Continuity and boundedness of infinitely divisible

Continuity and boundedness of infinitely divisible processes: a Poisson point process approach

Michael B. Marcus and Jan Rosi´ nski

Abstract Sufficient conditions for boundedness and continuity are obtained for stochastically continuous infinitely divisible processes, without Gaussian component, {Y (t), t ∈ T }, where T is a compact metric space or pseudo-metric space. Such processes have a version given by Y (t) = X(t) + b(t), t ∈ T where b is a deterministic drift function and X(t) =

Z

S

h

i

f (t, s) N (ds) − (|f (t, s)| ∨ 1)−1 ν(ds) .

Here N is a Poisson random measure on a Borel space S with σ–finite mean measure ν, and f : T × S 7→ R is a measurable deterministic function. Let τ : T 2 → R+ be a continuous pseudo–metric on T . Define the τ -Lipschitz norm of the sections of f by kf kτ (s) = D −1 f (t0 , s) + sup

u,v∈T

|f (u, s) − f (v, s)| . τ (u, v)

for some t0 ∈ T , where D is the diameter of (T, τ ). The sufficient conditions for boundedness and continuity of X are given in terms of the measure ν, kf kτ and majorizing measure and or metric entropy conditions determined by τ . They are applied to stochastic integrals of the form Z Y (t) = g(t, s) M (ds) t ∈ T S

where M is a zero-mean, independently scattered, infinitely divisible random measure without Gaussian component. Several examples are given which show that in many cases the conditions obtained are quite sharp.

In addition to obtaining conditions for continuity and boundedness, bounds are obtained for the weak and strong Lp norms of supt∈T |X(t)| and supτ (t,u)≤δ,t,u∈T |X(t) − X(u)| for all 0 < δ ≤ D. These results depend on inequalities for moments and related functions of the weak and strong ℓp norms of sequences {xj }, which are the events of Poisson point process M on R+ and are given in terms of the intensity measure of M . These results are of independent interest.

Keywords and phrases. Infinitely divisible processes, sample continuity, sample boundedness, majorizing measures, Poisson point processes, stochastic integrals. 2000 Mathematics Subject Classification. Primary: 60G17, 60G52, 60G55, 60H05. Secondary: 60G51, 60G57. 0

Continuity and boundedness of infinitely divisible processes: a Poisson point process approach Michael B. Marcus and Jan Rosi´ nski∗

1

Introduction

We obtain sufficient conditions for boundedness and continuity of infinitely divisible stochastic process. A stochastic process Y = {Y (t) : t ∈ T }, where T is a separable metric space, is said to be infinitely divisible if all its finite dimensional distributions are infinitely divisible. By this definition all Gaussian processes are infinitely divisible. Since boundedness and continuity of Gaussian processes are very well understood, we restrict our attention to infinitely divisible processes without a Gaussian component. This is a very broad and important class of processes which includes stable processes. The results we give here are interesting even when restricted to stable processes. We were drawn to this topic because there wasn’t a good theorem for continuity and boundedness of infinitely divisible processes. Kolmogorov’s theorem is a good general theorem for sample path behavior of stochastic processes, which is quite effective in many cases. However, it requires that the increments of the processes have good moment structure. In general infinitely divisible processes do not. A technique for circumventing this was introduced in [12] in the study of harmonizable stable processes (i.e., random Fourier series with stable coefficients and their generalizations). The idea is to symmetrize the process and write it as a Rademacher series of random terms. The marginal Rademacher The research of both authors is supported, in part, by grants from the National Science Foundation. ∗

1

processes have very strong moment properties, generally stronger even than Gaussian processes. Using generalizations of techniques that give sufficient conditions for continuity and boundedness of Gaussian process, on the Rademacher processes, very strong conditions can be obtained for the marginal Rademacher processes. One can then pass from the marginal process to the whole process. We don’t restrict ourselves to symmetric processes. However, by decomposing the original process into three terms, we can deal with the most complex of the terms by symmetrization. The other terms are dealt with using some new inequalities we obtain for Poisson point processes on the positive half line. Some of the ideas we use to extend the approach in [12] were introduced in [7, 13, 6]. However, none of these treat the problem in the generality considered in this paper. Also this paper employs a radically different approach from its predecessors. In this paper the infinitely divisible processes are represented as stochastic integrals with respect to Poisson point processes. This gives us a canonical representation of all infinitely divisible processes which can be analyzed efficiently. In the next section we state all our results on the continuity and boundedness of infinitely divisible processes. We derive estimates for the moments of suprema and moduli of continuity of such processes. Among other applications, they can be used in the study of limit theorems for such processes. In Section 3 we show how these results immediately give conditions on the continuity and boundedness of stochastic integrals with respect to independently scattered random measures. It is examples of this class of processes that are considered in [12, 7, 13, 6]. In Section 4 we show how these results give fairly sharp conditions for the continuity of harmonizable stable processes process and their generalizations. We deduce a recent result of Braverman [1] on the boundedness of highly dependent stable sequences as an application of our general results. It is interesting to note that this last result requires the use of majorizing measures and can not be obtained using metric entropy. We also give a simple criteria for the continuity of forward averages, to demonstrate how the results can be simplified for applications. In Section 5 we begin to develop the material used in the proofs. In Subsection 5.1 we show how symmetrized processes can be written as Rademacher series. This series expansion is different from the ones used in [12, 10, 23] and is new in this context. In Subsection 5.2 we consider different decompositions of infinitely divisible processes into deterministic and random components 2

