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ON THE Φ-VARIATION OF STOCHASTIC PROCESSES WITH EXPONENTIAL MOMENTS
Andreas Basse-O’Connor, Michel Weber
To cite this version:
Andreas Basse-O’Connor, Michel Weber. ON THE
Φ-VARIATION OF STOCHASTIC PROCESSESWITH EXPONENTIAL MOMENTS. 2015. �hal-01277693�
arXiv:1507.00605v1 [math.PR] 2 Jul 2015
ON THE Φ-VARIATION OF STOCHASTIC PROCESSES WITH EXPONENTIAL MOMENTS
ANDREAS BASSE-O’CONNOR AND MICHEL WEBER
Abstract. We obtain sharp sufficient conditions for exponentially integrable stochastic pro- cessesX = {X(t) :t ∈ [0,1]}, to have sample paths with bounded Φ-variation. When X is moreover Gaussian, we also provide a bound of the expectation of the associated Φ-variation norm ofX. For an Hermite processXof orderm∈Nand of Hurst indexH∈(1/2,1), we show thatX is of bounded Φ-variation where Φ(x) =x1/H(log(log 1/x))−m/(2H), and that this Φ is optimal. This shows that in terms of Φ-variation, the Rosenblatt process (corresponding to m= 2) has more rough sample paths than the fractional Brownian motion (corresponding to m= 1).
1. Introduction and Main Results.
Let Φ : R+ → R+ be a strictly increasing continuous function such that Φ(0) = 0 and limt→∞Φ(t) =∞. The Φ-variation of a functionf : [0,1]→Ris defined according to Young [35]
by
VΦ(f) := sup
0=t0<···<tn=1 n∈N
Xn
i=1
Φ |f(ti)−f(ti−1)|
.
The particular case Φ(t) = tp is classical and corresponds for p= 1 to the concept of bounded variation introduced by Jordan [12], and forp >1 to the one of boundedp-variation introduced by Wiener [32]. The Φ-variation is closely related to convergence of Fourier series (see [5, Chapter 11]), Hausdorff dimension (see e.g. [4, Theorem 8.4]), rough paths theory (see [8, Definition 9.15]) and integration theory (see [5, Chapter 3]). Ifdis a pseudo-metric on [0,1] (dhas all properties of a metric except for the implicationd(s, t) = 0⇒s=t), we will write
V(Φ, d) := sup
0=t0<···<tn=1 n∈N
Xn
i=1
Φ d(ti, ti−1) .
Furthermore, set log∗(x) = log(1 +x) and log∗2(x) = log∗(log∗x) for all x≥0. In several cases we only define a given function Φ :R+ →R explicitly on (0,∞), and in this case we always set Φ(0) = 0.
A classical result by P. L´evy states that the sample paths of a Brownian motion are of bounded p-variation if and only ifp >2. This result has been improved by Taylor [26, Theorem 1 and its Corollary] who derived that the optimal Φ-variation function of the Brownian motion is Φ(t) = t2/log∗2(1/t). Taylor’s result has been extended to the fractional Brownian motion by Dudley and Norvaiˇsa [5] and to other Gaussian processes with stationary increments by Kawada and Kˆono [14] and Marcus and Rosen [17, 18]. Furthermore, the ability to solve rough differential equations for the Brownian motion under minimal regularity assumptions relies on the above mentionned characterization of the Brownian sample paths of Taylor, see Theorems 10.41 and 13.15 in [8].
Let us recall and briefly discuss a quite general result by Jain and Monrad which has also motivated this work, and which is the key ingredient in the recent work Friz et al. [7] on rough path analysis of Gaussian processes. Assume that X ={X(t) :t∈[0,1]} is a centered Gaussian
1Key words and phrases: Φ-variation, Gaussian processes, Hermite processes, metric entropy methods.
AMS 2010 subject classifications: Primary 60G17, 60G15; secondary 60G18, 60G22.
July 3, 2015
1
process and letd(s, t) =kX(s)−X(t)kL2 for alls, t∈[0,1]. Let 1< p <∞(the casep= 1 being trivial). Jain and Monrad [11, Theorem 3.2] showed that if
(1.1) V(Ψ, d)<∞ where Ψ(t) =tp(log∗2(1/t))p/2, t >0
then X has sample paths of bounded p-variation almost surely. A closer examination of their proof shows that (1.1) implies the Lipschitz property
(1.2) |X(t, ω)−X(s, ω)| ≤C(ω)d(s, t)
log∗ 1 d(s, t)
1/2
∀0≤s, t≤1
with probability one, which in turn is shown under the weaker and also necessary assumption (1.3) V( ˜Ψ, d)<∞ where Ψ(t) =˜ tp, t≥0.
As (1.2) immediately implies that X has almost all sample paths of bounded Φ-variation, where Φ(t) =tp(log∗(1/t))−p/2,t >0, the double logarithm term in Ψ indicates which strengthening of the assumption (1.3) is necessary to get the finer property of having paths of boundedp-variation.
It also follows (as remarked, (1.3) implies (1.2), see notably Remark 2.3) that the latter property holds only if X has the Lipschitzian property (1.2). For d continuous with respect to the usual metric on [0,1], the fact thatX be of bounded p-variation almost surely, thus implies thatX is continuous almost surely for the usual metric, which isin contrast with the fact that a function f with bounded p-variation is not necessarily continuous (although both limits limy→x−f(y), limy→x+f(y) exist). Recall in addition thatf is continuous almost everywhere in the sense of the Lebesgue measure, see Bruneau [3] for these facts; and that by Jordan’s example [12], a function with finite variation may have positive jumps on each rational.
We consider exponentially integrable processes and study the Φ-variation of their sample paths.
We obtain general sufficient conditions for the sample paths to be of bounded Φ-variation. Our conditions are sharp. When applied to Gaussian processes, we recover and complete Jain and Monrad sufficient condition for bounded p-variation, but also extend it to general Φ-variation spaces.
Introduce for every 0< α <∞, the functionsφα(x) =e|x|α−1,x∈R. We consider processes X satisfying the following increment condition, in whichdis a given pseudo-metric on [0,1]:
(1.4) For some 0< α <∞, Eφα
X(t)−X(s) d(t, s)
≤2 for allt, s∈[0,1].
