# On the Lower Classes of Some Mixed Fractional Gaussian Processes with Two Logarithmic Factors

Authors: Charles El-Nouty
Journal: Journal of Applied Mathematics and Stochastic Analysis
 (2008)

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## Abstract

We introduce the fractional mixed fractional Brownian sheet and investigate the small ball behavior of its sup-norm statistic by establishing a general result on the small ball probability of the sum of two not necessarily independent joint Gaussian random vectors. Then, we state general conditions and characterize the sufficiency part of the lower classes of some statistics of the above process by an integral test. Finally, when we consider the sup-norm statistic, the necessity part is given by a second integral test.


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## Body

## 1. Introduction

Let{BHi(s),s≥0} be a fractional Brownian motion (FBM) with index 0<Hi<1,i∈ℕ*,
and {BHj,Hk(s,s′),s≥0,s′≥0} a fractional Brownian sheet (FBS) with index 0<Hj,Hk<1,j∈ℕ*,k∈ℕ*.
We refer to [1] for further information about the FBM and the FBS. Denote by λ1 and λ2 two real numbers such that λ1λ2≠0.Define a
fractional mixed fractional Gaussian process by a suitable combination of some
appropriate fractional Gaussian processes. In the sequel, we consider the
following three examples.Example 1.1.
The fractional mixed fractional Brownian motion (FMFBM) is defined by(1.1)X(w1,w2,s)=λ1sH2BH1(w1)+λ2sH1BH2(w2),
where
BH1 and BH2 are independent FBM with H1≠H2.Example 1.2.
The fractional mixed
fractional Brownian motion and fractional Brownian sheet (FMFBMFBS) are defined by(1.2)X(w1,w2,w3,s)=λ1sH2+H3BH1(w1)+λ2sH1BH2,H3(w2,w3),
where the FBM BH1 and the FBS BH2,H3 are independent.Example 1.3.
The fractional mixed
fractional Brownian sheet (FMFBS) is defined by(1.3)X(w1,w2,w3,w4,s)=λ1sH3+H4BH1,H2(w1,w2)+λ2sH1+H2BH3,H4(w3,w4),
where BH1,H2 and BH3,H4 are independent FBS with (H1,H2)≠(H3,H4).The motivation supporting this paper is threefold.(i)
The first
goal of the FMFBS deals with the potential applications. Since the  FMFBM, the
FMFBMFBS, and the FMFBS can be analyzed based on the large bodies of knowledge
on FBM and FBS, it can be used in the same fields, that is, natural time series
in economics, fluctuations in solid, hydrology, and, more recently, by new
problems in mathematical finance, telecommunication networks, and the environment
(see [2–4]).(ii)
A second
application deals with the small ball probability problem of the sum of two not
necessarily joint centered Gaussian random vectorsX and Y in a separable Banach space E with norm ∥⋅∥ (see [5]). The small ball behavior of the
FMFBS under the uniform norm can be investigated as a special case of the small
ball probability problem of the sum of two centered Gaussian random vectors,
having a log-type small ball factor (see [6]).(iii)
Last but
not least, this article extends El-Nouty's results [6–9] and consequently
answers some new questions. Recall first two definitions of the Lévy classes,
stated in [10]. Let {Z(t),t≥0} be a stochastic process defined on the basic
probability space (Ω,𝒜).Deffinition 1.4.
The functionf(t),t≥0,
belongs to the lower-lower class of the
process Z,(f∈LLC(Z)),
if,
for almost all ω∈Ω, there exists t0=t0(ω)
such that Z(t)≥f(t) for every t>t0.Deffinition 1.5.
The functionf(t),t≥0, belongs to the lower-upper class of the
process Z,(f∈LUC(Z)), if, for almost all ω∈Ω, there exists a sequence 0<t1=t1(ω)<t2=t2(ω)<⋯ with tn→+∞, as n→+∞, such that Z(tn)≤f(tn),n∈ℕ*.In the spirit
of [6–9, 11], the main aim of this paper is to characterize the lower classes
of the uniform norm of the FMFBS for any 0<H1,H2,H3,H4<1.
More precisely, we want to compare the influence of two FBSs and to measure the
weight of a log-type small ball factor versus another one.


