750–1771. �hal-01324693�

PERSISTENCE PROBABILITIES FOR STATIONARY INCREMENT PROCESSES

FRANK AURZADA, NADINE GUILLOTIN-PLANTARD, AND FRANC¸ OISE P`ENE

Abstract. We study the persistence probability for processes with stationary increments. Our results apply to a number of examples: sums of stationary correlated random variables whose scaling limit is fractional Brownian motion; random walks in random sceneries; random processes in Brownian scenery; and the Matheron-de Marsily model inZ²with random orientations of the horizontal layers. Using a new approach, strongly related to the study of the range, we obtain an upper bound of optimal order in the general case and improved lower bounds (compared to previous literature) for many processes.

1. Introduction

Persistence concerns the probability that a stochastic process has a long negative excursion. In this paper, we are concerned mainly with discrete-time processes. IfZ = (Zn)n∈Nis a stochastic process, we study the rate of the probability

P

k=1,...,Tmax Z_k ≤1

, asT →+∞.

In many cases of interest, the above probability decreases polynomially, i.e., asT^−θ+o(1), and it is the first goal to find the persistence exponent θ. For a recent overview on this subject, we refer to the survey [3] and for the relevance in theoretical physics we recommend [23, 7].

The purpose of this paper is to analyse the persistence probability for stationary increment processes, i.e. processes such that for any k ∈ N, (Z_n+k−Z_k)_n∈N

=L (Z_n)_n∈N, where ^=L means equality in law. Under rather general assumptions, we prove that

P

k=1,...,Tmax Z_k≤1

≈E

k=1,...,Tmax Z_k

/T asT →+∞,

where ≈ means up to a multiplicative term in T^o(1) (the multiplicative term is bounded by a constant for the upper bound and, for the lower bound, is larger than a function slowly varying at infinity under additional assumptions). We emphasize the fact that we obtain the exact order when the increments are bounded and that we obtain estimates even if increments admit no exponential moment.

Stationary increments are a feature shared by many stochastic processes that are important in theory and applications, and we shall treat here a number of examples. The first one will be sums of correlated sequences whose scaling limit is fractional Brownian motion. This is the natural analog of random walks, for which persistence probabilities have been studied extensively under the notion of fluctuation theory. For sums of correlated sequences, the persistence exponent is shown to be 1−H, where H is the scaling exponent, see Theorem 11 below. As a by- product, we obtain the following improvement for the bounds of the persistence probability of continuous-time fractional Brownian motion (improving [27, 1]).

2000Mathematics Subject Classification. 60F05; 60G52.

Key words and phrases. fractional Brownian motion ; Random walk in random scenery ; persistence . 1

Theorem 1(Fractional Brownian motion). LetH ∈(1/2,1). Let BH be a fractional Brownian motion of Hurst parameter H. Then, there exists c >0 such that

c⁻¹T^−(1−H)(logT)^−2H1 ≤P sup

t∈[0,T]

B_H(t)≤1

!

=O(T^−(1−H)).

The lower bound holds for anyH ∈(0,1).

We remark that the upper bound also holds for H < 1/2 with an additional logarithmic factor (logT)^{(2−H)/H+o(1)}, by [1], but we do not improve this result here. However we obtain the bound

P sup

t∈[0,T]

B_H(t)≤ −1

!

=O(T^−(1−H)) even whenH≤1/2.

A second class of examples is given by random walks in random sceneries (RWRS), and depending on the scaling properties of the random walk and of the environment, respectively, different exponents emerge.

Theorem 2 (Random walk in random scenery). Let d∈N^∗ and let S = (S_n)_n∈N be a random walk on Z^d. If S is recurrent, we assume moreover that (S_n/n^1/α)_n≥1 converges in distribution for some α∈[d,2]. Letξ = (ξ_ℓ)_ℓ∈Zd be a sequence of independent identically distributed random variables, which are centered and admit moments of every order. Assume that S and ξ are independent. Then,

a_nn^−1+o(1) ≤P max

m=1,...,n m

X

k=1

ξ_Sk ≤1

!

=O(a_n/n),

with a_n:=







n^1−2α1 if d= 1< α≤2,

√nlogn if α=d,

√n if S is transient.

Additionally,

• if ξ is Gaussian, then there exists c >0 such that P max

m=1,...,n m

X

k=1

ξ_Sk ≤1

!

≥ca_n/(np logn);

• if ξ is bounded, then there exists c >0 such that P max

m=1,...,n m

X

k=1

ξ_Sk ≤1

!

≥ca_n/n;

• if P(ξ0∈ {−1,0,1}) = 1, then there exists c >0 (specified in Theorem 16) such that P max

m=1,...,n m

X

k=1

ξ_Sk ≤ −1

!

∼ca_n/n, as n→+∞.

For RWRS with integer values, this result is strongly related to the behaviour of the expec- tation of the rangeRn of the RWRS.