The cases 1≤α <∞and 0< α <1 are of different nature since in the second case the functions φαare no longer convex. Accordingly, define for a random variableU
kUkφα= inf
∆>0 :Eφα
U
∆ ≤1
and writeU ∈Lφα, ifkUkφα <∞. Forα≥1,φαis an Orlicz function and the space (Lφα,k · kφα) is an Orlicz space and hence a Banach space. However, for 0< α <1, (Lφα,k ·kφα) is only a quasi- Banach space, i.e. satisfies all the axioms of a Banach space except that the triangle inequality is replaced by: there exists a constantKα>0 such that
kU +Ykφα ≤Kα kUkφα+kYkφα
for allU, Y ∈Lφα
(we may chooseKα= 21/αin our case). We refer to [24] for the theory of Orlicz spaces. The quasi- Banach space property of (Lφα,k · kφα) for 0< α <1 follows by [19, Section 2.1]. Furthermore, we refer to Talagrand [25] for sharp results on sample boundedness of stochastic processes satisfying (1.4).
The following theorem provides sufficient conditions for a class of stochastic processes with finite exponential moments to have sample paths of bounded Φ-variation.
Theorem 1.1. LetX ={X(t) :t∈[0,1]}be a separable stochastic process satisfying the increment condition (1.4)for someα >0. Suppose that there exists a continuous, strictly increasing function
Φ
Ψ : R+ → R+ with V(Ψ, d) < ∞. Suppose moreover that there exists C > 0 such that for all M >0,
X∞
m=0
2mΦ(M ym)e−xαm<∞ where (1.5)
xm= Φ−1(C2−m) KαR2−m
0 (log∗(2−mε ))1/αΨ−1(dε), ym= Z 2−m
0
log∗1 ε
1/α
Ψ−1(dε), (1.6)
and Kα is universal constant only depending on α. Then X has sample paths of bounded Φ- variation almost surely.
Here and in the next theorems, separability is understood with respect to the usual metric on [0,1], see Subsection 2.1. Forα= 2, a process satisfying (1.4) is called sub-Gaussian and this case includes all Gaussian processes if we setd(s, t) =kX(s)−X(t)kL2for alls, t∈[0,1]. On the other hand, processes satisfying (1.4) includes many non-Gaussian processes, for example all processes which are represented by multiple Wiener–Itˆo integrals of a fixed orderm ∈N, and in this case we can takeα= 2/mandd(s, t) =cmkX(s)−X(t)kL2 for a suitable constantcm>0. Processes satisfying (1.4) withα= 1 are usually called sub-exponential. From Theorem 1.1 we derive the following result:
Theorem 1.2. Suppose that X={X(t) :t∈[0,1]}is a separable stochastic process satisfying the increment condition (1.4) for someα >0. Then the following (1)–(4) holds:
(1) Suppose that V(Ψ, d)<∞for Ψ(t) =tp,p >0, and setΦ(t) =tp(log∗2(1/t))−p/α. Then X has sample paths of boundedΦ-variation almost surely.
(2) Suppose that V(Ψ, d)<∞ for Ψ(t) =tp(log∗2(1/t))−p/α and p >0. Then X has sample paths of boundedp-variation almost surely.
(3) For allβ >0 letΦβ(t) =e−t−1/β. Suppose thatV(Φβ0, d)<∞for some β0>1/α. Then for allβ <1−1/α+β0,X has sample paths of boundedΦβ-variation almost surely.
(4) For allc, r >0 letΦc,r(t) =e−r(log1t)c. Suppose thatV(Φc,r, d)<∞. Then for all v > r, X has sample paths of boundedΦc,v-variation almost surely.
It follows from Theorem 1.3 below that Theorem 1.2(1) gives the optimal result for all Hermite processes (including the fractional Brownian motion and the Rosenblatt process). In the special case whereX is a centered Gaussian process we recover the above cited result by Jain and Mon- rad [11, Theorem 3.2] from Theorem 1.2(2) if we set α = 2. The functions Φβ, defined in (3), play an important role in Fourier theory, where it follows from the Salem–Baernstein theorem that every continuous periodic function of bounded Φβ-variation has uniform convergent Fourier series if and only ifβ >1, see Remark 2.6 for more details. In particular, all continuous periodic functions of boundedp-variation for some p≥1 have an uniform convergent Fourier series. For α >1 we have 1−1/α >0 and hence we may chooseβ =β0in Theorem 1.2(3), which shows that V(Φβ, d)<∞implies thatX is of bounded Φβ-variation. This should be compared with Propo- sition 1.7, which shows that whenX is centered Gaussian (α= 2) and the function Φ satisfies the
∆2-condition (i.e. Φ(2x)≤CΦ(x) for all x), then the opposite implication holds, that is, if X is of bounded Φ-variation thenV(Φ, d)<∞. Notice, however, that the functions Φβ, considered in Theorem 1.2(3), do not satisfy the ∆2-condition. In the special case wherec= 1 andX is centered Gaussian, Theorem 1.2(4) implies the second statement of Jain and Monrad [11, Theorem 3.2].
In all of the above cited papers by Taylor [26], Dudley and Norvaiˇsa [5], Kawada and Kˆono [14]
and Marcus and Rosen [17, 18] the limiting Φ-variation of some Gaussian processes is moreover considered. To recall this notion we let, for any δ > 0, Πδ denote the set of all partitions π = {0 = t0 < · · · < tn = 1} of [0,1] such that |π| := maxi=1,...,nti −ti−1 ≤ δ, and for π∈Πδ set
vΦ(f, π) = Xn
i=1
Φ |f(ti)−f(ti−1)|
.
ThelimitingΦ-variation of a functionf : [0,1]→Ris defined by
(1.7) VΦ∗(f) := lim
δ→0
sup{vΦ(f, π) :π∈Πδ} .
LetX be a fractional Brownian motion with Hurst indexH ∈(0,1), and Φ denote the function Φ(t) = tH1
(log∗2(1/t))2H1 , t >0.
Dudley and Norvaiˇsa [5, Theorem 12.12] showed that with probability one
(1.8) lim
δ→0
sup{vΦ(X, π) :π∈Πδ}
= 22H1 ,
which characterizes the limiting Φ-variation of the fractional Brownian motion, and includes the Brownian motion case for H = 1/2, which goes back to the fundamental work Taylor [26, The- orem 1]. Eq. (1.8) implies that the fractional Brownian motion is of bounded Φ-variation, and that Φ is optimal for in the sense described in Theorem 1.3 below. Kawada and Kˆono [14] and Marcus and Rosen [17, 18] derived the limiting Φ-variation of other classes of Gaussian processes with stationary increments. On the other hand, the limiting Φ-variation of a symmetricα-stable L´evy processX withα∈(0,2) is characterized by Fristedt and Taylor [9, Theorem 2] and from their result it follows that there does not exists an optimal Φ-variation function for an α-stable L´evy process, in contrast to the Brownian motion.
