Incremental moments and Hölder exponents of multifractional multistable processes

DOI:10.1051/ps/2011151 www.esaim-ps.org

INCREMENTAL MOMENTS AND H ¨ OLDER EXPONENTS OF MULTIFRACTIONAL MULTISTABLE PROCESSES

Ronan Le Gu´ evel

and Jacques L´ evy V´ ehel

Abstract. Multistable processes, that is, processes which are, at each “time”, tangent to a stable process, but where the index of stability varies along the path, have been recently introduced as models for phenomena where the intensity of jumps is non constant. In this work, we give further results on (multifractional) multistable processes related to their local structure. We show that, under certain conditions, the incremental moments display a scaling behaviour, and that the pointwise H¨older exponent is, as expected, related to the local stability index. We compute the precise value of the almost sure H¨older exponent in the case of the multistable L´evy motion, which turns out to reveal an interesting phenomenon.

Mathematics Subject Classification. 60G17, 60G18, 60G22, 60G52.

Received September 2, 2010. Revised March 19, 2011.

1. Introduction

Multistable processes are stochastic processes which are “locally stable”, but where the index of stability varies with “time”. To be more precise, we need to recall the definition of alocalisable process[4,5]:Y ={Y(t) :t∈R} is said to beh-localisable atuif there exists anh∈Rand a non-trivial (i.e.finite and non-zero) limiting process Y_u such that

r→lim0⁺

Y(u+rt)−Y(u)

r^h =Y_u(t). (1.1)

(NoteY_u may and in general will vary withu). When the limit exits,Y_u ={Y_u(t) :t∈R} is termed thelocal form or tangent process of Y at u. The limit (1.1) may be taken in mainly two ways: convergence in finite dimensional distributions, or in distribution when the paths of the process are continuous or c`adl`ag (in which case the process is calledstronglyh-localisable).

A classical example of a strongly localisable process is multifractional Brownian motionY [1,2,10,18] which

“looks like” index-h(u) fractional Brownian motion close to timeubut whereh(u) varies, that is

r→lim0⁺

Y(u+rt)−Y(u)

r^h =B_h(u)(t) (1.2)

Keywords and phrases.Localisable processes, multistable processes, multifractional processes, pointwise H¨older regularity.

1 Universit´e Paris VI, Laboratoire de Probabilit´es et Mod`eles Al´eatoires 4 place Jussieu, 75252 Paris Cedex 05, France.

[email protected]

2 Regularity team, INRIA Saclay, Parc Orsay Universit´e 4 rue Jacques Monod, Bat P, 91893 Orsay Cedex, France.

[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2013

where B_h is index-h fractional Brownian motion. A generalization of mBm, where the Gaussian measure is replaced by an α-stable one, leads to multifractional stable processes, where the local form is an h(u)-self- similar linearα-stable motion [22,23].

Multifractional multistable processes provide a further step of generalization: they are localisable processes such that the tangent process is again an α-stable random process, but where αnow varies with time. Mul- tifractional multistable processes were constructed in [6–8,13] using respectively moving averages, sums over Poisson processes, the Ferguson–Klass–LePage series representation, and multistable measures. Section3.3 be- low provides several specific examples of such processes.

The aim of this work is twofold:

1. We show that, for a large class of (multifractional) multistable processes, a precise estimate for the incre- mental moments holds. More precisely, we prove in Section3.1 that there exists a natural scaling relation forE[|Y(t+ε)−Y(t)|^η] and εsmall. This class includes (multifractional) multistable processes considered in [6,13], in particular L´evy multistable motions and linear multistable multifractional motions. It also include certain moving average multistable processes such as the reverse multistable Ornstein–Uhlenbeck process of [8].

2. We then study the pointwise H¨older regularity of (multifractional) multistable processes. For the same class as above, we obtain an almost sure upper bound for this exponent. In the case of the L´evy multistable motion, we are able to compute its exact value. An interesting phenomenon occurs: when the functional parameterαis smooth, not surprisingly, the H¨older exponent is equal, at each point, almost surely, to the localisability index. However, when α is smaller than one and sufficiently irregular, the regularity of the process is governed by the one of the functionα: their H¨older exponents coincide almost surely. Note that a uniform statement,i.e.a statement like “almost surely, at each point”, cannot hold true in general. Indeed, it already fails for the case of a L´evy stable motion. The right frame in this respect is multifractal analysis, and results in this direction will be presented in a forthcoming work.