A third class of examples is given by the continuous-time analogues to RWRS, that is random process in Brownian scenery ∆_t:=R

RL_t(x)dW(x), with W a two-sided real Brownian motion and Lt(x) the local time (at position x and at time t) of a stable L´evy process with index

α ∈(1,2] (we also consider more general processes as defined in [12], see Section 5 below). In this context, we obtain the following result (improving [11, 12]).

Theorem 3 (Random process in Brownian scenery). Under the above assumptions, there exists c >0 such that

c⁻¹T^−2α1 (logT)^{−(2α−1)(1+α)} ≤P sup

t∈[0,T]

∆_t≤1

!

=O(T^−2α1 ).

The fourth example is the Matheron-de Marsily (MdM) modelM = (Mn)n∈Nwhich has been introduced in [26] (see also [5]) to model fluid transport in a porous stratified medium. M is a nearest neighbour random walk inZ², in which each horizontal line has been oriented either to the right or to the left (both with probability 1/2; for a detailed definition see Section 6). The second coordinate (M_n⁽²⁾)_n∈Nof this model is a (lazy) random walk, but the first one (M_n⁽¹⁾)_n∈N

is more complicated and has some analogy with a random walk in random scenery. For this process, we obtain the following precise estimate.

Theorem 4 (Persistence probability for the MdM model). There exists c >0 such that P

k=1,...,nmax M_k⁽¹⁾≤ −1

∼cn⁻¹⁴, as n→+∞.

This result was conjectured by Redner [28] and Majumdar [24]. We mention that the constant c appearing in Theorem 4 is given by an explicit formula involving the Kesten-Spitzer process (see Theorem 19).

The outline of the paper is as follows. The main general result, Theorem 5, is given in Sec- tion 2. Its proof is split into various intermediate results, outlined and proved also in Section 2.

This general result is then applied to the examples mentioned above: Sums of stationary se- quences are treated in Section 3, random walks in random scenery in Section 4, random processes in Brownian scenery in Section 5, and the Matheron-de Marsily model in Section 6.

2. General results for stationary increment processees

Our main general results are contained in the next theorem, in which, as usual, we write x₋ := max(−x,0). It states that the rate of the persistence probability can be obtained from the scaling of the maximum of the process.

Theorem 5. Let(Z_n)_n∈Nbe a centered process with stationary increments such thatZ₀ := 0and let (an)n∈N be a sequence of positive numbers that is regularly varying with exponent γ ∈ (0,1) such that the following limit exists

B := lim

n→+∞a⁻¹_n E

k=1,...,nmax Z_k

∈(0,∞).

Then

lim sup

n→+∞

n a_nP

ℓ=1,...,nmax Z_ℓ ≤ −1

≤B.

Further, if

(i) There exists a sub-σ-algebra F0 such that, given F0, the increments of Z are positively associated and that their common conditional distribution is independent of F0,

then there exists C >0 such that

∀n≥1, P

ℓ=1,...,nmax Z_ℓ≤1

≤Ca_n n. Finally, if (i) holds or, alternatively,

(ii) There exists p >1 such that k max

k=1,...,nZ_kkp =O(an), then

• If E[(Z₁)^β₋]<∞ for any β > 0 then

∀n≥1, P( max

ℓ=1,...,nZ_ℓ <0)≥a_nn^−1+o(1).

• If there exists ε >0 such that E[e^−εZ1]<∞ then there exists c >0 such that

∀n≥1, P

ℓ=1,...,nmax Z_ℓ <0

≥c an

nlogn.

• If there exists ε >0 such that E[e^ε(Z1)2−]<∞ (e.g. ifZ1 is Gaussian) then

∀n≥1, P

ℓ=1,...,nmax Z_ℓ<0

≥c a_n n√

logn.

• If (Z₁)₋ is bounded, then

∀n≥1, P

ℓ=1,...,nmax Z_ℓ<0

≥can

n .

Theorem 5 will appear as a consequence of Theorem 8 and Proposition 9 (for the upper bounds) and Theorem 10 (for the lower bounds) that are given in the next two subsections.

2.1. Upper bounds for the persistence probability. As explained in the introduction, we consider a stochastic process (Z_n)_n∈Nwith stationary increments. We start with a general upper bound that will be useful in the setup whereP(Z₁∈Z) = 1 and which is strongly related to the study of the range of (Z_n)_n∈^N.

Lemma 6. Let(Z_n)_n∈N be a centered process with stationary increments such that Z₀ := 0and let (a_n)_n∈^N be a sequence of positive real numbers converging to +∞ such that

lim sup

n→+∞ a⁻_n¹E

k=1,...,nmax Z_k− min

k=1,...,nZ_k

≤A.

Then

lim sup

n→+∞

n an

P

ℓ=1,...,nmin |Z_ℓ| ≥1

≤A. (1)

WhenZ1 isZ-valued, we consider T0 := inf{n≥1 : Zn= 0} the first return time ofZ to 0.

LetRn be the range ofZ up to time n, i.e. Rn:= #{Z₀, ..., Z_n}.