In the following we will recall the definition of Hermite processes (see Taqqu [28] or Tudor [30]):
LetB be an independently scattered symmetric Gaussian random measure on Rwith Lebesgue intensity measure. An Hermite process of order m∈N with Hurst parameter H ∈(1/2,1) is a stochastic processX ={X(t) :t∈[0,1]} of the form
(1.9) X(t) =c0
Z
Rm
Z t
0
Ym
i=1
(v−ui)−(1/2+(1−H)/m)
+ dv
dB(u1)· · ·dB(um).
Here c0 > 0 is a positive norming constant and the integral in (1.9) is a multiple Wiener–Itˆo integral, see e.g. Nualart [23]. Hermite processes appear as the limit in non-central limit theorems, see Taqqu [27, 28] or Tudor [30], they have stationary increments, are self-similar with index H, and for m ≥ 2 are non-Gaussian. An Hermite process X of order m = 2 is usually called a Rosenblatt process, and form= 1,X is the fractional Brownian motion. We may and do assume thatX has been chosen with continuous sample paths almost surely.
Our next result is the following theorem which characterizes the limiting Φ-variation of Hermite processes.
Theorem 1.3. LetX ={X(t) :t∈[0,1]}be an Hermite process of orderm∈Nwith Hurst index H ∈(1/2,1), and Φ = Φm,H be given by
(1.10) Φ(t) = tH1
(log∗2(1/t))2Hm t >0.
Then with probability one
(1.11) lim
δ→0
sup{vΦ(X, π) :π∈Πδ}
=σm,H
where the constant σm,H ∈ (0,∞) is defined in (1.12). In particular, we deduce that X has sample paths of bounded Φ-variation with probability one, and Φ is optimal in the sense that for all Φ :˜ R+ → R+ satisfying Φ(x)/Φ(x)˜ → ∞ as x → 0, we have VΦ˜(X) = ∞ a.s. Moreover, σm,H ≤ VΦ(X)<∞a.s.
We note that the functions Φ = Φm,H are decreasing inm, in particular, for all 1≤m < kwe have that Φk,H(x)/Φm,H(x)→0 asx→0, and hence Theorem 1.3 shows that whenmincreases then the sample paths of X becomes more “rough” when measured in terms of Φ-variation. In particular, the Rosenblatt process with Hurst index H has more “rough” sample paths than the fractional Brownian motion. The next remark concerns the constant σm,H appearing in Theorem 1.3.
Φ
Remark 1.4. The constantσm,H in (1.11) is defined by (1.12)
σm,H= 22Hm supn
Z
Rm
Q(u1, . . . , um)ξ(u1)· · ·ξ(um) du1· · ·dum
:ξ∈L2(R), kξk2≤1o1/H
where
Q(u1, . . . , um) =c0
Z 1 0
Ym
i=1
(v−ui)−(1/2+(1−H)/m)
+ dv.
We note that 0< σm,H <∞, and in fact we have the upper bound σHm,H≤2m2kQkL2(Rm)= 2m2(m!)−12kX(1)kL2
and
(1.13) kQkL2(Rm)=c0
hβ(1/2−1−Hm ,2−2Hm )m H(2H−1)
i1/2
where β(x, y) := R1
0 tx−1(1−t)y−1dt denotes the beta function evaluated in (x, y) ∈ (0,∞)2. Equality (1.13) follows by the proof of [30, Proposition 3.1]. By duality arguments it follows that σm,HH = 212kX(1)kL2 form= 1, whereasσHm,H<2m2kX(1)kL2 for m≥2.
The next results are related to Theorems 1.1–1.3. We will in particular estimate the associated Φ-variation norm ofX. Some notions and notation are necessary. LetBΦbe the class of all real functions f : [0,1]→Rsuch thatVΦ(f)<∞, namely having bounded Φ-variation. Recall some basic facts in the case where Φ is convex. Then Φ is also continuous. The classBΦis a symmetric and convex set, but is not necessarily a linear space. Musielak and Orlicz [22] haved proved that BΦis a vector space if and only if Φ verifies the ∆2-condition (Φ(2x)≤CΦ(x) for all x). In the latter case we define for anyf ∈ BΦ,
kfkΦ= inf{r >0 :VΦ(f /r)≤1},
and |||f|||Φ = |f(0)|+kfkΦ. We know by a theorem of Maligranda and Orlicz [15], that the space (BΦ,|||·|||Φ) is a Banach space, and in fact a Banach algebra. When Φ(x) = |x|p, p ≥1, we denote this space Bp, and we will write VΦ(·) = Vp(·) and k · kΦ =k · kp; we note that the latter takes the simple formkfkp =Vp(f)1/p. In the following we give explicit estimates on the expected Φ-variation norm of Gaussian processes under some additional assumptions on Φ and Ψ.
In particular, Corollary 1.6 completes Jain and Monrad’s result [11, Theorem 3.2] with an explicit estimate of the expectedp-variation norm. We note that in Jain and Monrad’s result, Φ and Ψ are given and conditionV(Ψ, d)<∞reads on d. Here we adopt the same point of view and we notice that our main result can also be read this way for Gaussian processes, namely with Φ and Ψ satisfying (1.5), nextX subject to satisfy conditionV(Ψ, d)<∞(the increment condition (1.4) is automatically satisfied withα= 2).
Theorem 1.5. Suppose that Φ,Ψ : R+ → R+ are strictly increasing continuous functions with Φ(0) = Ψ(0) = 0, limt→∞Φ(t) = limt→∞Ψ(t) = ∞ such that Φ and Ψ satisfy (1.5) of Theo- rem 1.1. In addition, suppose that Φis convex andΨ−1 is absolute continuous with a derivative (Ψ−1)′ satisfying
(1.14) (Ψ−1)′(xy)≤K0xq−1(Ψ−1)′(y), and that Φ(xy)≤K0xpΦ(y)
for all x, y ≥0 and some constants p, q, K0 >0. For all separable, centered Gaussian processes X ={X(t) :t∈[0,1]} with d(s, t) =kX(s)−X(t)kL2 for all s, t∈[0,1], such thatV(Ψ, d)<∞, the following estimate holds
EkXkΦ≤K
V(Ψ, d) +V(Ψ, d)p/2+1+V(Ψ, d)pq1/p
whereK is a finite constant not depending on process X.