The remainder of this work is organized as follows. In the next section, we recall the definition of multistable processes based on the Ferguson–Klass–LePage series representation used in [13] (this defines processes which are equal in distribution to the ones obtained in [6] through sums over Poisson processes). Our main results on incremental moments and upper bound for the pointwise H¨older exponents are described in Sections 3.1–3.3 applies these findings to linear multistable multifractional motion. In Section3.4, we state the result giving the exact value of the pointwise H¨older regularity of L´evy multistable motion. An exemple of a moving average multistable process (reverse multistable Ornstein–Uhlenbeck) is the topic of Section3.5. In Section4, we give intermediate results, some of which being of independent interest, which are used in the proofs of the main statements. Section5 gathers technical results followed by the proofs of the statements related with the incre- mental moments and upper bounds on the exponents. Section6 contains the computation of the exponent for the multistable L´evy motion. Finally, Section7 gives a list of the various technical conditions on multistable processes required by our approach so that their incremental moments and H¨older exponents may be estimated.

2. Multistable processes

We now define multistable processes using the Ferguson–Klass–LePage series representation. These are defined as “diagonals” of random fields that are described below. In the sequel, (E,E, m) will be a measure space, and U an open interval of R. We will assume that m is a finite or σ-finite measure. Let αbe a C¹ function defined onU and ranging in [c, d]⊂(0,2). Letf(t, u, .) be a family of functions such that, for all (t, u)∈U², f(t, u, .)∈ Fα(u)(E,E, m), where:

Fα≡ Fα(E,E, m) ={f :f is measurable andfα<∞},

and _αis the quasinorm (or norm if 1< α≤2) given by fα=

E|f(x)|^αm(dx)1/α

(α= 1)

E|f(x)|m(dx) +

E|f(x)β(x) ln|f(x)||m(dx) (α= 1).

(2.3)

By assumption on m, there existsr:E →R+ such that ˆm(dx) = _r(x)¹ m(dx) is a probability measure (see, e.g., [21], Prop. 3.11.3). Whenmis a finite measure, we always take r(x)≡m(E).

The following notations are used throughout in the sequel: (Γ_i)_i≥1 will be a sequence of arrival times of a Poisson process with unit arrival time. (V_i)_i≥1 will denote a sequence of i.i.d. random variables with distribution ˆm on E. Finally, (γ_i)_i≥1 will be a sequence of i.i.d. random variables with distribution P(γ_i= 1) =P(γ_i=−1) = 1/2. The three sequences (Γi)_i≥1, (V_i)_i≥1, and (γ_i)_i≥1 are independent.

As in [13], we will consider the following random field:

X(t, u) =C_α^1/α_(u)^(u) ∞ i=1

γ_iΓ_i^−1/α(u)r(V_i)^1/α(u)f(t, u, V_i), (2.4)

whereC_η=_∞

0 x^−ηsin(x)dx₋1

.

Note that when the functionαis constant, then (2.4) is just the Ferguson–Klass–LePage series representation of a stable random variable (see [3,9,14,15,20] and [21], Thm. 3.10.1, for specific properties of this representation).

Multistable processes

Multistable processes are obtained by taking diagonals on X, i.e. setting Y(t) = X(t, t): as shown in Theorems 3.3 and 4.5 of [13], provided both X and f fulfill certain conditions, Y is a localisable process whose local form is a stable process. In the sequel, we obtain, under some assumptions (which imply thatY is indeed localisable), estimates on the incremental moments and the pointwise H¨older regularity of Y. We will always assume that t→X(t, u) is localisable at anyuwith exponenth(u)∈(h₋, h+)⊂(0,1). The local form is denotedX_u(t, u). We assume in addition thatu→h(u) is aC¹ function.

3. Main results

The following two theorems apply to a diagonal process Y defined from the field X given by (2.4). For convenience, the conditions required onX and the kernelf that appears in (2.4) are gathered in Section7.

3.1. Moments of multistable processes

Theorem 3.1. Let t ∈ R and U be an open interval of R with t ∈ U. Let η ∈ (0, c). Suppose that f satis- fies (R1), (M1), (M2), (M3) and (H2). Then, whenεtends to 0,

E[|Y(t+ε)−Y(t)|^η]∼ε^ηh(t)E[|Y_t(1)|^η]. (f(x)∼g(x) whenx→0 means that lim_x→0f(x)/g(x) = 1).