Theorem 7. Assume we are in the situation of Lemma 6 and that additionally P(Z₁∈Z) = 1.

Then

lim sup

n→+∞

n

a_nP(T₀> n)≤A. (2)

Further, if (a_n)_n∈^N is a regularly varying sequence with exponent γ ∈(0,1) and if C₋ := lim inf

n→+∞

E[Rn] a_n >0,

and

C₊:= lim sup

n→+∞

E[Rn] an

<+∞, then

0<lim inf

n→+∞

n

a_nP(T₀> n)≤lim sup

n→+∞

n

a_nP(T₀ > n)<∞. (3) Moreover, if the sequence_E

[Rn] an

n≥0 converges to C as n→+∞, then

n→lim+∞

n

a_nP(T₀> n) =γC. (4)

In the particular case whereP(Z₁ ∈ {−1,0,1}) = 1and where (4) holds, we obtain that P

k=1,...,nmax Z_k ≤ −1

= 1

2P(T₀ > n)∼ γ a_n

2n C. (5)

The next fact is the crucial part in the upper bound in the general (i.e. not necessarily Z- valued) case.

Theorem 8. Let (Z_n)_n∈N be a centered process with stationary increments such that Z₀ := 0 and let(a_n)_n∈N be a sequence of positive real numbers such that

lim sup

n→+∞ a⁻_n¹E

k=0,...,nmax Z_k

≤B.

Then

lim sup

n→+∞

n a_nP

ℓ=1,...,nmax Z_ℓ ≤ −1

≤B. (6)

The next technical proposition allows to pass from the boundary −1 in the last fact to the boundary 1.

Proposition 9. Assume that there exists a sub-σ-algebraF0 such that, givenF0, the increments of Z are positively associated and that their common conditional distribution is independent of F0. Then, for any m >0 such that P(Z₁≤ −m)>0,

P( max

ℓ=1,...,nZ_ℓ ≤ −m)≤P( max

ℓ=1,...,nZ_ℓ≤1)≤ P(max_ℓ=1,...,nZ_ℓ≤ −m) (P(Z₁≤ −m))^⌊1/m⌋+2 .

2.2. Lower bounds for the persistence probability. The lower bound is much more com- plicated to establish in the general case and will require additional assumptions.

Theorem 10. Let (Z_n)_n∈N be a centered process with stationary increments such that Z₀ := 0.

Assume that ε >0 and that (a_n)_n∈N is a sequence of positive numbers such that D₁:= lim inf

n→+∞a⁻_n¹E

k=0,...,n+max⌊εn⌋Z_k

>lim sup

n→+∞ a⁻_n¹E

k=0,...,nmax Z_k

=:D₂.

Assume moreover that (b_n)_n∈N is a sequence of positive numbers for which one of the following assumptions holds true:

(i) (Z_n)_n∈^N satisfies the assumption of Proposition 9 and lim sup

n→+∞ nP(−Z₁> b_n)< 1 ε

1−D2

D₁

.

(ii) There exists p >1 such that lim sup

n→+∞ a⁻_n¹k sup

k=0,...,n+⌊εn⌋

Z_kkp(εnP(−Z1> bn))^1/q < D1−D2, where q is the conjugate of p (i.e. q is such that 1/p+ 1/q= 1).

Then

lim inf

n→+∞

nb_n an

P

k=1,...,nmax Z_k<0

>0.

2.3. Proofs of the intermediate results.

Proof of Lemma 6. SinceZ has stationary increments, nP

ℓ=1,...,nmin |Z_ℓ| ≥1

≤

n

X

k=1

P

ℓ=1,...,kmin |Z_ℓ| ≥1

=

n

X

k=1

P

ℓ=1,...,kmin |Z_n−k+ℓ−Z_n−k| ≥1

≤

n

X

k=1

P(∀ℓ= 1, ..., k, ⌊Z_n−k+ℓ⌋ 6=⌊Z_n−k⌋)

≤ E[#{⌊Z₀⌋, . . . ,⌊Z_n⌋}]

≤ E

k=1,...,nmax ⌊Z_k⌋ − min

k=1,...,n⌊Z_k⌋+ 1

≤ E

k=1,...,nmax Z_k− min

k=1,...,nZ_k+ 2

.

Proof of Theorem 7. SinceP(Z₁ ∈Z) = 1, P(T₀ > n) =P

ℓ=1,...,nmin |Z_ℓ| ≥1

and so (2) follows from Lemma 6.

Let us prove (3). Note that sinceZ_ℓ isZ-valued and using stationary increments, we have ERn = E[#{Z₁, . . . , Z_n}]

=

n

X

k=0

P(∀ℓ= 1, . . . , k:Z_n−k+ℓ 6=Z_n−k)

=

n

X

k=0

P(∀ℓ= 1, . . . , k:Z_ℓ6= 0)

=

n

X

k=0

P(T₀ > k). (7)

Since (P(T₀> k))_k is non increasing, for every 0< x <1< y, we have by (7) E[R_⌊yn⌋− Rn]

⌊yn⌋ −n ≤P(T₀> n)≤ E[Rn− R_⌊xn⌋] n− ⌊xn⌋ .