Corollary 1.6. Let p≥1 and set
Ψ(x) =xp[log∗2(1/(x∧1))]p/2 x >0.
For all separable, centered Gaussian processes X = {X(t) : t ∈ [0,1]} with d(s, t) = kX(s)− X(t)kL2 for all s, t∈[0,1], the estimate holds
(1.15) EkXkp≤KWp1/p(1 +Wp1/2) where Wp :=V(Ψ, d) andK is a finte constant not depending on process X.
Recall that in the setting of Corollary 1.6 the conditionWp<∞is Jain and Monrad’s sufficient conditions of bounded p-variation. In the case where Wp ≤1, (1.15) simplifies to the estimate EkXkp≤KWp1/p. The next result gives a necessary condition for a centered Gaussian process to be of bounded Φ-variation in the case where Φ satisfies the ∆2-condition, i.e., there exists a finite constantC such that Φ(2x)≤CΦ(x) for allx≥0.
Proposition 1.7. Suppose that X ={X(t) :t∈[0,1]} is a separable, centered Gaussian process, d(s, t) =kX(s)−X(t)kL2 for s, t∈[0,1],Φ is convex and satisfies the ∆2-condition. If X is of boundedΦ-variation thenV(Φ, d)<∞.
Next we discuss some related results. For p-variation of Markov processes we refer to Man- staviˇcius [16], for p-variation of stable processes we refer to Xu [34], and for results on the p- variation of the local time of stable L´evy processes in the space variable we refer to Marcus and Rosen [17, 18]. A result of Vervaat [31], states that if X is a self-similar process with index 0< H≤1 and has moreover stationary increments, then its sample paths have nowhere bounded variation, unlessX(t)≡tX(0). The weak variation of a class of Gaussian processes is characterized in Xiao [33], and various results on the moduli of continuity for Gaussian random fields are obtained in Meerschaert et al. [20]. In a different direction, the asymptotics of the renormalized quadratic variation of the Rosenblatt process is analyzed in Tudor and Viens [29] using Malliavin calculus.
The paper is organized as follows: Theorems 1.1–1.2 are proved in Section 2. Theorem 1.3 is proved in Section 3. We prove Theorem 1.5, Corollary 1.6 and Proposition 1.7 in Section 4.
Finally, some results used in the proofs are moved to the Appendix.
2. Proofs of Theorems 1.1 and 1.2.
2.1. Preliminaries. Recall some basic facts. Let d be a pseudo-metric on [0,1] with finite diameter D. Moreover, letX ={X(t) :t∈ [0,1]} be a stochastic process with basic (complete) probability space (Ω,A,P). We further say that X is separable (for the usual distance on [0,1]), if there exists a countable subset S of [0,1] (separation set) and a null set N of A, such that for any ω ∈ Nc and any t ∈ [0,1], there is a sequence {tn, n ≥ 1} ⊂ S verifying tn → t and X(t, ω) = limn→∞X(tn, ω). By [10, Ch. 4.2, Theorem 1], every stochastic process X ={X(t) : t∈[0,1]} admits a separable versionXe with values in the extended reals. The measurability of the functionalVΦ(X) follows from the separability assumption onX. For each continuous, strictly increasing function f : R+ → R, f−1 will denote its inverse. Recall that log∗(x) = log(x+ 1), log∗2(x) = log∗(log∗x) for all x ≥ 0, and that Φ : R+ → R+ denotes a continuous, strictly increasing function with Φ(0) = 0 and limt→∞Φ(t) =∞.
The following metric entropy result, Theorem 2.1, plays a key role in the proof of Theorem 1.1 both to obtain good modulus of continuity estimates on X, but also to estimate certain prob- abilities. It relies on the covering numbers N(T, d, ε), ε > 0, of a pseudo-metric space (T, d), which are defined to be the smallest number of open balls of radius εto coverT. Note also that φ−1α (x) = (log∗x)1/α, x≥0.
Theorem 2.1. Let (T, d) be a pseudo-metric space with diameter D, and letX ={X(t) :t∈T} be a separable stochastic process such that the increment condition (1.4)is fulfilled for some 0<
α <∞and with[0,1]replaced byT. Further assume that Z D
0
(log∗N(T, d, ε))1/αdε <∞.
Φ
Let δ(ε) =Rε
0(log∗N(T, d, u))1/αdu,ε∈[0, D]. Then there exists a finite constantKαdepending onαonly such that
sup
s,t∈T
|X(s)−X(t)|
δ(d(s, t)) φα
≤ Kα.
Remarks 2.2. (i): It is possible to deduce Theorem 2.1 from Kwapie´n and Rosi´nski [13], Corol- lary 2.2 and Remark 1.3(2), by modifying the functionsφα in a suitable way in the caseα <1 to obtain a Young function. The result of [13] relies on the majorizing measure method, and hence does also apply under the metric entropy integral condition by a well known existence result. For the reader’s convenience we give a direct proof of Theorem 2.1 using metric entropy methods and which holds for anyα >0 in Appendix A.
(ii): Theorem 2.1 implies that sup
s,t∈T
|X(s)−X(t)|
φα ≤ Kα
Z D 0
(log∗N(T, d, ε))1/αdε, a classical result. And at this regard, it is a natural complement of this one.
2.2. Proof of Theorem 1.1. Put for 0≤t≤1, F(t) = sup
0=t0<···<tn=t n∈N
Xn
i=1
Ψ d(ti, ti−1) . The basic observation is that ifs≤t, then Ψ d(s, t)
≤F(t)−F(s). Hence, (2.1) d(t, s)≤Ψ−1 F(t)−F(s)
0≤s≤t≤1,
and in particular D ≤ Ψ−1(F(1)) (notice that F(1) = V(Ψ, d)). Introduce for all m ∈ Z and j= 1, . . . ,⌈F(1)2m⌉, the sets
Im,j =](j−1)2−m, j2−m], Sm,j ={s∈[0,1] :F(s)∈Im,j}.
Put also
Zm=♯
j= 1, . . . ,⌈F(1)2m⌉: sup
s,t∈Sm,j
Φ(|X(t)−X(s)|)> C2−m ,
where♯Adenotes the number of elements in a setA, and⌈x⌉denotes the least integer larger thanx.