## 2. Main Results

Our first result is given in the following theorem.Theorem 2.1.
LetX and Y be any two joint Gaussian random vectors in a
separable Banach space with norm ∥⋅∥.
Assume that there exist CX≥1 and CY≥1 such that one has, for any ϵ>0 small enough,
(2.1)−CX≤ϵ1/α(log⁡(1/ϵ))βlogP(∥X∥≤ϵ)≤−1CX,−CY≤ϵ1/α(log⁡(1/ϵ))βlogP(∥Y∥≤ϵ)≤−1CY,
with
0<α,α<+∞,0≤β,β<+∞ and (α,β)≠(α,β).
If(α<α) or (α=α and β>β),
then there exists KX≥CX depending on CX only such that one has, for any ϵ>0 small enough,
(2.2)−KX≤ϵ1/α(log⁡(1/ϵ))βlogP(∥X+Y∥≤ϵ)≤−1KX.Since the study
of the lower classes of the FMFBM (resp., FMFBMFBS) under the uniform norm was
investigated in [8] (resp., [9]), we focus our attention to the FMFBS. Set
(2.3)Y(t)=sup⁡0≤s≤tsup⁡0≤w1,w2,w3,w4≤s|X(w1,w2,w3,w4,s)|,t≥0.
Note first
that, by the scaling property, we have, for any ϵ>0,
(2.4)P(Y(t)≤ϵtH1+H2+H3+H4)=P(sup⁡0≤s≤1sup⁡0≤w1,w2,w3,w4≤s|X(w1,w2,w3,w4,s)|≤ϵ)=P(Y(1)≤ϵ):=ϕ(ϵ),
where ϕ is named the small ball function and γ:=H1+H2+H3+H4 the scaling factor.Recall that the
small ball behavior of the FBS under the uniform norm was studied in [12, 13].Setα=min(H1,H2,H3,H4),
which is in ]0,1[.
We introduce the number β taking its values in {0,1+1/α}.
As a direct consequence of Theorem 2.1, we have the following corollary.Corollary 2.2.
There is a constantK0,0<K0≤1, depending on H1,H2,H3,H4,λ1 and λ2 only, such that one has, for any ϵ>0 small enough,
(2.5)exp⁡(−(log⁡(1/ϵ))βK0ϵ1/α)≤ϕ(ϵ)≤exp⁡(−K0(log⁡(1/ϵ))βϵ1/α).Recall that we
suppose(H1,H2)≠(H3,H4).
In the sequel, there is no loss of generality to assume also that
(2.6)H1≤H2,H3≤H4.
Thus when (H1=H2,H3<H4 and H1≤H3), (H1=H2,H3=H4 and H1<H3),
or (H1<H2,H3=H4 and H3≤H1),
we emphasize that β=1+1/α,
that is, we have a log-type small ball factor.Note first that
the minimumα plays a key role. This is not really
surprising. Indeed, this phenomenon was already observed in [8, 9].It appears
that the sufficiency part of the lower classes ofY can be stated in a general framework. Roughly
speaking, we follow the same lines as those of [6, 7].Let{Y0(t),t≥0} be a real-valued statistic of the two
independent FBSs, BH1,H2 and BH3,H4,
such that Y0(t) is a nondecreasing function of t≥0.The following
notation is needed. If𝕂 is a Hausdorff compact space, we denote, by C(𝕂),
the space of all continuous functions from 𝕂 to ℝ equipped with the classical sup-norm. Let 𝕏=C([0,1]2)×C([0,1]2) be the product space equipped with the product
topology. Denote,  by L (BH1,H2,BH3,H4),
the Gaussian measure associated to BH1,H2 and BH3,H4 and defined on 𝔹,
the Borel σ-field of 𝕏.We assume thatY0 satisfies the following three conditions:(i)
The scaling condition. There exists γ0>0 such that
(2.7)P(Y0(t)≤ϵtγ0)=P(Y0(1)≤ϵ):=ϕ(ϵ).(ii)
The convexity condition. There exists