Hence, writingC₋:= lim inf_n→+∞^E[_a^R_n^n] and C₊:= lim sup_n→+∞^E[_a^R_n^n], we obtain y^γC₋−C₊

y−1 ≤lim inf

n→+∞

n

a_nP(T0> n)≤lim sup

n→+∞

n

a_nP(T0 > n)≤ C₊−x^γC₋ 1−x . WhenC₊=C₋=C, (4) is deduced by letting the variables xand y converge to one.

Assume now thatP(Z₁∈ {−1,0,1}) = 1. Observe that by centering, the random variable Z₁ must be symmetric. Therefore,

P

k=1,...,nmax Z_k≤ −1

= P(T₀> n, Z₁<0)

= P(T₀> n, Z₁>0)

= 1

2P(T₀> n).

Now, by observing that

E[Rn] =E

"

sup

k=1,...,n

Z_k− inf

k=1,...,nZ_k+ 1

#

∼C an, relation (5) directly follows from (4).

Proof of Theorem 8. LetZ_n,m^∗ := max_k=n,...,mZ_k. Since the increments ofZ are stationary and Z₀= 0, we deduce that Z_k+1,k+n^∗ −Z_k has the same distribution as Z_1,n^∗ and so

nP(Z_1,n^∗ ≤ −1) ≤

n−1

X

k=0

P(Z_1,n^∗ _−k ≤ −1)

=

n−1

X

k=0

P(Z_k+1,n^∗ ≤Z_k−1)

=

n−1

X

k=0

P(1 +Z_k+1,n^∗ ≤Z_k)

≤ E[Z_0,n^∗ ],

since #{k= 0, ..., n−1 : 1 +Z_k+1,n^∗ ≤Z_k} ≤Z_0,n^∗ −Z_n and E[Z_n] = 0.

Proof of Proposition 9. LetK ∈N be such that Km > 1, so that −(K+ 1)m+ 1 ≤ −m. Let n∈N^∗. Given F0, the increments of (Z_n)_n∈^N are positively associated, hence

P

ℓ=1,...,n+K+1max Z_ℓ ≤ −m F0

≥P

∀k= 1, ..., K+ 1, (Z_k−Z_k−1)≤ −m, max

ℓ=K+2,...,n+K+1(Z_ℓ−ZK+1)≤1 F0

≥

K+1

Y

k=1

P(Z_k−Z_k−1 ≤ −m| F0)

! P

ℓ=K+2,...,n+K+1max (Z_ℓ−Z_K+1)≤1 F0

≥

K+1

Y

k=1

P(Z₁ ≤ −m)

! P

ℓ=K+2,...,n+K+1max (Z_ℓ−Z_K+1)≤1 F0

,

since the conditional distribution ofZ_k−Z_k−1 givenF0 is the distribution ofZ1. So P

ℓ=1,...,n+K+1max Z_ℓ≤ −m

≥ (P(Z₁ ≤ −m))^K+1 P

ℓ=K+2,...,n+K+1max (Z_ℓ−Z_K+1)≤1

≥ (P(Z₁ ≤ −m))^K+1 P

ℓ=1,...,nmax Z_ℓ ≤1

,

since the increments ofZ are stationary.

Proof of Theorem 10. Since the increments ofZ are stationary andZ₀ = 0, we can deduce that P(max_k=1,...,nZ_k<0) =P(Z_k+1,n+k^∗ < Z_k).

Letε >0 and M_n:= #{k= 0, ...,⌊εn⌋ −1 : Z_{k+1,n+⌊εn⌋}^∗ < Z_k}. Note that

⌊εn⌋P(Z_1,n^∗ <0) ≥

⌊εn⌋−1

X

k=0

P

Z_1,^∗_{⌊(1+ε)n⌋−k} <0

=

⌊εn⌋−1

X

k=0

P

Zk+1,⌊(1+ε)n⌋^∗ < Z_k

= E[M_n]. (8)

We setR_Mn+1:= inf{k=⌊εn⌋, ..., n+⌊εn⌋ : Z_k> Z_{k+1,n+⌊εn⌋}^∗ } and

∀i= 1, ..., M_n, R_i := sup{k= 0, ..., R_i+1−1 : Z_{k+1,n+⌊εn⌋}^∗ < Z_k}. Note that Z_R1 =Z_0,n+^∗ _⌊εn⌋ and Z_RMn+1 =Z_⌊^∗_{εn⌋,n+⌊εn⌋}. Hence

Z_0,n+^∗ _⌊εn⌋−Z_⌊^∗_{εn⌋,n+⌊εn⌋} = Z_R1 −Z_RMn+1 =

Mn

X

i=1

(Z_Ri−Z_Ri+1)

≤

Mn

X

i=1

(Z_Ri−Z_Ri+1)

≤ sup

k=1,...,⌊εn⌋

(Z_k−1−Z_k)