We haved(s, t)≤Ψ−1(2−m) for alls, tsuch thatF(s), F(t)∈Im,j. Let 0< ρ <2−m/2. Consider a subdivision ofIm,j of size 2ρ, thus of lengthN=N(ρ) bounded by
2−m/ρ
+ 1≤2−m+1/ρ. It induces a partitionU1, . . . , UN(ρ)ofSm,j. For any non-empty elementU from this partition, any s, t∈U, we have Ψ−1(|s−t|)≤Ψ−1(2ρ). Pickuarbitrarily inU, we see that thed-ball centered at uand of radius Ψ−1(2ρ), contains U. Moreover, repeating this operation for each non-empty element U1, . . . , UN(ρ), we manufacture like this a covering of Sm,j byd-balls of radius Ψ−1(2ρ), centered in Sm,j of size at mostN. Thus N(Sm,j, d,Ψ−1(2ρ))≤2−m+1/ρ. Lettingρ= Ψ(ε)/2, we deduce
(2.2) N(Sm,j, d, ε)≤2−m+2
Ψ(ε) , 0< ε <Ψ−1(2−m).
In particular, form= 0 and by considering ˜F(t) =F(t)/F(1) and ˜Ψ(t) = Ψ(t)/F(1) instead ofF and Ψ we obtain that
(2.3) N([0,1], d, ε)≤ 4F(1)
Ψ(ε) , 0< ε <Ψ−1(F(1)).
By Remark 2.2(ii) and (2.2), sup
s,t∈Sm,j
|X(s)−X(t)|
φα ≤ Kα
Z Ψ−1(2−m) 0
log∗2−m+2 Ψ(ε)
1/α
dε≤Kα41/αym
where we use the estimate
(2.4) log∗(yx)≤(y∨1) log∗(x) for ally, x≥0.
With Cgiven as in the theorem andxmdefined in (1.5) we have Pn
sup
s,t∈Sm,j
Φ(|X(s)−X(t)|)> C2−mo
≤ P n
sup
s,t∈Sm,j
|X(s)−X(t)|> xm
sup
s,t∈Sm,j
|X(s)−X(t)|
φα
o
≤ 1
φα(xm). And this holds forj= 1, . . . ,⌈F(1)2m⌉. By lettingr0=φ−1α (1) and ˜xm=xm∨r0 we have
(2.5) Pn
sup
s,t∈Sm,j
Φ(|X(s)−X(t)|)> C2−mo
≤ 1
φα(˜xm) since the left-hand side of (2.5) is always less than or equal to one.
Fix now a partitionπ={0 =t0<· · ·< tn = 1}of [0,1]. Letm0∈Zbe any number satisfying max1≤i≤nF(ti)−F(ti−1)≤2−m0. Introduce the following sets
Λm = n
1≤i≤n: 2−m−1< F(ti)−F(ti−1)≤2−mo
, m≥m0,
E(ω) = n
1≤i≤n: Φ |X(ti, ω)−X(ti−1, ω)|
> C[F(ti)−F(ti−1)]o . (2.6)
We have Xn
i=1
Φ |X(ti, ω)−X(ti−1, ω)|
= X
i∈E(ω)
Φ |X(ti, ω)−X(ti−1, ω)|
+ X
i /∈E(ω)
Φ |X(ti, ω)−X(ti−1, ω)|
≤ X
i∈E(ω)
Φ |X(ti, ω)−X(ti−1, ω)|
+CF(1).
(2.7)
In order to control the remainding subsum, we start with the following simple bound, X
i∈E(ω)
Φ |X(ti, ω)−X(ti−1, ω)|
= X∞
m=m0
X
i∈E(ω)∩Λm
Φ |X(ti, ω)−X(ti−1, ω)|
≤ X∞
m=m0
sup
i∈E(ω)∩Λm
Φ |X(ti, ω)−X(ti−1, ω)|
♯ E(ω)∩Λm . (2.8)
We now need some auxiliary estimates. We first claim that
(2.9) ♯ E(ω)∩Λm
≤9Zm−2(ω) for allm≥m0.
For i ∈ E(ω)∩Λm choose an integer j = 1, . . . , . . . ,⌈F(1)2m⌉ such that F(ti) ∈ Im,j. Then F(ti−1)∈Im,j−1∪Im,j =J, say. For somek= 1, . . . ,⌈F(1)2m−2⌉, we haveJ ⊆Im−2,k which by definition of ΛmandE(ω) implies that
(2.10) sup
s,t∈Sm−2,k
Φ(|X(t, ω)−X(s, ω)|)> C(F(ti)−F(ti−1))≥C2−(m−2). On the other hand, for eachk= 1, . . . ,⌈F(1)2m−2⌉we have
(2.11) ♯{i∈E(ω)∩Λm:F(ti)∈Im−2,k} ≤9.
Indeed, to show (2.11) let i1 and i0 be the largest and least integers in Ak :={i∈E(ω)∩Λm: F(ti)∈Im−2,k}. SinceF(ti0), F(ti1)∈Im−2,k, by definition, we have
2−(m−2) ≥ F(ti1)−F(ti0) = X
i∈Ak:i>i0
[F(ti)−F(ti−1)]
≥ X
i∈Ak:i>i0
2−m−1≥2−m−1 ♯Ak−1 ,
which shows (2.11). Combining (2.10) and (2.11) we obtain (2.9). By (2.8) and (2.9), X
i∈E(ω)
Φ |X(ti, ω)−X(ti−1, ω)|
≤ 9 X∞
m=m0
Zm−2(ω) sup
i∈E(ω)∩Λm
Φ |X(ti, ω)−X(ti−1, ω)|
.
Φ
Besides, by Theorem 2.1 there exists a finite constantKα>0 depending onαonly such that if Θ = sup
s,t∈[0,1]
|X(s)−X(t)|
δ(d(s, t)) , (2.12)
thenkΘkφα ≤Kα (δis defined in Theorem 2.1 withT = [0,1]). Now, ifi∈E(ω)∩Λm, we have by (2.1),d(ti, ti−1)≤Ψ−1 F(ti)−F(ti−1)
≤Ψ−1(2−m), and hence Φ |X(ti, ω)−X(ti−1, ω)|
≤ Φ Θ(ω)δ d(ti, ti−1)
≤Φ Θ(ω)δ◦Ψ−1(2−m) . By (2.3) and (2.4) we have withcd= [(4F(1))∨1]1/α that
δ◦Ψ(2−m)≤
Z Ψ(2−m) 0
log∗4F(1) Ψ(ε)
1/α
dε
≤cd
Z Ψ(2−m) 0
log∗ 1 Ψ(ε)
1/α
dε=cdym, and hence
(2.13) X
i∈E(ω)
Φ |X(ti, ω)−X(ti−1, ω)|
≤9 X∞
m=m0
Φ Θ(ω)cdym
Zm−2(ω).