a convex and 𝔹-measurable function g: (𝕏,L(BH1,H2,BH3,H4))→ℝ such that, for any t≥0,Y0(t)=g(BH1,H2(s1t,s2t),  BH3,H4(s3t,s4t);0≤s1,s2,s3,s4≤1), and Y0(t)<+∞,
with probability 1.(iii)
The log-type small ball condition. There exist α0∈]0,γ0],β0∈ℝ and a constant K,0<K≤1,
depending on H1,H2,H3,H4,γ0,α0 and β0 only such that we have, for any ϵ>0 small enough,
(2.8)exp⁡(−(log⁡(1/ϵ))β0Kϵ1/α0)≤ϕ(ϵ)≤exp⁡(−K(log⁡(1/ϵ))β0ϵ1/α0).Note that these
conditions generalize those of [6, 7]. The small ball function still plays a
key role. The convexity of the function ψ defined by ψ(ϵ)=−logϕ(ϵ),0<ϵ<1,
is ensured by (ii) (see [14, 15]).Our second
result is given in the following theorem.Theorem 2.3.
Letf(t) be a positive nondecreasing function of t≥0.
Assume that there exists m>0 such that (f(t)/tγ0−α0)(logtγ0/f(t))−β0α0≥m.
If(2.9)f(t)tγ0isboundedand∫0+∞f(t)−1/α0t(γ0/α0)−1(logtγ0f(t))β0ϕ(f(t)tγ0)dt<+∞,
then one has
(2.10)f∈LLC(Y0).The sup-norm
statisticY clearly satisfies the three above conditions
with γ0=γ=H1+H2+H3+H4,α0=α=min(H1,H2,H3,H4)=min(H1,H3),β0=β∈{0,1+1/α}, and K=K0.
Now, we characterize the necessity part of the lower classes of the FMFBMFBS.
Our main result is stated in the following theorem.Theorem 2.4.
Letf(t) be a positive nondecreasing function of t≥0 such that f(t)/tγ is a nonincreasing function of t>0.
If(2.11)f∈LLC(Y),
then we have
(2.12)lim⁡t→+∞f(t)tγ=0,∫0+∞f(t)−1/γϕ(f(t)tγ)dt<+∞.First, we can
notice that Theorem2.3 involves γ0,α0 and β0.
If β0=0,
Theorem 2.3 looks like [7, Theorem 1] or else like [6, Theorem 1.1].
Theorem 2.4 has the same form as the necessity part of [11, Theorem 1.2] or
as the theorems obtained in [6–9]. In his previous works on the study of the
lower classes, the author showed that the methodology in [11] led to two
integral tests, these tests are actually identical when γ=α and β=0.
Here, α=min(H1,H3)<γ=H1+H2+H3+H4.
This is why the integral tests of Theorems 2.3 and 2.4 have different forms.
Moreover, since α<γ,
we must assume, as in [6–9], that f(t)/tγ is not only bounded, but also a nonincreasing
function of t>0.
This last assumption will play a key role in some proofs. Finally, although
they are two different integral tests, Theorems 2.3 and 2.4 are sharp. Indeed,
set, if β=0,
(2.13)f(t)=λtγ(log⁡log⁡t)α,t≥3,⁢λ>0,
or else (i.e., β=1+(1/α))
(2.14)f(t)=tγ(λlog⁡log⁡log⁡t)1+α(log⁡log⁡t)α,t≥16,⁢λ>0.
If λ is small enough, then Theorem 2.3 yields f∈LLC(Y),
or else if λ is large enough, then f∈LUC(Y) by applying Theorem 2.4.In Section3,
we prove Theorem 2.1. The proof of Theorem 2.4 is postponed to Sections 4 and 5. In the latter, we establish some key small ball estimates. Note also that
these estimates can be of independent interest. The proofs, which are
modifications of those of [6, 7], will be consequently omitted, in particular,
the proof of Theorem 2.3.