! M_n,

and so, forb_n>0, E[M_n] ≥ Eh

M_n1_{supk=1,...,⌊εn⌋(Zk−1−Zk)≤bn}

i

≥ (b_n)⁻¹Eh

(Z_0,n+^∗ _⌊εn⌋−Z_⌊^∗_{εn⌋,n+⌊εn⌋})1_{supk=1,...,⌊εn⌋(Zk−1−Zk)≤bn}

i

≥ (b_n)⁻¹ E

h

Z_0,n+^∗ _⌊εn⌋1_{supk=1,...,⌊εn⌋(Zk−1−Zk)≤bn}

i−E[Z_0,n^∗ ] , sinceE[Z_⌊^∗_{εn⌋,n+⌊εn⌋}] =E[Z_0,n^∗ ].

Assumption (i). Then E[Mn] ≥ (bn)⁻¹ E

"

E[Z_{0,n+⌊εn⌋}^∗ |F0]P sup

k=1,...,⌊εn⌋

(Z_k−1−Z_k)≤bn

F0

!#

−E[Z_0,n^∗ ]

!

≥ (b_n)⁻¹ Eh

E[Z_0,n+^∗ _⌊εn⌋|F0](1−εnP(−Z₁> b_n))i

−E[Z_0,n^∗ ]

= (b_n)⁻¹ Eh

Z_0,n+^∗ _⌊εn⌋i

(1−εnP(−Z₁> b_n))−E[Z_0,n^∗ ]

, (9)

where we used the fact that the increments of Z are positively associated conditionally to F0, and that their common conditional distribution is independent of F0. Let η be such that lim sup_n→+∞nP(−Z1 > bn) < η < ¹_ε

1−^D_D²₁

. For nlarge enough, εnP(−Z1> bn)≤εη. Due to (8) and (9), we obtain

εnP(Z_1,n^∗ <0)≥(b_n)⁻¹E h

(1−εη)Z_{0,n+⌊εn⌋}^∗ −Z_0,n^∗ i . Therefore

lim inf

n→+∞

εn b_n

a_n P(Z_1,n^∗ <0)≥(1−εη)D₁−D₂ >0.

Assumption (ii). Let q be such that 1/p+ 1/q = 1 and letη be such that lim sup

n→+∞ a⁻¹_n kZ_0,n+^∗ _⌊εn⌋kp(εnP(−Z₁> b_n))^1/q < η < D₁−D₂. Then, fornlarge enough,

E

hZ_0,n+^∗ _⌊εn⌋1_{supk=1,...,⌊εn⌋(Zk−1−Zk)>bn}

i

≤ kZ_0,n+^∗ _⌊εn⌋kp P sup

k=1,...,⌊εn⌋

(Z_k−1−Z_k)> b_n

!!¹_q

≤ kZ_{0,n+⌊εn⌋}^∗ kp(εnP(−Z₁ > b_n))^1q ≤ηa_n and so

E[Mn]≥(bn)⁻¹ E

h

Z_{0,n+⌊εn⌋}^∗ −Z_0,n^∗ i

−ηan

; from which we obtain

lim inf

n→+∞

εn b_n an

P(Z_1,n^∗ <0)≥D₁−D₂−η >0.

Proof of Theorem 5. The upper bounds follow from Theorem 8 and Proposition 9.

For the lower bounds, we assume thatE[G((Z₁)₋)]<∞ with either G(t) =e^at or G(t) =t^β′ or G(t) =e^at2 for somea >0 or some β^′>1. Due to the Markov inequality,

P(−Z₁ > b_n) =P((Z₁)₋> b_n)≤ E[G((Z1)₋)]

G(b_n) .

Note that we can apply Theorem 10 withb_n=c₀log(n) ifG(t) = e^at, b_n=c₀n^β1′ if G(t) = t^β′ and b_n = c₀p

log(n) if G(t) = e^at2, for a suitable c₀. If (Z₁)₋ is bounded, we choose b_n :=

k(Z₁)₋k∞.

3. Sums of stationary sequences and fractional Brownian motion

Let (X_i)_i≥0 be a stationary centered Gaussian sequence with variance 1 and correlations r(j) :=E[X₀X_j] =E[X_kX_j+k] satisfying asn→+∞,

n

X

i,j=1

r(i−j) =n^2Hℓ(n), (10)

whereH ∈(0,1) andℓis a slowly varying function at infinity. We are interested in the persistence exponent of the sum of the sequenceZn:=P_n

i=1Xi forn≥1 and Z0 := 0.

We recall that the scaling limit of (Zn)n∈Nis the fractional Brownian motion BH with Hurst parameterH, (see [30], [33, Theorem 4.6.1]):

n^−Hℓ(n)^−1/2Z_[nt]

t≥0

=L⇒

n→∞(B_H(t))_t≥0, (11)

and thatB_H is a real centered Gaussian process with covariance function E[B_H(t)B_H(s)] = 1

2(t^2H +s^2H − |t−s|^2H).