In the following we will show that the right-hand side of (2.13) is finite almost surely. For allǫ >0 chooseM such thatP(Θ≤M)≥1−ǫ. Forω∈ {Θ≤M},
X∞
m=m0
Φ Θ(ω)cdym
Zm−2(ω)≤ X∞
m=m0
Φ M cdym
Zm−2(ω) =:ZM(ω).
Suppose that m0 is the greatest integer such that F(1) ≤2−m0, and hence for all m ≥ m0 we have thatF(1)2m≥1/2, which implies that⌈F(1)2m⌉ ≤2F(1)2m. By definition ofZmand (2.5) we have for allm≥m0,
EZm =
⌈F(1)2m⌉
X
j=1
Pn sup
s,t∈Sm,j
Φ(|X(s)−X(t)|)> C2−mo
≤ ⌈F(1)2m⌉ 1
φ(˜xm) ≤2F(1)2m 1
φα(˜xm) ≤4F(1)2me−xαm (2.14)
where we have used that 1/φα(˜xm)≤2e−xαm in the last inequality. Hence EZM =
X∞
m=m0
Φ M cdymEZm−2≤4F(1) X∞
m=m0
2mΦ M cdym
e−xαm <∞
by assumption (1.5). That is, the random variable in (2.13) is finite with probability greater than 1−ǫfor allǫ >0 and hence finite almost surely. By (2.7)
(2.15)
Xn
i=1
Φ |X(ti, ω)−X(ti−1, ω)|
≤9 X∞
m=m0
Φ Θ(ω)cdym
Zm−2(ω) +CF(1).
This inequality holds for allωin a measurable set Ω0of probability 1 and all partitionsπ. Note also that the right-hand sidedoes notdepend on the given partitionπ; it only depends on the process X. So that we can take supremum over all partitions πon the left-hand side. And recalling in view of the separability assumption thatVΦ(X) is measurable, we have shown that for allω∈Ω0,
VΦ(X(·, ω))≤9 X∞
m=m0
Φ Θ(ω)cdym
Zm−2(ω) +CF(1),
which shows thatX has sample paths of bounded Φ-variation almost surely, and completes the
proof.
Remark 2.3. Suppose that V(Ψ, d) <∞ with Ψ(x) = xp, and letδ(ε) be defined as in Theo- rem 2.1 withT = [0,1]. Then the estimate (2.3) shows that
δ(ε)≤ Z ε
0
log∗ 4 εp
1/α
du≤Cp,α ε log∗1 ε
1/α
. Specify this forX Gaussian (α= 2). We deduce from Theorem 2.1, that
sup
s,t∈T
|X(s)−X(t)|
d(s, t)(log∗d(s,t)1 )1/2 φ2
≤ K,
which justifies the implication (1.3) ⇒(1.2) invoked in the Introduction. Furthermore, the above estimates remain valid under the conditions considered in Theorem 1.2(2).
Remark 2.4. The following example provides a good illustration of Jain and Monrad result (see Introduction). Taking d(s, t) = d1(|s−t|) where d1(u) ≤ u1/plog∗2(1/u)−1/2 and Ψ(t) = tp(log∗2(1/t))p/2 for t > 0, we see that V(Ψ, d) < ∞, so that X has sample paths of bounded p-variation almost surely. Further, from Remark 2.3
|X(t, ω)−X(s, ω)| ≤C(ω)|s−t|1/p (log∗|s−t|1 )1/2
(log∗2 |s−t|1 )1/2 ∀0≤s, t≤1.
thereby implying that has sample paths of bounded Ψ1-variation almost surely, where Ψ1(x) =
|s−t|p(log∗2|s−t|1 )−p/2(log∗|s−t|1 )p/2, which is clearly weaker than boundedp-variation. Thus, we can not obtain boundedp-variation only from the modulus of continuity ofX.
Next we will show Theorem 1.2 by suitable applications of Theorem 1.1.
2.3. Proof of Theorem 1.2. Cases (1)–(4) follow from Theorem 1.1, once we have shown that the sum (1.5) is finite. The two sequences (xm)m≥1 and (ym)m≥1 are defined in (1.6). In the following K with subscript will denote a finite constant only depending on the subscript, but might vary throughout the proof. Given two sequences (am)m≥1and (bm)m≥1, we writeam∼bm
asm→ ∞ifam/bm→1 asm→ ∞, and similar for functions.
Case (1): Set Ψ(t) =tp and Φ(t) =tp(log∗2(1/t))−p/α for all t >0 and Φ(0) = Ψ(0) = 0. Since Ψ−1(t) =t1/p we have that
ym= Z 2−m
0
log∗1 ε
1/α
Ψ−1(dε) = (1/p) Z 2−m
0
log∗1 ε
1/α
ε1/p−1dε∼(1/p)m1/α2−m/p asm→ ∞since the integrand is regularly varying. For allm≥1, we have by substitution that
Z 2−m 0
log∗2−m ε
1/α
Ψ−1(dε) =Kα,p2−m/p.
We note that Φ−1(t)∼t1/p(log∗2(1/t))1/αast→0, and hence, for allC >0 and allmlarger than somem0≥1
xm= Φ−1(C2−m) KαR2−m
0 (log∗(2−mε ))1/αΨ−1(dε) ≥Kα,pC1/p2−m/plog(m)1/α
2−m/p =Kα,pC1/plog(m)1/α. For allM >0,
X∞
m=m0
2mΦ(M ym)e−xαm ≤MpKα,p
X∞
m=m0
2m2−mmp/αe−Kα,pα Cα/plogm
=MpKα,p
X∞
m=m0
mp/α−Kα,pα Cα/p <∞ forC= (p/α+ 2)p/αKα,p−p. This shows (1.5) and completes the proof of (1).