## 3. Proof of Theorem2.1

Recall, first,
a Gaussian correlation inequality stated in [5].Theorem A.
Letμ be a centered Gaussian measure on a separable
Banach space E.
Then for any 0<λ<1,
and any symmetric convex sets A and B in E,
(3.1)μ(A∩B)μ(λ2A+(1−λ2)B)≥μ(λA)μ((1−λ2)1/2B).
In particular,
(3.2)μ(A∩B)≥μ(λA)μ((1−λ2)1/2B).Rougly speaking,
the proof follows the same lines as those in [15] and will be split into two
parts: the lower bound and the upper one.Part I. The Lower Bound
Theorem A implies, for any0<δ<1 and 0<λ<1,
(3.3)P(∥X+Y∥≤ϵ)≥P(∥X∥≤(1−δ)ϵ,∥Y∥≤δϵ)≥P(∥X∥≤λ(1−δ)ϵ)P(∥Y∥≤(1−λ2)1/2δϵ).Then we get, by
using (2.1),
(3.4)log⁡P(∥X+Y∥≤ϵ)≥−CX(log⁡(1/λ(1−δ)ϵ))β(λ(1−δ)ϵ)1/α−CY(log⁡(1/(1−λ2)1/2δϵ))β((1−λ2)1/2δϵ)1/α.
Hence, since (α<α) or (α=α and β>β),
there exists CX′≥CX depending on CX only such that we have, for any ϵ>0 small enough,
(3.5)ϵ1/α(log⁡(1/ϵ))βlogP(∥X+Y∥≤ϵ)≥−CX′(λ(1−δ))1/α,
and the lower
bound follows by taking δ→0 and λ→1.Part II. The Upper Bound
A new use of Theorem A implies, for any0<δ<1 and 0<λ<1,
(3.6)P(∥X∥≤ϵ(1−δ)λ)≥P(∥X+Y∥≤ϵλ,∥Y∥≤δϵ(1−δ)λ)≥P(∥X+Y∥≤ϵ)P(∥Y∥≤(1−λ2)1/2δϵ(1−δ)λ).Then combining
(2.1) with the fact that (α<α) or (α=α and β>β),
there exists CX′′≥CX depending on CX only such that we have, for any ϵ>0 small enough,
(3.7)ϵ1/α(log⁡(1/ϵ))βlogP(∥X+Y∥≤ϵ)≤−(λ(1−δ))1/αCX′′,
and the upper
bound follows by taking δ→0 and λ→1.By choosingKX=max(CX′,CX′′), we complete the proof of Theorem 2.1.Remark 3.1.
WhenX and Y are independent, there is a simple proof
without using the correlation inequality in the spirit of [5]. A direct proof
of Corollary 2.2 can also be done as in [9].


## 4. Proof of Theorem2.4, Part I

To simplify
the reading of our paper, we introduce the following notation. Setat=f(t)/tγ and bt=ϕ(at).Suppose here
that, with probability 1,f(t)≤Y(t) for all t large enough. We want to prove that limt→+∞at=0 and ∫0∞at−1/γbt(dt/t)<+∞.In the sequel,
there is no loss of generality to assume thatf(t) is a continuous function of t≥0.Lemma 4.1.
One has(4.1)lim⁡t→+∞at=0.To prove Theorem2.4, we will show that f∈LUC(Y) when ∫0∞at−1/γbt(dt/t)=+∞ and limt→+∞at=0.Following the
same lines as those in [11], our aim is to construct a suitable subset J of ℕ such that we have the following property for
an appropriate family of sets (Ei)i∈J in a basic probability space:
given ϵ>0,
there exist a number K and an integer p such that
(4.2)∀n∈J,n≥p⟹∑m∈J,m>nP(En∩Em)≤P(En)(K+(1+ϵ)∑m∈J,m>nP(Em)).Lemma 4.2.
When∫0∞at−1/γbt(dt/t)=+∞ and limt→+∞at=0,
we can find a sequence {tn,n≥1}
with the two following properties:
(4.3)tn+1≥tn(1+atn1/γ),∑n=1∞btn=+∞.Remark 4.3.
The condition “f(t)/tγ is a nonincreasing function of t>0” is essential to prove Lemma 4.2
(see [7, page 373]).To continue
the construction of the setJ,
we need the following definition and notation.Definition 4.4.
Consider the intervalAk=[2k,2k+1[,k∈ℕ. If ati−1/γ∈Ak,i∈ℕ*,
then one notes u(i)=k.Next, setIk={i∈ℕ*,u(i)=k∈ℕ} which is finite by Lemma 4.1 and
(4.4)Nk=exp⁡(K0(γlog⁡2)βkβ2γ(k−1)/α),
where K0 was defined in Corollary 2.2.Notation
(i)
Um,k={i∈ℕ*,i∈Ik,i<m,card(Ik∩[i,m[)≤Nk},m∈ℕ*,k∈ℕ;(ii)
k0=inf{n∈ℕ,2γn/α≥2γ/α/(K0(2γ/α−1)(γlog2)β)+22γ/α/K02(2γ/α−1)},
(k0  depends on γ,α,β and K0 only);(iii)
Vm=⋃k∈ℕUm,k,
where m is fixed, u(m)=k1, and k≥k1+k0;(iv)
W=⋃m≥1Vm.Now, we can
define the setJ as follows:
(4.5)J=N*−W.
Since it is
assumed that f(t)/tγ is a nonincreasing function of t>0 (being a particular case of the condition “f(t)/tγ is bounded”),(4.6)i<m⟹k=u(i)≤u(m)=k1.
Moreover, since k0≥1,
we get k<k1+k0.
Hence Vm is always an empty set (by construction). Thus
we obtain J=ℕ*.We have, by Lemma4.2,
(4.7)∑n∈Jbtn=+∞.Lemma 4.5.
n∈J,m∈J,n<m,
such that card (Iu(m)∩[n,m])>exp(K02u(m)−1),
one has
(4.8)tmtn≥exp⁡(exp⁡(K042min⁡(u(n),u(m)))).Proof.
Setk=u(n),k1=u(m) and G=Ik1∩[n,m]={i1,i2,. . .,iz}, where n≤i1<i2<⋯<iz≤m.
We have
(4.9)tmtn=tmtiztiztiz−1⋯ti1tn.
Note that, when i∈Ik1,
we have ti+1≥ti(1+ati1/γ)≥ti(1+2−k1−1).
Moreover, since card(G)>exp(K02k1−1) by hypothesis, (4.9) implies
(4.10)tmtn≥exp⁡(exp⁡(K02k1−1)log⁡(1+2−k1−1))≥exp⁡(exp⁡(K042k1)),
when n hence k1 are large enough.
Thus, sincek≤k1,
(4.10) implies (4.8).
The proof of Lemma4.5 is now complete.