A sequence satisfying relation (10) is said to have long-range dependence if H >1/2. We refer to [29] for a recent overview of the field.

In this setup the following theorem is valid.

Theorem 11. Assume (X_i)_i≥0 is a stationary centered Gaussian sequence such that (10) holds with H ∈ (0,1) and ℓ slowly varying. Then there is some constant c >0 such that, for every n≥1,

c⁻¹ n^−(1−H) pℓ(n)

√logn ≤P

k=1,...,nmax Z_k<0

and P

k=1,...,nmax Z_k≤ −1

≤c n^−(1−H)p

ℓ(n). (12) If moreover, inf_nPn

i=1r(i−1)>0, then there is some constant c >0 such that

∀n≥1, c⁻¹n^−(1−H) pℓ(n)

√logn ≤P

k=1,...,nmax Z_k≤1

≤n^−(1−H)e^c√logn.

If moreover the correlation functionr is non-negative (which implies that H ≥1/2) then there is some constant c >0 such that

∀n≥1, c⁻¹n^−(1−H) pℓ(n)

√logn ≤P

k=1,...,nmax Z_k≤1

≤c n^−(1−H)p ℓ(n).

Note that (12) gives a lower bound for the boundary 0 and an upper bound for the boundary

−1. These estimates will follow more or less directly from Theorem 5. In order to change the boundaries (to boundary +1, or in fact to any finite constant), one has to proceed differently according to whether the sequence (X_i) is positively correlated (which is easier) or whether it may also be negatively correlated (in which case we need the extra condition on the sum of the correlations).

Proof of Theorem 11. Assume (10). We setσ_n:= sup_k=1,...,na_kwherea_n:=kZ_nk2 =n^Hp ℓ(n).

Due to Karamata’s characterization of slowly varying functions [18], there exists c₁ >0 and a functionεsuch that lim_t→+∞ε(t) = 0 and such thatc⁻₁¹b_n≤a_n≤c₁b_nwithb_n:=n^He^{R0nε(t)t dt}. Let us denote byD_n the Dudley integral

Dn:=

Z σn

0

plogN(n, t)dt, (13)

whereN(n, t) is the smallest number of open balls of{0, ..., n}of radiustfor the pseudo-metric d(k, ℓ) =a_|k−ℓ| which form a covering of {0, ..., n}. Note that if ar ≤c1br ≤ t for some r < n then N(n, t) ≤ 2n/r. Further, trivially N(n, t) ≤n+ 1 for any t and N(n, t) = 1 for t > an.

Therefore, forϑ∈(0,1), for nlarge enough, Dn ≤

Z bn/√

log(n+1) 0

plog(n+ 1) dt+ Z c1bn

bn/√

log(n+1)

plogN(n, t) dt

≤ b_n+

n

X

k=n^ϑ

Z c1b_k c1bk−1

plog(2n/(k−1))dt

≤ b_n+

n

X

k=n^ϑ

c₁(b_k−b_k−1)p

log(2n/(k−1)).

But, due to the form of b_k, ^bk−_b^bk−1

k = ^kH−(k−1)_kH ^H +

(k−1)^H

1−e

Rk−1 k

ε(t) t dt

k^H = O(k⁻¹). Let υ ∈ (0, H). There exists c₂ >0 such that

D_n ≤ b_n+c₂b_n1 n

n

X

k=n^ϑ

nb_k kb_n

plog(2n/(k−1))

≤ b_n+c₂b_n1 n

n

X

k=n^ϑ

(k/n)^H−1−υ =O(a_n), (14) fornlarge enough (where we used again the expression of b_n). Due to [22, Corollary 2, p.181],

∀u >0, P sup

k=1,...,n

Z_k > u+ 4√ 2D_n

≤P(Z₁ > u/σ_n). Therefore,

Eh sup

k=1,...,n

Z_k²i

≤ 32D²_n+ Z _∞

32D_n²

P sup

k=1,...,n|Z_k|>√ u

du

= 32D²_n+ 2 Z _∞

4√ 2Dn

P sup

k=1,...,n

Z_k> x 2xdx

= 32D²_n+ Z _∞

0

P(Z1 > u/σn) 4(4√

2Dn+u) du

= 32D²_n+σ_n Z _∞

0

P(Z₁ > v) 4(4√

2D_n+vσ_n) dv

≤ 32D²_n+C(σ_nD_n+σ²_n) =O(a²_n), due to (14) and the fact thatσ_n∼a_n asnis large.

Thus, we have proved thatkmax_k=1,...,nZ_kk2 =O(a_n), which combined with (11) yields 1

n^Hℓ(n)^1/2E

k=1,...,nmax Z_k

→E

"

sup

t∈[0,1]

B_H(t)

#

∈(0,∞).

Hence, Theorem 5 implies both the first upper bound and the (last) estimate corresponding to the case when r is non-negative (since in this case the increments of Z are then positively associated).