Φ
Case (2): The proof of this case is analogous with the one of case (1). We set Φ(t) = tp and Ψ(t) = tp(log∗2(1/t))p/α. As t → 0 we have Ψ−1(t) ∼ t1/p/(log∗2(1/t))1/α and (Ψ−1)′(t) ∼ (1/p)t1/p−1/(log∗2(1/t))1/α. Hence
ym= Z 2−m
0
log∗1 ε
1/α
(Ψ−1)′(ε) dε≤Kα,p2−m/pm1/p, and therefore
Φ(M ym)≤MpKα,p2−mmp/α
for allmlarge enough. Further,xm≥Kα,pC1/p(logm)1/α for allmlarge enough. Indeed, Z 2−m
0
log∗2−m ε
1/α
Ψ−1(dε) = 2−m Z 1
0
log∗ 1 v
1/α
(Ψ−1)′(2−mv) dv
≤ Kα,p2−m Z 1
0
log1 v
1/α
(2−mv)1p−1 dv (log∗2 2vm)1/α
≤ Kα,p 2−m/p (logm)1/α
Z 1 0
log1 v
1/α
v1p−1dv
≤ Kα,p 2−m/p (logm)1/α. Thus
xm= Φ−1(C2−m) KαR2−m
0 log∗(2−mε )1/α
Ψ−1(dε) ≥Kα,pC1/p 2−m/p
2−m/p (logm)1/α
=Kα,pC1/p(logm)1/α
formlarge enough. Hence (1.5) follows by settingC= (p/α+ 2)p/αKα,p−p.
Case (3): Let Φβ(t) =e−t−1/β and note that Φ−1β (t) = (log(1/t))−β,t >0. Letβ0>1/αand set Ψ = Φβ0, Φ = Φβ. For all m≥1,
ym≤y1= Z 1
0
log∗1 ε
1/α
Ψ−1(dε)
=
Z Ψ−1(1) 0
log∗ 1 Ψ(ε)
1/α
dε≤Kα,β0
Z Ψ−1(1) 0
ε−1/(β0α)dε <∞.
Moreover, Z 2−m
0
log∗2−m ε
1/α
Ψ−1(dε) = Z 2−m
0
log∗2−m ε
1/α log1
ε
−1−β0
ε−1dε
= Z 1
0
log∗4 ε
1/α
log2m ε
−1−β0
ε−1dε≤Kα,β0m−1−β0.
SetC= 1. Since Φ−1(C2−m)≥m−β we have thatxm≥m1+β0−β, and hence X∞
m=1
2mΦ(M ym)e−xαm ≤Φ(M y1) X∞
m=1
exp
mlog(2)−mα(1+β0−β)
<∞ (2.16)
sinceα(1 +β0−β)>1.
Case (4): Recall that Φc,r(t) = e−r(log1t)c and note that Φ−1c,r = Φ1
c,r′ with r′ =r−1/c. We set Ψ = Φc,r and Φ = Φc,v wherev > r. Sinceε7→(log(1/ε))c/α is slowly varying at zero we have as
n→ ∞
ym= Z 2−m
0
log∗1 ε
1/α
Ψ−1(dε) =
Z Ψ−1(2−m) 0
log∗ 1 Ψ(ε)
1/α
dε
= Z Φ1
c,r′(2−m) 0
log1 ε
αc dε
∼Φ1
c,r′(2−m)
log 1
Φ1
c,r′(2−m) αc
= (r′)α1e−r′(mlog 2)1/c(mlog 2)α1. SetC= 1. Further, as we always have thatxm≥KαΦ−1(C2−m)y−1m, it follows that
xm≥1
2Kαexph
(r′−v′)(mlog 2)1/ci
(mlog 2)−1α ≥exp(δm1/c)
for allm≥m0, for somem0≥1, whereδ= (r′−v′)(log 2)1/c/2. For allM >0 we have X∞
m=m0
2mΦ(M ym)e−xαm ≤Φ(M y1) X∞
m=m0
exph
mlog(2)−exp
αδm1/ci
<∞.
Remarks 2.5.
1. In the two first cases of the proof of Theorem 1.2 the series in (1.5) converges slowly, whereas in the two last cases, the above series converges fastly.
2. Concerning Case 3, we observe from (2.16) that by the assumption made on β, namely β <
1 +β0−1/α, the convergence of the series in (1.5) is controlled by the sole sequence (xm), the fact that (ym) be bounded indeed suffices. Ifβ < β0−1/α, then the convergence of the above series in (1.5) is already granted by the sequence (ym).
3. Concerning Case 4, let 0< r′′< r′. Notice, although not used here thatym≤Ke−r′′(mlog 2)1/c, and so
Φ(M ym)≤expn
−v
log 1
M Ke−r′′(mlog 2)1/c co
≤K′expn
−v(r′′)c(mlog 2)o ,
where K, K′ depend on r′, r′′, c, M. And so when 0 < r < 1, Φ(M ym) ≤K′′2−ρm for suitable constants K′′ and ρ > 1. Consequently, P∞
m=m02mΦ(M ym) <∞, which of course suffices for concluding.
Remark 2.6. For β > 0 let Φβ(t) = e−t−1/β for all t > 0 and Φβ(0) = 0. Moreover, for any r > 0 let ˜Φβ,r(t) = Rt
0Φβ(s)rds, t ≥ 0. Fix an r > 0. From the Salem–Baernstein theorem, [5, Theorems 11.8 and 11.13], it follows that every continuous periodic function of bounded ˜Φβ,r- variation has uniform convergent Fourier series if and only if β >1 (argue as on p. 568 in [5]).
The same result holds with ˜Φβ,rreplaced by Φβ, which follows from the estimate Φ˜β,1(t)≤Φβ(t)≤Φ˜β,12(t)
which holds for allt close enough to zero.
3. Proof of Theorem 1.3.
Letn≥0 be a fixed positive integer andHn:R→Rbe then-th Hermite polynomial which is defined by
Hn(x) = (−1)n
n! ex2/2 dn
dxne−x2/2, n≥1, andH0(x) = 1. For eachf ∈L2(R) setB(f) =R
Rf(s) dB(s) and letHn be theL2-closure of the linear span of
nHn(B(f)) : f ∈L2(R),kfkL2 = 1o .
Φ
Next choose a real numberan>0 such that
(3.1) na1/nn +
X∞
k=n+1
(ka2/nn 2/n)k
k! ≤2.