Proof. See Section5.

Remark: Under the conditions listed in the theorem, Theorems 3.3 and 4.5 of [13] imply that Y is h(t)−localisable att.

3.2. Pointwise H¨ older exponent of multistable processes

γ : lim

r→0

|Y(t+r,ω)−Y(t,ω)|

|r|^γ = 0

denote the H¨older exponent of the (non-differentiable) processY at t.

Theorem 3.2 (upper bound). Suppose that there exists a functionhdefined onU such that (M1), (M2), (M3), (M4), (M5), (M6) and (M7) hold. Assuming (R1), (H1), (H3), (H4) and (H5), one has, for allt, almost surely:

Ht≤h(t).

Proof. See Section5.

3.3. Example: linear multistable multifractional motion

In this section, we apply the results above to the “multistable version” of a classical process known as linear stable multifractional motion, which is itself a extension of linear stable fractional motion. In the sequel, M will always denote a symmetric α-stable (0 < α < 2) random measure on R with control measure Lebesgue measureL. The linear stable fractional motion is defined as follows [21]:

L_α,H,b⁺_,b−(t) =

∞

−∞

f_α,H(b⁺, b⁻, t, x)M(dx) wheret∈R,H ∈(0,1),b⁺, b⁻∈R, and

f_α,H(b⁺, b⁻, t, x) =b⁺

(t−x)^H−₊ ^1/α−(−x)^H−+ ^1/α

+b⁻

(t−x)^H−₋ ^1/α−(−x)^H−₋ ^1/α

,

where (x)+ = max{0, x} and (x)₋ =−(−x)+. When b⁺ =b⁻ = 1, this process is called well-balanced linear fractionalα-stable motion and denotedL_α,H.

The localisability of linear fractional α-stable motion simply stems from the fact that it is 1/α-self-similar with stationary increments [5].

The multifractional multistable version of this process was defined in [6,13] (note that the choice of ˆm is arbitrary as long as it fulfills the required condition, and is made purely for convenience). Its incremental moments and regularity are described by the following theorems:

Theorem 3.3 (linear multistable multifractional motion). Let α :R → [c, d]⊂(0,2) and H : R→ (0,1) be continuously differentiable. Let (Γ_i)_i≥1, (V_i)_i≥1 and (γ_i)_i≥1 be as described in Section 2, where we take the distribution of (V_i)_i≥1 to bem(dx) =ˆ _π³2

+∞

j=1j⁻²1_{[−j,−j+1[∪[j−1,j[}(x)dxonR. Define:

X(t, u) =C_α^1/α(u)_(u) ∞ i,j=1

π²j²

3

1/α(u)

γ_iΓ_i^−1/α(u)

|t−V_i|^{H(u)−1/α(u)}− |Vi|^{H(u)−1/α(u)}

1_{[−j,−j+1[∪[j−1,j[}(V_i) (3.5) and the linear multistable multifractional motion

Y(t) =X(t, t).

Then for allt∈Randη < c, whenεtends to 0, E[|Y(t+ε)−Y(t)|^η]∼ 2^η−1Γ(1−_α(t)^η )

η_∞

0 u^−η−1sin²(u)du

R

|1−x|^{H(t)−α(t)1} − |x|^{H(t)−α(t)1 α(t)}dx _α(t)^η

ε^ηH(t).

Proof. See Section5.

Theorem 3.4. Let Y be the linear multistable multifractional motion defined onRwithH−_α¹ a non-negative function. For all t∈R, almost surely,

Ht≤H(t).

Proof. See Section5.

3.4. Example: L´ evy multistable motion

In the case of the L´evy multistable motion, we are able to provide a more precise result, to the effect that, at each point, the exact almost sure value of the H¨older exponent is known. Let us first recall some definitions.

WithM again denoting a symmetricα-stable (0< α <2) random measure onRwith control measure Lebesgue measureL, we write

L_α(t) :=

t 0 M(dz) forα-stable L´evy motion.