It remains to prove the upper bound of the second point. We are now going to use Propo- sition 1.6 in [2]. For this purpose, let us denote by H the reproducing kernel Hilbert space of the process (Z_k), i.e. the kernel Hilbert space belonging to the kernel K(n, m) := E[Z_nZ_m], n, m ∈ N. Assume that κ := infnP_n

i=1r(i−1) = infnK(1, n) > 0. Consider the function

f(n) := 2K(1, n)/κ∈ H. Note that f(n)≥2 and that kfk²_H=hf, fiH= 2

κhK(1, .), fiH= 2

κf(1) = (2/κ)²K(1,1) = (2/κ)². Using these two properties and Proposition 1.6 in [2] we can conclude that

cn^−(1−H)ℓ(n)^1/2 ≥P(max

k≤nZ_k ≤ −1)

=P(∀k≤n : Z_k+f(k)≤ −1 +f(k))

≥P(∀k≤n : Z_k+f(k)≤1)

≥P(∀k≤n : Z_k≤1) exp(− q

2kfk²_Hlog(1/P(∀k≤n : Z_k≤1))− kfk²_H/2) Inserting now the lower bound given by (12) and the value for kfkH we obtain, for every α > H−1,

cn^−(1−H)ℓ(n)^1/2 ≥P(∀k≤n : Z_k ≤1) exp(−p

8ακ⁻²logn−2/κ²),

fornlarge enough. This shows the upper bound for the desired probability.

Now let us finally prove the consequences for the (continuous-time) fractional Brownian mo- tion.

Proof of Theorem 1. A simple example in Theorem 11 is fractional Gaussian noise: If B_H denotes a (continuous-time) fractional Brownian motion, define X_i := B_H(i+ 1)− B_H(i), i = 0,1, . . .. Theorem 11 holds and gives that there exists some constant c > 0 such that for large enoughn

P

k=1,...,nmax B_H(k)≤ −1

≤c n^−(1−H). WhenH ∈(¹₂,1), we obtain

P

k=1,...,nmax B_H(k)≤1

≤c n^−(1−H), while forH ∈(0,¹₂) we get

P

k=1,...,nmax B_H(k)≤1

≤c n^−(1−H)e^c√logn.

From these computations, we can deduce an alternative proof of the upper bound for the persistence probability of the continuous-time fractional Brownian motion because trivially:

P sup

t∈[0,T]

B_H(t)≤1

!

≤P

k=1,...,[Tmax ]B_H(k)≤1

.

The estimate for the lower bound are more involved. Due to Theorem 10, there existsc >0 such that

P

k=1,...,nmax BH(k)≤0

> cn^−(1−H)

√logn . Moreover,

tmax∈[0,n]B_H(t)− max

k=1,...,n⌊(Clogn)^1/(2H)⌋B_H(n_k)≤ max

k=1,...,n⌊(Clogn)^1/(2H)⌋A_k, with

A_k:= max

j∈[nk,nk+1](B_H(j)−B_H(n_k))

wheren_k:= ^k−1

⌊(Clogn)^1/(2H)⌋. Then, using the self-similarity of the fractional Brownian motion, P

max

k=1,...,n⌊(Clogn)^1/(2H)⌋A_k>1

≤ n(Clogn)^1/(2H)P

tmax∈[0,1]B_H(t⌊(Clogn)^1/(2H)⌋⁻¹)>1

≤ n(Clogn)^1/(2H)P

tmax∈[0,1]B_H(t)>⌊(Clogn)^1/(2H)⌋^H

≤ n(Clogn)^1/(2H)e^−aClogn

for some constant a > 0 (see [27] for instance). We choose C so that this last quantity is in o(n^−(1−H)(logn)^−2H1 ). Then

P

t∈[0,n]max BH(t)≤1

≥ P

max

k=1,...,n⌊(Clogn)^1/(2H)⌋B_H

k

⌊(Clogn)^1/(2H)⌋

≤0

−o(n^−(1−H)(logn)^−2H1 )

≥ P

⌊(Clogn)^1/2H⌋^−H max

k=1,...,n⌊(Clogn)^1/(2H)⌋B_H(k)≤0

−o(n^−(1−H)(logn)^−2H1 )

≥ P

max

k=1,...,n⌊(Clogn)^1/(2H)⌋B_H(k)≤0

−o(n^−(1−H)(logn)^−2H1 )

≥ cn^−(1−H)(logn)^−2H1 .

4. Random walks in random sceneries

Random walks in random sceneries were introduced independently by H. Kesten and F. Spitzer [19] and by A. N. Borodin [6]. Let d∈N^∗ and S = (S_n)_n∈N be a random walk in Z^d starting at 0, i.e.,S₀ = 0 andX_n:=S_n−S_n−1, n≥1 is a sequence of i.i.d.Z^d-valued random variables.

Let ξ = (ξ_x)_x∈Zd be a field of i.i.d. real valued random variables independent of S. The field ξ is called the random scenery. The random walk in random scenery (RWRS) Z := (Z_n)_n∈^N is defined by settingZ₀ := 0 and, forn∈N^∗, Z_n:= Pn

i=1ξ_Si. We will denote by P the joint law ofS and ξ.