that facilitates their analysis. Subsection 5.3 contains many sharp inequalities involving the norms of real valued Poisson point processes. These results are of independent interest. Subsection 5.4 gives a pointwise bound, on the probability space, for the modulus of continuity of stochastic processes in exponential Orlicz spaces. This result is well known for Gaussian processes, see e.g. [3, 14]. Various extensions to bounds for the modulus of continuity of general processes have been given. (See “Notes and References” in Chapter 11, [9]). Nevertheless, we do not know a reference for the extension to majorizing measures in the form that we need. Sections 6 and 7 are devoted to the proofs of results on Poisson point processes and stochastic integrals with respect to independently scattered random measures respectively. We are grateful to Tom Kurtz and the referee of the first draft of this paper for pointing our research in a fruitful direction.

3

2

Processes given by stochastic integrals with respect to Poisson random measures

Let Y={Y (t) : t ∈ T } be a stochastically continuous infinitely divisible stochastic process without a Gaussian component. It follows from Maruyama [17], that Y has a version given by (2.1)

Y (t) = b(t) + X(t)

where b : T 7→ R is a continuous deterministic function and (2.2)

X(t) =

Z

S

h

i

f (t, s) N(ds) − (1 ∨ |f (t, s)|)−1ν(ds) .

Here N is a Poisson random measure on a Borel space S with a σ–finite intensity measure ν and f : T × S 7→ R is measurable deterministic function such that for every t ∈ T Z 

(2.3)

S



|f (t, s)|2 ∧ 1 ν(ds) < ∞.

Note that (2.3) is necessary and sufficient for the integral (2.2) to exist (see, e.g. [5] ). When EY (t) < ∞ for each t ∈ T it is natural to take X to be zero–mean process represented by X(t) =

(2.4)

Z

S

f (t, s) [N(ds) − ν(ds)].

In this case we require that (2.5) If (2.6)

Z  S



|f (t, s)|2 ∧ |f (t, s)| ν(ds) < ∞ Z

S

(|f (t, s)| ∧ 1) ν(ds) < ∞

∀ t ∈ T. ∀t ∈ T

the integral (2.2) exists without a compensator. In this case we represent X by Z X(t) = f (t, s) N(ds). (2.7) S

Of course, the different representations of X yield different deterministic drift functions b. 4

Throughout this paper, X = {X(t) : t ∈ T } denotes a stochastically continuous process, indexed by a precompact metric space T , defined on a probability space (Ω, P ). In general we take X to be of the form (2.2) but we also consider the special cases (2.4) and (2.7) to make it easier to use our results when these additional conditions are satisfied. Our objective in this paper is to give relatively simple, usable sufficient conditions for the boundedness and continuity of X. Our results are expressed in terms of a weak modulus of continuity for f (·, s). Let τ : T × T 7→ R+ denote a continuous pseudo–metric on T and D := supu,v∈T τ (u, v) denote the diameter of (T, τ ). For each s ∈ S we refer to f (·, s) : T → R as a section of f . Fix t0 ∈ T . We define a Lipschitz type pseudo–norm on the sections of f by (2.8)

|f (t, s) − f (u, s)| . τ (t, u) τ (t,u)6=0

kf kτ (s) = D −1 |f (t0 , s)| + sup

t,u∈T

To ensure the measurability of kf kτ , as well as of the related quantity supt∈T |f (t, ·)|, we assume that there is a countable set T0 ⊂ T such that for every t ∈ T and s ∈ S there exists {ti } ⊂ T0 such that limi f (ti , s) = f (t, s). To avoid trivialities, we assume that τ is such that kf kτ (s) < ∞, ν– almost surely, which implies continuity of sections of f . This condition is only relevant in Theorem 2.1 since all the other results contain hypotheses which imply it. (Continuity of sections of f is a necessary condition for sample continuity of non Gaussian infinitely divisible processes, [21]). Throughout the paper we assume that there are at least two points say u, v ∈ T such that τ (u, v) > 0. The questions we consider are trivial when this is not the case. Let m be a probability measure on T and Bτ (t, r) be a closed ball of radius r in (T, τ ), centered at t. Set (2.9)