The localisability of L´evy motion is a consequence of the fact that it is 1/α-self-similar with stationary increments [5]. Its multistable version and incremental moments are described in the following theorem:

Theorem 3.5 (symmetric multistable L´evy motion). Let α: [0,1] → [c, d] ⊂ (1,2) be continuously differen- tiable. Let(Γ_i)_i≥1,(V_i)_i≥1and(γ_i)_i≥1 be as described in Section2, where we take the distribution of (V_i)_i≥1 to bem(dx) = dxˆ on [0,1]. Define

X(t, u) =C_α^1/α(u)_(u) ∞ i=1

γ_iΓ_i^−1/α(u)1_[0,t](V_i) (3.6) and the symmetric multistable L´evy motion

Y(t) =X(t, t).

Then for allt∈(0,1)and η < c, whenεtends to 0,

E[|Y(t+ε)−Y(t)|^η]∼ 2^η−1Γ(1−_α^η_(t)) η_∞

0 u^−η−1sin²(u)du ε^α(t)η .

Proof. See Section5.

Theorem 3.6. Let Y be the symmetric multistable L´evy motion defined on[0,1]withα: [0,1]→[c, d]⊂(0,2).

For allt∈(0,1), almost surely,

Ht≤ 1 α(t)·

Proof. See Section5.

Theorem 3.7. Let u∈U ⊂(0,1).

1. If 0< α(u)<1, almost surely,

Hu= min 1

α(u),H_u^α

· provided _α(¹_u)=H^α_u, whereH_u^αdenotes the H¨older exponent of αatu.

2. If 1≤α(u)<2, and αisC¹, almost surely,

Hu= 1 α(u)·

Proof. See Section6.

Thus, in the case 0< α(u)<1, the regularity of multistable L´evy motion is the smallest number between _α(u)¹ and the regularity of the functionαat u. This is very similar to the case of multifractional Brownian motion, where the H¨older exponent is the minimum between the functional parameterhand its regularity [10,11]. We conjecture that the same result holds whenα≥1.

3.5. Example: reverse multistable Ornstein–Uhlenbeck

We consider in this section an exemple of a different nature, since the multistable process that we deal with is obtained by generalizing a moving average process rather than a self-similar one, as was the case of L´evy motion and linear fractional stable motion. This shows that conditions of Section7 required for our results to hold, although somewhat numerous and technical, are indeed natural in our frame.

Letλ > 0 and 1< α ≤2 and let M be anα-stable measure onR with control measureL. The stationary process

Y(t) =

∞ t

exp(−λ(x−t))M(dx) (t∈R)

is called reverse Ornstein–Uhlenbeck process. By Proposition 2.2 of [8], it has a version inD(R) that is 1/α- localisable at allu∈RwithY_u=L_α.

A multistable version is obtained by taking r(x) =

+∞

j=1

2^j+11_{[−j,−j+1[∪[j−1,j[}(x) (3.7) and

f(t, w, x) = e^−λ(x−t)1_[t,+∞)(x) (3.8) in (2.4) and considering as usual the diagonal process. Using the results of [13], it is easy to prove that Y is 1/α(u)-localisable at allu∈Rwith local formY_u =L_α(u). We then have:

Theorem 3.8 (reverse Ornstein–Uhlenbeck multistable process). Let α : R → [c, d] ⊂ (1,2). Let X(t, u) be the random field of (2.4) with r given by (3.7) and f given by (3.8). Define the reverse Ornstein–Uhlenbeck multistable process as

Y(t) =X(t, t).

Then Theorems3.1 and3.2applies toY at allt∈R.

Proof. See Section5.

4. Intermediate results

Let ϕ_X denote the characteristic function of the random variable X. We first state the following almost obvious fact, which will be used in the proof of Theorem3.2:

Proposition 4.9. Assume that for a givent∈Rthere existsε0>0 such that supt∈U sup

r∈(0,ε0) +∞ 0

ϕY(t+r)−Y(t) rh(t) (v)

dv <+∞,

where Y is a symmetrical process. Then there existsK >0 such that for all t ∈U, for all r∈(0, ε0)and all x >0,

P(|Y(t+r)−Y(t)|< x)≤K x r^h(t)·

Proof. This is a straightforward consequence of the inversion formula. Letx >0 and 0< r < ε0. Since Y is a symmetrical process,ϕ_Y(t+r)−Y(t) is an even function and