AllY ∈ Hn satisfy the following equivalence of moments inequality kYkp≤pn/2kYk2 for allp >2, cf. e.g. [1, Eq. (4.1)], and hence for allY ∈ Hn withkYk2≤an we have
Eexp(|Y|2/n) = Xn
k=0
1
k!E[|Y|2k/n] + X∞
k=n+1
1
k!E[|Y|2k/n]
≤na1/nn + X∞
k=n+1
(ka2/nn 2/n)k
k! ≤2.
(3.2)
Throughout the rest of this sectionX ={X(t) :t ∈[0,1]} will denote an Hermite process of the form (1.9), and we letQt denote the kernel
Qt(u1, . . . , um) =c0
Z t 0
Ym
i=1
(v−ui)−(1/2+(1−H)/m)
+ dv, u1, . . . , um∈R.
SinceX(t)∈ Hm for all t∈[0,1], (3.2) shows that X satisfies (1.4) with α= 2/mand d(s, t) = a−1mkXt−Xsk2 for all s, t∈ [0,1], wheream is given by (3.1). By self-similarity and stationary increments we have thatd(s, t) = ˜c0|s−t|H for alls, t∈[0,1] and a suitable constant ˜c0. We let Φ be given by (1.10) and Ξ :R+→R+denote the function
Ξ(x) =xH(log∗2(1/x))m/2 x >0, Ξ(0) = 0.
We note that Ξ is an asymptotic inverse to Φ in the sense that Φ(Ξ(x))∼Ξ(Φ(x))∼xasx→0.
3.1. Proof of the upper bound in Theorem 1.3. In the following we will show the upper bound
(3.3) VΦ∗(X)≤σm,H a.s.
To show (3.3) we will need the following two-sided Law of the Iterated Logarithm forX. Lemma 3.1. For each t >0 we have with probability one,
(3.4) lim
δ→0
sup
u,v≥0 0<u+v<δ
|X(t+u)−X(t−v)|
Ξ(u+v)
≤σm,HH .
Key elements in the proof is a large deviation result by Borell [2] and methods of Dudley and Norvaiˇsa [5, Lemma 12.21].
Proof. Fixt > 0 and consider the process Z(u, v) =X(t+u)−X(t−v). For all δ ∈(0,1) set S(δ) ={(u, v)∈R2+: 0< u+v≤δ}, and note thatS(δ) =δS(1). By the stationary increments and self-similarity ofX it follows that Z is self-similar of indexH, and hence
(3.5) P
sup
(u,v)∈S(δ)
|Z(u, v)|> z
=P sup
(u,v)∈S(1)
|Z(u, v)|> zδ−H . By Borell [2], we have that
(3.6) t−2/mlogP
sup
(u,v)∈S(1)
|Z(u, v)| ≥t
→ −1
2ςZ2/m ast→ ∞ where
ςZ = sup
ξ∈L2(R) kξkL2(R)≤1
(u,v)∈S(1)
Z
Rm
Qt+u(u1, . . . , um)−Qt−v(u1, . . . , um)
ξ(u1)· · ·ξ(um) du1· · ·dum
.
Using the scaling property of the kernel {Qt: t ∈ [0,1]} we obtain that ςZ = 2−m/2σHm,H. Fix ǫ ∈ (0,1/2) and choose ˜ǫ > 0 such that (1−ǫ)˜2(1 +ǫ)2/m > 1. For n ∈ N set δn = e−n1−˜ǫ, Ξn=δHnςZ(2 log∗2(1/δn))m/2,Sn =S(δn) and
En=n sup
(u,v)∈Sn
|Z(u, v)| ≥(1 +ǫ)Ξn
o.
By (3.5),
P(En) = P sup
(u,v)∈S(1)
|Z(u, v)|>(1 +ǫ)Ξnδn−H
= P sup
(u,v)∈S(1)
|Z(u, v)|>(1 +ǫ)ςZ 2(1−˜ǫ) log(n)m/2 . According to (3.6), there existsT >0 such that for all t≥T
P sup
(u,v)∈S(1)
|Z(u, v)| ≥t
≤exp
−t2/m(1−˜ǫ) 2ςZ2/m
.
Hence there exists a positive integerN ≥1 such that for alln≥N we have P(En)≤exp
−(1−˜ǫ)(1 +ǫ)2/m(1−˜ǫ) log(n)
=n−β
where β := (1−˜ǫ)2(1 +ǫ)2/m. Since β >1, the Borel–Cantelli lemma shows that there exists a measurable set Ω1 withP(Ω1) = 1 and for allω∈Ω1 there exists an integern0(ω) such that for alln≥n0(ω) we have that
sup
(u,v)∈Sn
|Z(u, v, ω)| ≤(1 +ǫ)Ξn. For allω∈Ω1,δ≤δn wheren≥n0(ω) we have
sup
u,v≥0 0<u+v<δ
|Z(u, v, ω)|
Ξ(|u+v|) ≤ sup
j:j≥nsupn|Z(u, v, ω)|
Ξ(|u+v|) : (u, v)∈Sj\Sj+1
o
≤ sup
j:j≥n
supn|Z(u, v, ω)|
Ξ(δj+1) : (u, v)∈Sj\Sj+1
o
≤ sup
j:j≥n
(1 +ǫ)Ξj
Ξ(δj+1) = (1 +ǫ)ςZ2m/2 sup
j:j≥n
δj
δj+1
H log∗2(1/δj) log∗2(1/δj+1)
m/2 . Since
δj
δj+1 →1 and log∗2(1/δj)
log∗2(1/δj+1) →1 asj → ∞ we can for allω∈Ω1 choosen≥n0(ω) such that for allδ < δn we have
(3.7) sup
u,v≥0 0<u+v<δ
|Z(u, v, ω)|
Ξ(|u+v|) ≤(1 +ǫ)2ςZ2m/2.
Since ǫ > 0 was arbitrary chosen and σHm,H = ςZ2m/2, (3.7) implies (3.4), and the proof is
complete.
Proof of Theorem 1.3 (the upper bound). We will show the upper bound (3.3). To this aim recall from the beginning of this section that X satisfies (3.2) with d(s, t) = ˜c0|s−t|H andα= 2/m.
Set Ψ(t) =t1/H for all t ≥0, and recall that Φ is defined in (1.10). According to the proof of Theorem 1.2(i) we know that Φ and Ψ satisfy (1.5) for a suitable large constantC >0, which will be fixed throughout the proof. Furthermore, since V(Ψ, d) <∞ the conditions of Theorem 1.1 are satisfied. By definition of dand Ψ above we have that F(t) = ˜c1/H0 t for all t ∈ [0,1]. Set