Iq (m, τ ; δ) = sup t∈T

and (2.10)

Z

0

δ

1 log m(Bτ (t, r))

I∞ (m, τ ; δ) = sup t∈T

Z

0

δ

log log

!1/q

dr

2≤q c) c>0

!1/p

< ∞.

Theorem 2.1 Let X = {X(t), t ∈ T } be a separable infinitely divisible process as given in (2.2). Let m be a probability measure and τ a pseudo–metric on T such that (2.12) Iq (m, τ ; D) < ∞ for some 2 < q ≤ ∞. Let kf kτ ∧ 1 ∈ Lp,∞ (S, ν)

(2.13)

where 1/p + 1/q = 1, 2 ≤ q < ∞ and p = 1 when q = ∞. Then, when the sections f (·, s) are bounded for ν–almost all s ∈ S, X has bounded paths almost surely. And when, in addition, the sections f (·, s) are continuous for ν–almost all s ∈ S and (2.14) lim Iq (m, τ ; δ) = 0 δ→0

X has continuous paths almost surely. Throughout this paper the numbers p and q are related as Theorem 2.1. Note that for the continuity or boundedness of X, one needs only consider the values of the τ –norm near zero. In the next theorem we give results on the size of the supremum of X and on its modulus of continuity. Now we must consider the large values of the τ –norm as well. Theorem 2.2 Let 1 ≤ p < 2. Let X = {X(t), t ∈ T } be a separable infinitely divisible process given by (2.4) when 1 < p < 2 and by (2.2) when p = 1. Suppose that (2.12) holds but instead of (2.13) we have the stronger condition (2.15) kf kτ ∈ Lp,∞ (S, ν). Then, when 1 < p < 2,

(2.16)

Λp (P, sup |X(t)|) ≤ Cp Λp (ν, kf kτ )Iq (m, τ ; D) t∈T

6

and for every δ ∈ (0, D] (2.17)

Λp (P, sup |X(t) − X(u)|) ≤ Cp Λp (ν, kf kτ )Iq (m, τ ; δ). τ (t,u)≤δ

t,u∈T

If X is symmetric then (2.16)–(2.17) also holds with p = 1. In general, when (2.15) holds with p = 1 Λ1 (P, sup |X(t)|) t,u∈T

(2.18)

h

≤ C1 Λ1 (ν, kf kτ ) I∞ (m, τ ; D) + D log+ (DΛ1 (ν, kf kτ )) ∨ 1

and for every δ ∈ (0, D]

h

i

Λ1 (P, sup |X(t) − X(u)|) ≤ C1 Λ1 (ν, kf kτ ) I∞ (m, τ ; δ) τ (t,u)≤δ

t,u∈T

(2.19)

i

+3δ[log+ (DΛ1 (ν, kf kτ )) + log+ (δΛ1 (ν, kf kτ ))−1 ] .

Throughout this paper Ck denotes a numerical constant that depends only on a parameter k and may be different in each occurrence. Note that in concrete cases one can seldom evaluate kf kτ exactly but can only obtain upper bounds for it. The preceding results remain valid if kf kτ is replaced by an upper bound. Condition (2.15) requires that both the small and large values of kf kτ are in Lp,∞ . However, in general, we need more flexibility in describing the behavior of the small and large values of kf kτ . The next theorem, which we present after making some preliminary observations, gives us this flexibility. Let h be a nonnegative function on S. Let (2.20)

n

o

c∗ = c∗ (h) := inf c > 0 : ν({s : h(s) ≥ c}) ≤ 1 .

We call c∗ a separating number, (of the range of h relative to ν). If c∗ ∈ (0, ∞) then ν(h > c∗ ) ≤ 1 ≤ ν(h ≥ c∗ ). So, restricted to {h > c∗ }, ν is a sub– probability measure. Let (2.21)

X0 (t) =

Z

h≤c∗

f (t, s) [N(ds) − ν(ds)] + 7

Z

h>c∗

f (t, s) N(ds).

and (2.22)

b0 (t) = X(t) − X0 (t).