P(|Y(t+r)−Y(t)|< x) = 1 π

+∞

0 ϕ_{Y(t+r)−Y(t)}

v r^h(t)

sin

vx r^h(t)

dv

v

≤ 1 π

x

r^h(t) sup

r∈B(0,ε0) +∞ 0

ϕY(t+r)−Y(t) rh(t) (v)

dv

≤K x

r^h(t)·

The next proposition will also be used in the proof of Theorem3.2:

Proposition 4.10. Suppose that there exists a functionhdefined on U such that (H1), (H3), (H4), and (H5) hold. Assuming (R1), (M4), (M5), (M6), and (M7) one has:

supt∈U sup

r∈(0,ε0) +∞ 0

ϕY(t+r)−Y(t) rh(t) (v)

dv <+∞.

Proof. The expression of the characteristic functionϕY(t+r)−Y(t)

rh(t) is given in [13]:

ϕY(t+r)−Y(t)

rh(t) (v) = exp

⎛

⎝−2

R +∞ 0 sin²

⎛

⎝vC_α(t+r)^1/α(t+r)f(t+r, t+r, x)

2r^h(t)y^1/α(t+r) −vC_α(t)^1/α(t)f(t, t, x) 2r^h(t)y^1/α(t)

⎞

⎠dy m(dx)

⎞

⎠.

Forv≤1,ϕY(t+r)−Y(t)

rh(t) (v)≤1. Forv ≥1, we fixε < _d¹. Lemma (5.14) entails that there existsK_U >0 such that

ϕY(t+r)−Y(t)

rh(t) (v)≤exp

⎛

⎜⎝−

R ^{KU v}

1−εdd r

vC_α^1/α_(t+^(t_r⁺₎^r)f(t+r, t+r, x)

2r^h(t)y^1/α(t+r) −vC_α^1/α_(t)^(t)f(t, t, x) 2r^h(t)y^1/α(t)

2

dy m(dx)

⎞

⎟⎠.

Let

N(v, t, r) =

R ^{KU v}_r^1−εdd

vC_α(t+r)^1/α(t+r)f(t+r, t+r, x)

2r^h(t)y^1/α(t+r) −vC_α(t)^1/α(t)f(t, t, x) 2r^h(t)y^1/α(t)

2

dy m(dx).

Using Lemma5.15, there existK_U >0 andε0>0 such that for all v≥1, N(v, t, r)≥K_Uv^{2+1−εdd (1−2c)}. The inequality becomes

ϕY(t+r)−Y(t)

rh(t) (v)≤exp

−KUv^{2+1−εdd (1−2c)}

, and

+∞

0 ϕY(t+r)−Y(t)

rh(t) (v)dv≤1 +

∞ 1 exp

−KUv^{2+1−εdd (1−2c)}

dv

<+∞·

We now consider multistable processes. The next proposition will be used in the proof of Theorem3.1.

Proposition 4.11. Assuming (R1), (M1), (M2) and (M3), there existsK_U >0 such that for allu∈U,v∈U andx >0,

P(|X(v, v)−X(v, u)|> x)≤K_U

⎛

⎝|v−u|^d x^d

1 +

log|v−u|

x

d

+|v−u|^c x^c

1 +

log|v−u|

x

c⎞

⎠·

Proof. See Section5.

5. Proofs and technical results 5.1. Proof of Proposition 4.11

First case:r(x)≡1.

We proceed as in [13]. Note that condition (M1) implies that there existsδ∈(^d_c −1, δ) such that:

supt∈U R

w∈Usup

|f(t, w, x) log|f(t, w, x)||^α(w)1+δ

m(dx)<∞. (5.9)

Since (M1) is true with δ in place of δ, we write in the sequel δ for δ. The function u → C_α^1/α_(u)^(u) is a C¹ function sinceα(u) ranges in [c, d]⊂(0,2). We shall denotea(u) = (m(E))^1/α(u)C_α^1/α_(u)^(u). The functionais thus alsoC¹. Let (u, v)∈U².We estimate:

X(v, v)−X(v, u) = ∞ i=1

γ_i(Φ_i(v)−Φ_i(u)) + ∞ i=1

γ_i(Ψ_i(v)−Ψ_i(u)), where

Φ_i(u) =a(u)i^−1/α(u)f(v, u, V_i) and

Ψ_i(u) =a(u)

Γ_i^−1/α(u)−i^−1/α(u)

f(v, u, V_i).