## 5. Proof of Theorem2.4, Part II

Consider, now,
the eventsEn={Y(tn)<f(tn)}.
We have directly ℙ(En)=btn,
and, by (4.7), ∑n∈Jbtn=+∞. To prove (4.2), we remark that, given n∈J,J can be rewritten as follows: J=J′∪(⋃k∈ℕJk)∪J′′,
where J′={m∈J,tn≤tm≤2tn},Jk={m∈J∩Ik,tm>2tn,card(Ik∩[n,m])≤exp(K02k−1)}, and J′′=J−(J′∪(⋃k∈ℕJk)).Our first key small ball estimate is given in the
following lemma.Lemma 5.1.
Consider0<t<u, and θ,ν>0. Then one has
(5.1)P({Y(t)≤θtγ}∩{Y(u)≤ν})≤exp⁡(K5)P(Y(t)≤θtγ)exp⁡(−K5(u−t)ν1/γ),
where
K5 depends on H1,H2,H3,H4,λ1, and λ2 only.Proof.
SetF1={Y(t)≤θtγ} and F2={Y(u)≤ν}.
We have
(5.2)P(F1∩F2)=P(F1∩{sup⁡0≤s≤usup⁡0≤w1,w2,w3,w4≤s|X(w1,w2,w3,w4,s)|≤ν})≤P(F1∩{sup⁡t≤s≤usup⁡0≤w1,w2,w3,w4≤s|X(w1,w2,w3,w4,s)|≤ν})≤P(F1∩{sup⁡t≤s≤usup⁡0≤w1,w2≤s|λ1sH3+H4BH1,H2(w1,w2)|≤ν}).
Denote, by [x],
the integer part of a real x.
Let δ>0.
We consider the sequence tk,k∈{0,…,n},
where t0=t,tk+1=tk+δ, and n=[(u−t)/δ].
Consider also the rectangles Rj=[tj,tj+1]×[tj,tj+1],
where j∈{0,…,n−1}.
Their area is (tj+1−tj)2=δ2.
Let Gj be the event defined by
(5.3)Gj=F1∩{sup⁡t≤s≤tjsup⁡0≤w1,w2≤s|λ1sH3+H4BH1,⁢H2(w1,w2)|≤ν}.
We have F1∩F2⊂Gj.
Moreover, we
have also(5.4)Gj+1⊂Gj∩{Zj≤4ν},
where
(5.5)Zj=λ1tj+1H3+H4(BH1,H2(tj+1,tj+1)−BH1,H2(tj,tj+1)−BH1,H2(tj+1,tj)+BH1,H2(tj,tj)).
Before
rewritting Zj,
we recall the integral representation of BHk,Hl,k∈ℕ*,l∈ℕ*,
given by
(5.6)BHk,Hl(sk,sl)=∫−∞sk∫−∞slgHk(sk,uk)gHl(sl,ul)W(d(uk,ul)),
where W(uk,ul),uk∈ℝ,ul∈ℝ,
is a standard Brownian sheet, gH(s,u)=k2H−1(max(s−u,0)H−1/2−max(−u,0)H−1/2),
and k2H is a normalizing constant.
HenceZj can be rewritten by (5.6) as follows: Zj=Zj,1+Zj,2,
where
(5.7)Zj,1=λ1k2H1−1k2H2−1tj+1H3+H4∫tjtj+1∫tjtj+1(tj+1−u1)H1−1/2(tj+1−u2)H2−1/2W(d(u1,u2)).
Note also that Zj,1 and Zj,2 are independent.
Sinceℙ(|Zj,1+x|≤4ν) is maximum at x=0 and Zj,1 and Gj are independent, we have
(5.8)P(Gj+1)≤P(Gj)P(|Zj,1|≤4ν).
The integral
representation of Zj,1 implies that 𝔼(Zj,1)=0 and
(5.9)VarZj,1=λ12k2H1−2k2H2−24H1H2δ2(H1+H2)tj+12(H3+H4)≥λ12k2H1−2k2H2−24H1H2δ2γ:=L2δ2γ.
Denote, by Φ,
the distribution function of the absolute value of a standard Gaussian random