Thus, if X is given by (2.2) b0 (t) =

Z

h≤c∗

Z

f (t, s)[1−(1∨|f (t, s)|)−1] ν(ds)−

h>c∗

f (t, s)(1∨|f (t, s)|)−1 ν(ds).

(2.23) To obtain bounds for the moments of the supremum and the modulus of continuity of X we combine the corresponding bounds for X0 and b0 . We consider the deterministic function b0 in Subsection 5.2. At this point we only note that if X is symmetric then b0 = 0. Theorem 2.3 Let X0 = {X0 (t), t ∈ T } be a separable infinitely divisible process as given in (2.21) where h is a nonnegative function such that (2.24)

kf kτ (s) ≤ h(s)

Suppose that (2.25)

ν − a.s.

hI{h≤1} ∈ Lp,∞ (S, ν),

for some 1 ≤ p < 2, and (2.12) holds for 2 < q ≤ ∞ (1/p + 1/q = 1). If hI{h>1} ∈ Lr,∞ (S, ν)

(2.26)

for some r > 0, then for every δ > 0 (2.27)



Λr P, sup |X0 (t) − X0 (u)| τ (t,u)≤δ

t,u∈T

n









≤ Cr,p Λp ν, hI{h≤c∗ } + Λr ν, hI{h>c∗ }

o

Iq (m, τ ; δ)

where c∗ is the separating number (2.20). If instead of (2.26), hI{h>1} ∈ Lr (S, ν)

(2.28)

for some r > 0, then for every δ > 0 (2.29)

h

E sup |X0 (t) − X0 (u)|r τ (t,u)≤δ

t,u∈T

n



i1/r 

≤ Cr,p Λp ν, hI{h≤c∗ } +

Z

h>c∗

hr dν

1/r o

Iq (m, τ ; δ).

Bounds for the supremum of X0 have a similar form. One only needs to replace supτ (t,u)≤δ |X0 (t) − X0 (u)| on the left–hand side of (2.27) and (2.29) t,u∈T

by supt∈T |X0 (t)| and take δ = D on the right–hand side. 8

For 1 ≤ p ≤ 2, condition (2.13) in Theorem 2.1 is weakest when p = 2. Therefore we consider versions of Theorems 2.1–2.3 for p = q = 2. However, in this case the analogue of (2.13) uses L2 not L2,∞ . Theorem 2.4 Theorem 2.1 holds as stated when p = q = 2 and when condition (2.13) is replaced by kf kτ ∧ 1 ∈ L2 (S, ν).

(2.30)

Analogous to Theorem 2.2 we have Theorem 2.5 Let X be a separable process as given in (2.4) and assume that (2.12) holds with q = 2 and that kf kτ ∈ L2 (S, ν).

(2.31) Then



(2.32)

E sup |X(t)|2 t∈T

1/2

≤ C2

Z

kf k2τ dν

1/2

I2 (m, τ ; D)

and for every δ ∈ (0, D] (2.33)



E sup |X(t) − X(u)| τ (t,u)≤δ

t,u∈T

 2 1/2

≤ C2

Z

kf k2τ

dν

1/2

I2 (m, τ ; δ).

Analogous to Theorem 2.3 we have Theorem 2.6 Let X0 = {X0 (t), t ∈ T } be a separable infinitely divisible process as given in (2.21) with h satisfying (2.24) and hI{h≤1} ∈ L2 (S, ν).

(2.34)

Assume that (2.12) holds with q = 2. If, in addition, (2.26) holds then for every δ > 0 (2.35)



Λr P, sup |X0 (t) − X0 (u)| τ (t,u)≤δ

t,u∈T

≤ Cr

n Z

h≤c∗

h2 dν

1/2

9

 

+ Λr ν, hI{h>c∗ }

o

I2 (m, τ ; δ).

where c∗ is the separating number (2.20). If instead of (2.26) condition (2.28) holds, then for every δ > 0 (2.36)

h

E sup |X0 (t) − X0 (u)|r τ (t,u)≤δ

t,u∈T

≤ Cr

n Z

h≤c∗

h2 dν

i1/r

1/2

+

Z

h>c∗

hr dν

1/r o

I2 (m, τ ; δ).