Thanks to the assumptions onaandf,Φ_i andΨ_i are differentiable and one computes:

Φ_i(u) =a(u)i^−1/α(u)f(v, u, V_i) +a(u)i^−1/α(u)f_u(v, u, V_i) +a(u)α(u)

α(u)²log(i)i^−1/α(u)f(v, u, V_i), and

Ψ_i(u) =a(u)

Γ_i^−1/α(u)−i^−1/α(u)

f(v, u, V_i) +a(u)

Γ_i^−1/α(u)−i^−1/α(u)

f_u(v, u, V_i) +a(u)α(u)

α(u)²

log(Γ_i)Γ_i^−1/α(u)−log(i)i^−1/α(u)

f(v, u, V_i).

Using the mean value theorem, there exists a sequence of independent random numbersw_i∈(u, v) (or (v, u)) and a sequence of random numbersx_i∈(u, v) (or (v, u)) such that:

X(v, u)−X(v, v) = (u−v) ∞

i=1

(Z_i¹+Z_i²+Z_i³) + (u−v) ∞ i=1

(Y_i¹+Y_i²+Y_i³), (5.10)

where

Z_i¹=γ_ia(w_i)i^−1/α(wi)f(v, w_i, V_i), Z_i²=γ_ia(w_i)i^−1/α(wi)f_u(v, w_i, V_i), Z_i³=γ_ia(w_i)α(w_i)

α(w_i)²log(i)i^−1/α(wi)f(v, w_i, V_i), Y_i¹=γ_ia(x_i)

Γ_i^−1/α(xi)−i^−1/α(xi)

f(v, x_i, V_i), Y_i²=γ_ia(x_i)

Γ_i^−1/α(xi)−i^−1/α(xi)

f_u(v, x_i, V_i), Y_i³=γ_ia(x_i)α(x_i)

α(x_i)²

log(Γ_i)Γ_i^−1/α(xi)−log(i)i^−1/α(xi)

f(v, x_i, V_i).

Note that each w_i depends on a, f, α, u, v, V_i, and each x_i depends on a, f, α, u, v, V_i, Γ_i but not on γ_i. This remark will be useful in the sequel.

In [13], it is proved that each series ^∞

i=1Z_i^j and ^∞

i=1Y_i^j, j = 1,2,3, converges almost surely. Let x > 0. We considerP

^∞

i=1Z_i^j > x

andP ^∞

i=1Y_i^j > x

forj= 1,2,3.

Letη∈(0,min(²_d^c −1,^c_d(δ+ 1)−1)). Markov inequality yields

P

∞ i=1

Z_i^j > x

≤ 1 x^dE

⎡

⎣

∞ i=1

Z_i^j

d⎤

⎦

≤ 1 x^d

⎛

⎝E

⎡

⎣

∞ i=1

Z_i^j

d(1+η)⎤

⎦

⎞

⎠

1+η1

.

The random variablesZ_i^j are independent with mean 0 thus, by Theorem 2 of [24]:

E

⎡

⎣

+∞

i=1

Z_i^j

d(1+η)⎤

⎦≤2

+∞

i=1

E[|Z_i^j|^d(1+η)].

Forj= 1, E

|Z_i¹|^d(1+η)

=E

|a(w_i)|^d(1+η)i⁻

d(1+η)

α(wi)|f(v, w_i, V_i)|^d(1+η)

≤ K_U i^1+ηE

⎡

⎣

sup

w∈B(u,ε)|f(v, w, V_i)|^{α(w) d(1+η)}_α(

wi)⎤

⎦

≤ K_U i^1+ηE

⎡

⎣

w∈Bsup(u,ε)|f(v, w, V1)|^α(w) 1+η

+

w∈Bsup(u,ε)|f(v, w, V1)|^{α(w) d}

c(1+η)⎤

⎦

≤ K_U i^1+η ·

Forj= 2, E

|Z_i²|^d(1+η)

≤ K_U i^1+ηE

⎡

⎣

sup

w∈B(u,ε)|f_u(v, w, V1)|^α(w) 1+η

+

sup

w∈B(u,ε)|f_u(v, w, V1)|^α(w)

^d_c(1+η)⎤

⎦

≤ K_U i^1+η · Forj= 3,

E

|Z_i³|^d(1+η)

=E a(w_i)α(w_i) α(w_i)²

^d(1+η)|f(v, w_i, V_i)|^d(1+η)(logi)^d(1+η) i^{d(1+η)α(wi)}

!