variable. Then we obtain
(5.10)P(Zj+1)≤P(Zj)Φ(4νLδγ),
and therefore, ℙ(F1∩F2)≤ℙ(F1)Φ(4ν/Lδγ)n.
Choosingδ=ν1/γ,
we get K5=−logΦ(2/L).
Lemma 5.1 is proved.Lemma 5.2.
∑m∈J′ℙ(En∩Em)≤K′btn and ∑m∈(∪kJk)ℙ(En∩Em)≤K′′btn,
where
K′ and K′′
are numbers.Proof.
Settingu=tm,t=tn,θ=atn and ν=f(tm),
Lemma 5.1 implies
(5.11)P(En∩Em)≤exp⁡(K5)btnexp⁡(−K5(tm−tn)f(tm)1/γ).
Consider, first,
the case when m∈J′.
Lemma4.2 implies that, for all i≥n,
we have ti+1−ti≥tiati1/γ=f(ti)1/γ≥f(tn)1/γ.
Then we can establish
(5.12)tm−tn≥(m−n)f(tn)1/γ,f(tm)≤f(tn)(tmtn)γ≤2γf(tn).
Combining (5.11)
with (5.12), we get
(5.13)P(En∩Em)≤exp⁡(K5)btnexp⁡(−K5(m−n)2),
which is the
first part of Lemma 5.2.
Consider, now,
the casem∈Jk.
Combining (5.11)
with the definition of Jk,
we have
(5.14)P(En∩Em)≤exp⁡(K5)btnexp⁡(−K52(atm)1/γ).
Since u(m)=k,
we get
(5.15)P(En∩Em)≤exp⁡(K5)btnexp⁡(−K52k−1),
and
consequently, by noting that cardJk≤card(Ik∩[n,m])≤exp(K02k−1) and by assuming K0<K5,
we have
(5.16)∑m∈JkP(En∩Em)≤exp⁡(K5)btnexp⁡((K0−K5)2k−1).
Lemma 5.2 is,
therefore, proved.To deal with
the setJ′′,
we first state a standard large deviation result and a technical lemma (see
[6]).Lemma A.
LetX={X(s1,s2),(s1,s2)∈[0,1]2} be a separable real-valued centered Gaussian
process such that X(0,0)=0 with probability 1 and satisfying, for any
[s1,s1+h1]×[s2,s2+h2]⊂[0,1]2,(5.17)(EX([s1,s1+h1]×[s2,s2+h2])2)1/2≤κ(h1,h2)≤cκh1α1h2α2,α1>0,α2>0,
where
(5.18)X([s1,t1]×[s2,t2])=∫[s1,t1]×[s2,t2]X(d(u1,u2)).
Then one has, for
cκ−1δ>1,(5.19)P(sup⁡(s1,s2)∈[0,1]2|X(s1,s2)|≥δ)≤1Cexp⁡(−C(cκ−1δ)2),
where
C
is a positive constant independent of
cκ
and
δ.Lemma B.
One has, forϵ1>ϵ/2, where ϵ is small enough,
(5.20)exp⁡(−K3|ϵ1−ϵ|(log⁡(1/ϵ))βϵ1+1/α)≤ϕ(ϵ1)ϕ(ϵ)≤exp⁡(K3|ϵ1−ϵ|(log⁡(1/ϵ))βϵ1+1/α),
where
K3>0.Building on Lemmas A and B, one can establish our last key small ball estimate in the following
result.Lemma 5.3.
Letλ
be a
real number such that
1/2<λ<1.
Set
(5.21)r=min⁡(1−max⁡(H1,H2,H3,H4)3,(1−λ)α3).
Then one has, for
u≥2t,(5.22)P(Y(t)≤θtγ,Y(u)≤νuγ)≤ϕ(θ)ϕ(ν)exp⁡(2(tu)rK3((log⁢(1/θ))βθ1+(1/α)+(log⁢(1/ν))βν1+(1/α)))+3(1C12,2exp⁡(−C12,24λ12KH1,22(ut)r)+1C34,2exp⁡(−C34,24λ22KH3,22(ut)r))+3(1C12,1exp⁡(−C12,14λ12KH1,12(ut)r)+1C34,1exp⁡(−C34,14λ22KH3,12(ut)r)),