Bounds for the supremum of X0 have a similar form. One only needs to replace supτ (t,u)≤δ |X0 (t) − X0 (u)| on the left–hand side of (2.35) and (2.36) t,u∈T

by suptT |X0 (t)| and take δ = D on the right–hand side. Condition (2.12) of Theorem 2.1 becomes weaker as q increases or p decreases. The opposite holds for (2.13) which is the most restrictive when p = 1. In fact, if one passes from L1,∞ in (2.13) to L1 , the boundedness of the process X follows from a simple argument which does not require majorizing measures. We give these results for the sake of completeness. Proposition 2.1 Let X = {X(t), t ∈ T } be a separable infinitely divisible process as given by (2.7). Assume that the sections of f (·, s) are bounded for ν–almost all s ∈ S and that sup |f (t, ·)| ∧ 1 ∈ L1 (S, ν).

(2.37)

t∈T

Then X has bounded paths a.s. If, in addition to the above, the sections f (·, s) are uniformly continuous for ν–almost all s ∈ S, then X has continuous paths a.s. Proposition 2.2 Let X = {X(t), t ∈ T } be a separable infinitely divisible process as given by (2.7). Let h be a function satisfying (2.24) and such that hI{h≤1} ∈ L1 (S, ν).

(2.38)

If (2.26) holds, then for every δ > 0 (2.39)



Λr P, sup |X(t) − X(u)| τ (t,u)≤δ

t,u∈T

≤ Cr δ

nZ

h≤c∗

10

 

h dν + Λr ν, hI{h>c∗ }

o

.

where c∗ is the separating number (2.20). If instead of (2.26) condition (2.28) holds, then for every δ > 0 h

E sup |X(t) − X(u)|r

(2.40)

τ (t,u)≤δ

t,u∈T

≤ Cr δ

nZ

h≤c∗

i1/r

h dν +

Z

h>c∗

hr dν

1/r o

.

Bounds for the supremum of X have a similar form. One only needs to replace supτ (t,u)≤δ |X(t) − X(u)| on the left–hand side of (2.39) and (2.40) by t,u∈T

suptT |X(t)| and take δ = D on the right–hand side. Remark 2.1 1. Consider the metric entropy integrals Z

Jq (τ, δ) = and J∞ (τ, δ) =

δ

0

Z

0

δ

(log N(τ, r))1/q dr

log(1 + log N(τ, r)) dr

q 0 and all u sufficiently small, then Jq (τ, D) < ∞. By the first part of this remark, there exists a majoring measure m such that Iq (m, τ ; δ) ≤ Kq Jq (τ, δ). In particular (2.14) holds. When T is well behaved, such as a cube, a ball, etc., then m can be taken to be normalized Lebesgue measure on T .

3

Processes given by stochastic integrals with respect to infinitely divisible random measures

It is often useful, especially in applications, to express infinitely divisible processes as stochastic integrals with respect to L´evy processes, or more generally, with respect to independently scattered random measures. We show how to adapt the results of Section 2 to processes expressed in this way. Furthermore, by considering these stochastic integrals we can better understand the significance of the conditions imposed in Section 2. Let us begin by considering the zero–mean stochastic process given by (3.1)

X(t) =

Z

U

g(t, u) dZ(u) 12

t∈T

where {Z(u) : u ∈ U} is a mean zero L´evy process without Gaussian component, U is a bounded or unbounded subinterval of R, and g : T × U 7→ R is a measurable function. The characteristic function of the increments of Z is given by n

E exp(iv[Z(u2 ) − Z(u1)]) = exp (u2 − u1 )

Z

R

(eivx − 1 − ivx) θ(dx)

o

R

for every (u1 , u2 ] ⊂ U, where θ, the L´evy measure of Z, satisfies (|x|2 ∧ |x|)θ(dx) < ∞. A necessary and sufficient condition for the existence of the stochastic integral (3.1) in L1 ((Ω, P ); (see e.g. [20]) is that (3.2)

Z Z U

R

(|xg(t, u)|2 ∧ |xg(t, u)|) θ(dx)du < ∞.

Z also has an Itˆo representation with respect to a Poisson random measure N on U × R with mean measure ν(du, dx) = duθ(dx) given by Z(u2 ) − Z(u1 ) =

Z

(u1 ,u2 ]×R

x [N(du, dx) − duθ(dx)].

(See e.g. [8]). By a change of measure in (3.1) we can write (3.3)

X(t) =

Z

U ×R

h

i

xg(t, u) N(du, dx) − duθ(dx) .