≤K_U(logi)^d(1+η) i^1+η · Finally, sup

v∈U +∞ i=1E

|Z_i^j|^d(1+η)

<+∞thus P

∞ i=1

Z_i^j > x

≤K_U x^d · We consider nowP

∞ i=1Y_i^j

> x

forj= 1,2,3.

P

∞ i=1

Y_i^j > x

≤P

|Y1^j| ≥x 2

+P

∞ i=2

Y_i^j ≥ x

2

· Since P

∞ i=2Y_i^j

≥ ^x₂

≤ _x^2dd

E

∞ i=2Y_i^j

^d(1+η)

!_1+η¹

, we want to apply Theorem 2 of [24] again. Let S_m=^m

i=1Y_i^j and writeY_i^j=γ_iW_i^j. Note thatγ_i is independent ofW_i^j andS_i−1. E

Y_m^j₊₁|S_m

=E

E(Y_m^j₊₁|S_m, W_m+1)|S_m

=E

E(γ_m+1W_m+1^j |S_m, W_m+1)|S_m

=E

W_m+1^j E(γ_m+1|S_m, W_m+1)|S_m

=E

W_m+1^j E(γm+1)|Sm

= 0.

We apply Theorem 2 of [24] with (d(1 +η)<2), E

⎡

⎣

∞ i=2

Y_i^j

d(1+η)⎤

⎦≤2 ∞ i=2

E|Y_i^j|^d(1+η), and

P

∞ i=1

Y_i^j > x

≤P

|Y₁^j| ≥x 2

+2^d x^d

2

∞ i=2

E|Y_i^j|^d(1+η) _1+η¹

.

Forj= 1,

P

|Y₁¹| ≥ x 2

=P

⎛

⎝|a(x1)|^α(x1)

1

Γ₁^1/α(x1)−1

α(x1)

|f(v, x1, V1)|^α(x1)≥x^α(x1) 2^α(x1)

⎞

⎠

≤P

⎛

⎝

1

Γ₁^1/α(x1) −1

α(x1)

sup

w∈B(u,ε)|f(v, w, V1)|^α(w)≥K_Ux^α(x1)

⎞

⎠.

Forx <1,

P

|Y1¹| ≥ x 2

≤P

⎛

⎝

1

Γ₁^1/α(x1)−1

α(x1)

w∈Bsup(u,ε)|f(v, w, V1)|^α(w)≥K_Ux^d

⎞

⎠

≤P

sup

w∈B(u,ε)|f(v, w, V1)|^α(w)≥K_Ux^d

"

∩ {Γ1>1}

+P 1

Γ1

Γ₁^1/α(x1)−1|^α(x1) sup

w∈B(u,ε)|f(v, w, V1)|^α(w)≥K_Ux^d

"

∩ {Γ1≤1}

.

P

sup

w∈B(u,ε)|f(v, w, V1)|^α(w)≥K_Ux^d

"

∩ {Γ1>1}

≤ K_U x^d E

sup

w∈B(u,ε)|f(v, w, V1)|^α(w)

≤ K_U x^d ·

LetW(v, x) = sup_w∈B(u,ε)|f(v, w, x)|^α(w) andF_v,V1 be the distribution ofW(v, V1).

P# 1

Γ1

Γ₁^1/α(x1)−1|^α(x1)W(v, V1)≥K_Ux^d

$

∩ {Γ1≤1}

≤P

W(v, V1)≥K_Ux^dΓ1

=

+∞

0 P

z≥K_Ux^dΓ1

F_v,V1(dz)

=

+∞ 0

1−e^{−KU xdz}

F_v,V1(dz)

≤ ^+∞

0

z

K_Ux^dF_v,V1(dz)

≤ K_U x^d · Forx≥1,

P

|Y1¹| ≥ x 2

≤P

⎛

⎝

1

Γ₁^1/α(x1)−1

α(x1)

w∈Bsup(u,ε)|f(v, w, V1)|^α(w)≥K_Ux^c

⎞

⎠ P

|Y1¹| ≥ x 2

≤ K_U x^c ·