where
KH1,1,KH1,2>0
depend on
H1(H1≤H2)
only,
KH3,1,KH3,2>0
depend on
H3(H3≤H4)
only,
K3>0
is
defined as in Lemma B, and
C12,1,C12,2,C34,1,C34,2>0
are defined as in Lemma A.Proof.
SetQ=ℙ(Y(t)≤θtγ,Y(u)≤νuγ).
Setv=ut. If t=o(u), then t=o(v) and v=o(u).
Based on (5.6), BH1,H2 and BH3,H4 can be split as follows:
(5.23)BH1,H2=BH1,H2,1+BH1,H2,2,BH3,H4=BH3,H4,1+BH3,H4,2,
where we have,
for (i,j)∈{(1,2),(3,4)},
(5.24)BHi,Hj,1(wi,wj)=∫|xi|≤v∫−∞wjgHi(wi,xi)gHj(wj,xj)W(d(xi,xj)).
Note that BH1,H2,1 and BH1,H2,2 are independent as BH3,H4,1 and BH3,H4,2.
Equation (5.23) implies that the FMFBS X can be rewritten as follows: X=X1+X2, where
(5.25)Xi(w1,w2,w3,w4,s)=λ1sH3+H4BH1,H2,i(w1,w2)+λ2sH1+H2BH3,H4,i(w3,w4).
Set
(5.26)Yi(t)=sup⁡0≤s≤tsup⁡0≤w1,w2,w3,w4≤s|Xi(w1,w2,w3,w4,s)|,t≥0,i∈{1,2}.
Then, given δ>0,
we have (see [11])
(5.27)Q≤ϕ(θ+2δ)ϕ(ν+2δ)+3P(Y2(t)>δtγ)+3P(Y1(u)>δuγ).
Equation (5.20) implies
(5.28)ϕ(θ+2δ)≤ϕ(θ)exp⁡(2δK3((log⁢(1/θ))βθ1+(1/α))),
and, consequently,
(5.29)ϕ(θ+2δ)ϕ(ν+2δ)≤ϕ(θ)ϕ(ν)exp⁡(2δK3((log(1/θ))βθ1+(1/α)+(log(1/ν))βν1+(1/α))).
If we choose δ=(t/u)r,
then we get the first term of the RHS of Lemma 5.3.
Next, we want
to obtain an upper bound of(5.30)P(Y2(t)>δtγ)=P(sup⁡0≤s≤1sup⁡0≤w1,w2,w3,w4≤s|λ1sH3+H4LH1,H2,2(w1,w2)+λ2sH1+H2LH3,H4,2(w3,w4)|>δ),
where we have, for (i,j)∈{(1,2),(3,4)}, (5.31)LHi,Hj,2(wi,wj)=∫|xi|≥v/t∫−∞wjgHi(wi,xi)gHj(wj,xj)W(d(xi,xj)).
We can show, by
standard computations, that
(5.32)P(Y2(t)>δtγ)≤P(sup⁡0≤w1,w2≤1|LH1,H2,2(w1,w2)|>δ2|λ1|)+P(sup⁡0≤w3,w4≤1|LH3,H4,2(w3,w4)|>δ2|λ2|).
Denote, by σH,
the covariance function of a FBM BH.
Set σH,2,
the covariance function of the process {BH,2(w),0≤w≤1}, defined by
(5.33)BH,2(w)=∫|x|≥v/tgH(w,x)W(dx),
where W(x),x∈ℝ, is a Wiener process.
Since(5.34)E(LH1,H2,2(w1,w2)LH1,H2,2(w1′,w2′))=σH1,2(w1,w1′)×σH2(w2,w2′),
we have, for any
[w1,w1+h1]×[w2,w2+h2]⊂[0,1]2,(5.35)E(LH1,H2,2([w1,w1+h1]×[w2,w2+h2])2)=E(∫[w1,w1+h1]×[w2,w2+h2]LH1,H2,2(d(x1,x2))×∫[w1,w1+h1]×[w2,w2+h2]LH1,H2,2(d(x1′,x2′)))≤∫w1w1+h1∫w1w1+h1|σH1,2(x1,x1′)|dx1dx1′×∫w2w2+h2∫w2w2+h2|σH2(x2,x2′)|dx2dx2′:=I×II.