Thus X is of the form (2.4) with S = U × R, f (t, (u, x)) = g(u, t)x and ν(du, dx) = duθ(dx). Furthermore, the condition in (3.2) is just (2.5). In this change of variables 

|g(t1 , u) − f (t2 , u)| (3.4) kf kτ (u, x) = |x| D |g(u, t0)| + sup τ (t1 , t2 ) τ (t1 ,t2 )6=0 −1

t1 ,t2 ∈T



= |x|kgkτ (u). Therefore, we can apply the results of Section 2 to determine the boundedness and continuity of X. We actually apply the results of Section 2 in much greater generality because the class of processes in (3.1) is too restricted. We consider stochastic integrals with respect to non–homogeneous independent increment processes in the general framework of random measures. Let {W (A) : A ∈ A} be an independently scattered mean zero infinitely divisible random measure. 13

Here A is a σ–ring of subsets of an arbitrary set U with the property that S U = Un for some sequence {Un } ⊂ A. We assume that for each A ∈ A, W (A) has no Gaussian component. The characteristic function of W can be written as (3.5)

EeivW (A) = exp

Z

A×R

[eivx − 1 − ivx] ν(du, dx)



A∈A

where ν is a σ–finite measure on σ(A) × B(R) such that Z

(3.6)

A×R

(x2 ∧ |x|) θ(du, dx) < ∞

for every A ∈ A, (see e.g. [20] for details). We now consider stochastic process Y (t) =

(3.7)

Z

U

g(t, u) W (du)

t∈T

where g : T × U 7→ R is a deterministic function such that for each t ∈ T Z Z U

R

(|xg(t, u)|2 ∧ |xg(t, u)|) ν(du, dx) < ∞.

The only essential difference between (3.7) and (3.1) is that here ν is arbitrary while in (3.1) the mean measure is a product measure. In particular, the process Y is equal, in distribution, to an integral of the form of (3.3) with duθ(dx) replaced by ν(du, dx). Therefore, without loss of generality we may assume that Y in (3.7) is given by (2.4) with S = U × R and f (t, (u, x)) = g(u, t)x. The process X in (3.1) is a special case of Y . Let us be more specific about how the results in Section 2 apply to Y . Assume that (3.8) ν(du, dx) = θ(dx, u)w(du) where {θ(·, u) : u ∈ U} is a measurable family of L´evy measures on R and w is a σ–finite measure on (U, σ(A)). We also assume that for some p, r > 0 and all λ > 0 (3.9)

θ({x : |x| ≥ λ}, u) ≤ a(u)λ−p I(λ ≤ k(u)) + b(u)λ−r I(λ > k(u))

where a(u), b(u), and k(u) are positive functions. We have such a bound if and only if (3.10) lim sup λp θ({x : |x| ≥ λ}, u) < ∞ λ→0

14

and (3.11)

lim sup λr θ({x : |x| ≥ λ}, u) < ∞. λ→∞

Note that when (3.10)–(3.11) hold one can always take k(u) = 1, a(u) = sup λp θ({x : |x| ≥ λ}, u) λ 1. Let kgkτ be as defined in (2.8) and assume that (2.12) holds for some probability measure m on T . Then, when p ≥ r (3.12)





Λr P, sup |Y (t)| ≤ Cp,r t∈T

 Z

U

akgkpτ dw +

Z

U

1/p

bkgkrτ

dw

1/r 

Iq (m, τ ; D).

Moreover, for every δ > 0 (3.13)





Λr P, sup |Y (t) − Y (u)| ≤ Cp,r τ (t,u)≤δ

t,u∈T

+

Z

 Z

U

U

akgkpτ dw

bkgkrτ

dw

1/p

1/r 

Iq (m, τ ; δ).

These bounds hold also for r = 1 when Y is symmetric. When p < r (3.14)





Λr P, sup |Y (t)| ≤ Cp,r t∈T

+

 Z

Z

U

U

(a ∨ bk r−p )kgkpτ dw

(ak

15

p−r

∨

b)kgkrτ

dw

1/p

1/r 

Iq (m, τ ; D).

Moreover, for every δ > 0 (3.15)





Λr P, sup |Y (t) − Y (u)| ≤ Cp,r τ (t,u)≤δ

t,u∈T

+

Z

U

 Z

U

(a ∨ bk r−p )kgkpτ dw

(ak p−r ∨ b)kgkrτ dw

1/r 

1/p

Iq (m, τ ; δ).

If the right hand side of (3.9) is continuous in λ, a(u) = k r−p (u)b(u). In this case the right–hand sides of (3.12) and (3.14) are the same, as are the right–hand sides of (3.13) and (3.15). As one can see by examining the proof of Theorem 3.1, adapting results of Section 2 to processes of the form (3.7) is straightforward. On the other hand, in some concrete cases of processes of the form (3.7), it may be possible to get sharper estimates, than the ones given by Theorem 3.1, by applying the results of Section 2 directly.