Limit theorems for RWRS have a long history, we refer to [16] for a complete review. As in [19], we consider the case when the distribution ofξ₀ is in the normal domain of attraction of a stable distribution of indexβ ∈(1,2] and, ifS is recurrent, we assume that the distribution of X₁ is in the normal domain of attraction of a stable distribution of orderα∈[d,2]. We assume without any loss of generality that the support of the distribution of X₁ is not contained in a proper subgroup ofZ^dand that the closed subgroup generated by the support of the distribution of ξ₀ is either Z or R. Under the previous assumptions, the following weak convergence holds in the space D([0,+∞)) of c`adl`ag real-valued functions defined on [0,∞), endowed with the Skorokhod topology (with respect to the classical J₁-metric):

n^−α1S_⌊nt⌋

t≥0

=L⇒

n→∞(U(t))_t≥0 if S is recurrent and

n^−β1 P_⌊nt⌋

k=1ξ_±ke1

t≥0=⇒^Ln→∞(Y(±t))_t≥0 , where U(·), Y(·), Y(−·) are independent L´evy processes of respective orderα, β, β, withe₁ = (1,0, . . . ,0)∈Z^d. (When α=d= 2, the process U corresponds to the two-dimensional Brownian motion). Whend= 1< α, in [19], Kesten and

Spitzer proved the following convergence in distribution in (D([0,+∞), J1), (n^{−1+α1−αβ1} Z_[nt])t≥0 L

=⇒

n→∞ ∆t:=

Z

[0,+∞[

Lt(x) dY(x) + Z

[0,+∞[

Lt(−x) dY(−x)

!

t≥0

, where (L_t(x))_x∈R,t≥0 is a continuous version with compact support of the local time of the processU (see [25]).

When S is transient, ((n^−β1Z_[nt])_t≥0)_n∈N∗ converges in distribution (with respect to the M₁- metric), to (∆_t:=c₀Y(t))_t≥0for somec₀ >0 (see [9]). Whenα =d, ((n^−β1(logn)^1β−1Z_[nt])_t≥0)_n∈N∗

converges in distribution (with respect to theM₁-metric), to (∆_t:=c₁Y(t))_t≥0 for somec₁ >0 (see [9]). Hence in any of the cases considered above, ((Z_⌊nt⌋/an)t≥0)n∈N converges in distribu- tion (with respect to theM₁-metric) to some process ∆, with

a_n:=







n^{1−α1+αβ1} if d= 1, α∈(1,2].

n^β1(logn)^1−β1 if α=d.

n^β1 if S is transient.

(15)

For every y ∈Z^d and every integer n≥1, we write N_n(y) for the number of visits of the walk S to sitey before time n, i.e.

N_n(y) := #{k= 1, ..., n : S_k=y}.

We also write R_n := #{S₁, ..., S_n} for the range of S up to time n. Note that Z and can be rewritten as follows:

Zn= X

y∈Zd

ξyNn(y).

When Z₁ takes its values in Z, we define the range Rn of the RWRS Z, i.e. the number of sites visited byZ before time n, by

Rn:= #{Z₀, . . . , Z_n}.

Remark 12 (Transient RWRS). Assume that β ∈ (0,1) and that P(ξ₁ ∈ Z) = 1. Then the RWRS is transient (see for instance [10]) and, due to an argument by Derriennic [34, Lemma 3.3.27], (Rn/n)_n∈N∗ converges P-almost surely to P[Z_j 6= 0, ∀j ≥ 1].(There, we consider the ergodic dynamical system (Ω, µ, T) given by Ω := (Z^d)^Z ×Z^Z, µ := (P_S

1)^⊗Z ⊗(P_ξ

1)^⊗Z and T((α_k)_k,(ǫ_k)_k) := ((α_k+1)_k,(ǫ_k+α0)_k) (see for instance [17] for its ergodicity, p. 162). We set f((α_k)_k,(ǫ_k)_k) = ǫ0. With these choices, (Zj)j≥1 has the same distribution under P as (Pj

k=1f◦T^k)j≥1 underµ.)

We assume from now on that β > 1. Then the RWRS Z is recurrent (see top of the page 2083 in [10]). Let us check that the assumptions of Theorem 5 are satisfied with (a_n)_n∈N given in (15). First, sinceξ is a sequence of i.i.d. random variables independent ofS, the increments (ξ_Sk)_kofZ_nare positively associated conditionally toF0 theσ-algebra generated bySand their conditional distribution givenS is the distribution ofξ0.

Proposition 13. Assume β >1, then

max_k=0,...,nZ_⌊nt⌋−min_k=0,...,nZ_⌊nt⌋ an

−→L sup

t∈[0,1]

∆_t− inf

t∈[0,1]∆_t and

n→lim+∞

E[max_k=0,...,nZ_k]

a_n =E

"

sup

t∈[0,1]

∆_t

# .