Consider II
first. We get, by the inequality of Cauchy-Schwarz,
(5.36)II≤∫w2w2+h2∫w2w2+h2x2H2x2′H2dx2⁢dx2′≤h22.
Consider I now.
A straight computation implies that there existsKH1,2>0 depending on H1 such that
(5.37)E(BH1,2(x1))2≤KH1,22x12(v/t)2H1−2,
and, consequently, by the inequality of Cauchy-Schwarz,
(5.38)|σH1,2(x1,x1′)|≤KH1,22x1x1′(v/t)2H1−2.
So we get
(5.39)I≤KH1,22(v/t)2H1−2h12.
Hence,
combining (5.35) with (5.36) and (5.39), we have
(5.40)E(LH1,H2,2([w1,w1+h1]×[w2,w2+h2])2)≤KH1,22(v/t)2H1−2h12h22.
An application of Lemma A with
α1=α2=1, cκ=KH1,2(v/t)H1−1, and cκ−1δ>1 implies that
(5.41)P(sup⁡0≤w1,w2≤1|LH1,H2,2(w1,w2)|>δ2|λ1|)≤1C12,2exp⁡(−C12,2KH1,22(v2/t2)H1−1δ24λ12).
Similarly, we can establish
(5.42)P(sup⁡0≤w3,w4≤1|LH3,H4,2(w3,w4)|>δ2|λ2|)≤1C34,2exp⁡(−C34,2KH3,22(v2/t2)H3−1δ24λ22).
Set δ=(t/u)r.
Recall  that v2=ut and r≤(1−max(H1,H2,H3,H4))/3.
Combining (5.32) with (5.41) and (5.42), we get
(5.43)P(Y2(t)>δtγ)≤1C12,2exp⁡(−C12,24λ12KH1,22(ut)r)+1C34,2exp⁡(−C34,24λ22KH3,22(ut)r),
that is the
second term of the RHS of Lemma 5.3.
Finally, we
can establish a similar result forℙ(Y1(u)>δuγ),
that  is,
(5.44)P(Y1(u)>δuγ)≤1C12,1exp⁡(−C12,14λ12KH1,12(ut)r)+1C34,1exp⁡(−C34,14λ22KH3,12(ut)r),
which achieves the proof of Lemma 5.3.Finally, we
state the last technical lemma.Lemma 5.4.
There exists an integerp such
that if n>sups≤p(supIs), then for m∈J′′,m>n, given ϵ>0,
one has ℙ(En∩Em)≤(1+ϵ)btnbtm.Proof.
Letu(n)=k′ and u(m)=k1.
We have, by Lemma 4.5,
(5.45)tmtn≥exp⁡(exp⁡(K042min⁡(k′,k1))).
Let p∈ℕ.
Then k′>p and k1>p.
Thus we have min(k′,k1)>p.
Sett=tn,u=tm,θ=atn and ν=atm.
Note that log(1/θ)≤1/θ≤2(k′+1)α,log(1/ν)≤1/ν≤2(k1+1)α, and 1/btnbtm=exp(ψ(θ)+ψ(ν)).
By using Lemma5.3 and letting p→+∞,
we end the proof of Lemma 5.4.Lemmas5.2 and 5.4 yield that (4.2) holds. Combining Borel-Cantelli's second lemma with (4.2)
and (4.7), we show that, given ϵ>0,
(5.46)∑n∈JP(En)≥1+2Kϵ⟹11+2ϵ≤P(⋃n∈JEn)=P(⋃n∈J{Y(tn)≤f(tn)}),
and, consequently, f∈LUC(Y).
The proof of Theorem 2.4 is now complete.



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*Source: 160303-2008-03-17.xml*