4 4.1

Some applications Harmonizable processes

Consider a harmonizable processes (4.1)

V (t) =

Z

∞

−∞

eitu W (du)

where W = W1 +iW2 is a complex rotationally invariant, independently scattered infinitely divisible random measure. V has four components involving cosine and sine functions as integrands and W1 and W2 as integrators. We will investigate just one of them (4.2)

Y (t) =

Z

∞

−∞

cos(ts) W1 (ds), t ∈ [−π, π].

It should be clear that the estimates obtained for Y apply also to the other components of V . The process Y is symmetric because W1 is symmetric. Suppose that 

EeivW1 (A) = exp −2

Z

A×R

[1 − cos(vx)] ν(ds, dx) 16



A ∈ B(R)

where ν(ds, dx) = θ(dx, s)w(ds) and θ(·, s) is a symmetric L´evy measure on R. Assume that θ({x : |x| ≥ λ}, s) ≤

(4.3)

λ k(s)

!−p

λ + k(s)

I(λ ≤ k(s))

!−r

I(λ > k(s))

for some p ∈ (1, 2), r ≥ 1, and a positive function k. By varying k we can assign different type of jumps to different frequencies s in (4.2). We will now apply Theorem 3.1 to the process Y . Consider the metric τ on T = [−π, π] given by c τ (t, u) = log | t − u|

(4.4)

!−(1+ǫ)/q

where ǫ > 0. (Recall that q = p/(p−1)). Here c := cq,ǫ = 2π exp(1+(1+ǫ)/q. This ensures that (log(c/|s|))−(1+ǫ)/q is concave on [0, 2π] and thus that (4.4) is a metric on [−π, π]. Let m be the normalized Lebesgue measure on T . It is not difficult to see that ǫ Iq (m, τ ; δ) ≤ Cq,ǫ δ 1+ǫ . (4.5) (We use the fact that m(Bτ (t, r) ≤ Cτ −1 (r) ≤ C ′ exp(−(r −q/(1+ǫ) )).) We now compute kgkτ for g(t, s) = cos(ts). We have

(4.6)

c |g(t, s) − g(u, s)| ≤ (|s||t − u| ∧ 2) log τ (t, u) |t − u|

!(1+ǫ)/q

≤ Cq,ǫ (log(e + |s|))(1+ǫ)/q .

To verify this note that x(log xc )(1+ǫ)/q is increasing for x ∈ [0, 2π]. Let |s| ≥ 1/π. Then for |s||t − u| ≤ 2, or |t − u| ≤ 2/|s| ≤ 2π, we have |s||t − 

c u| log |t−u|



(1+ǫ)/q

(1+ǫ)/q



≤ 2 log 2c |s|

(1+ǫ)/q



c . If |s−t| > 2/|u|, 2 log |t−u|

(1+ǫ)/q

≤

2 log 2c |s| . Thus we have (4.6) when |s| ≥ 1/π. For |s| < 1/π the middle term in (4.6) is uniformly bounded, so that (4.6) hold trivially. Consequently, (4.7) kgkτ (s) ≤ Cq,ǫ (log(e + |s|))(1+ǫ)/q . 17

Since the right hand side of (4.3) is continuous in λ, Theorem 3.1 gives the same bounds for p < r and p ≥ r. Using (4.5)–(4.7) we get 

Λr P, sup |Y (t) − Y (u)| τ (t,u)≤δ

t,u∈T

≤ Cp,r,ǫ +

Z

 Z ∞

−∞

∞ −∞



k p (s) (log(e + |s|)) (1+ǫ)p/q w(ds)

k r (s) (log(e + |s|))(1+ǫ)r/q w(ds)

1/p

1/r 

ǫ

δ 1+ǫ .

Substituting (log( δc ))−(1+ǫ)/q for δ we obtain the following estimate for the modulus of continuity of Y 

Λr P,

sup

|Y (t) − Y (u)|

|t−u|≤δ

t,u∈[−π,π]

≤ Cp,r,ǫ +

Z

 Z

∞

−∞

∞

−∞



k p (s) (log(e + |s|)) (1+ǫ)p/q w(ds) (1+ǫ)r/q

r

k (s) (log(e + |s|))

w(ds)

1/p

1/r  

c log( ) δ

−ǫ/q

.

The same bound holds for other components of the harmonizable process R∞ R∞ R∞ V in (4.1), i.e. −∞ sin(ts) W1 (ds), −∞ cos(ts) W2 (ds) and −∞ sin(ts) W2 (ds). Consequently, we have the following corollary: Corollary 4.1 Let V = {V (t) : t ∈ [−π, π]} be the separable harmonizable process given by (4.1). Suppose that 

E exp(iℜ(¯ v W (A))) = exp −2

Z

A×C

[1 − cos(ℜ(¯ v x))] ν(ds, dx)



A ∈ B(R), v ∈ C, where ν(ds, dx) = θ(dx, s)w(ds), w is a measure on R and {θ(·, s)} is a measurable family of rotationally invariant L´evy measures on C such that (4.3) holds, (with |x| replaced by |v|, for v ∈ C). Then, for every ǫ > 0 and r, p ≥ 1 

(4.8) Λr P,

sup

|V (t) − V (u)|

|t−u|≤δ

t,u∈[−π,π]

≤ Cp,r,ǫ +

Z

 Z

∞

−∞

∞

−∞



k p (s) (log(e + |s|)) (1+ǫ)p/q w(ds) (1+ǫ)r/q

r

k (s) (log(e + |s|)) 18

w(ds)

1/p

1/r  

c log( ) δ

−ǫ/q

.

In particular, if the two integrals are finite, V has uniformly continuous sample paths almost surely. In [12] necessary and sufficient conditions are given for the continuity of p–stable harmonizable processes. In this case θ({x : |x| ≥ λ}) = λ−p . By Corollary 4.1 we have the following estimate on the modulus of continuity 

Λr P,

sup |t−u|≤δ

|V (t) − V (u)|

t,u∈[−π,π]

≤ Cp,r,ǫ

Z

∞ −∞



(log(e + |s|))

(1+ǫ)p/q

w(ds)

1/p 

c log( ) δ

−ǫ/q

.

In particular, V has uniformly continuous sample paths when (4.9)

Z

∞

∞

(log(e + |s|))(1+ǫ)p/q w(ds) < ∞.

If ǫ is replaced by zero in (4.9), it is no longer a sufficient condition for continuity. This can be seen by considering a special case of a p–stable harmonizable process which is the random Fourier series (4.10)

V (t) =

∞ X

aj ξj eitj

j=0

t ∈ [−π, π].

Here {ξj } are independent identically distributed rotationally invariant complex p–stable random variables. V can be represented by (4.1) with w supported on the integers, w({j}) = Cp apj . In this case condition (4.9) is (4.11)

∞ X

j=0

apj (log(e + j))(1+ǫ)p/q < ∞.

When |aj | is non–increasing for j ≥ j0 , for some j0 (4.12)

∞ X

k=2

(

P∞

j=k

apj )1/p

k (log k)1/q

0. The only explanation for (4.14) in these cases is that the {ξn } must be highly dependent. Define Z 1 ξn Xn = = (4.16) gn (u) dZ(u) (log n)1/q 0 γn (u) , n ≥ 2 and X1 ≡ 0. We apply Theorem 3.1 in (log n)1/q the case where ν(du, dx) = θ(dx)w(du), θ({x : |x| ≥ λ}) = λ−p and w is

where gn (u) =

20

Lebesgue measure on [0, 1]. We define a metric τ on T = N by   (log n)−1/q

+ (log k)−1/q

n 6= k, n, k > 1 k = 1, n > 1  (log n) 0 n = k. D = (log 2)−1/q + (log 3)−1/q is the diameter of (N, τ ). Clearly (4.17)

τ (n, k) =

−1/q

|gn (u) − gk (u)| ≤ 1. τ (n, k) Therefore (4.18)

kgkτ ≤ 1.

We choose the probability measure m on N by m({1}) = 1/2 and m({n}) = C/n2 , n ≥ 2, for an appropriate constant C. Note that if 0 < r < (log n)−1/q , Bτ (n, r) contains only the singleton {n}. However when r ≥ (log n)−1/q , Bτ ({n}, r) contains {n} ∪ {1}. Consequently (4.19)

Iq (m, τ ; D) ≤ sup n≥2

Z

(log n)−1/q

0

+

Z

D

(log n)−1/q

(2 log n + log C)1/q dr 

(log 2)1/q dr = Cq < ∞.

Using (3.12) of Theorem 3.1 we see that (4.20)

Λp

|ξn | P, sup 1/q n≥2 (log n)

!

!

= Λp P, sup |Xn | ≤ Cp < ∞. n∈N

This gives (4.14). If is easy to verify that in this case the metric entropy integral Jq (τ, D) = ∞; (see Remark 2.1).

4.3

A simple continuity condition for forward averages

With a view towards practical applications we show how Corollary 2.1 can be used to obtain a simple sufficient condition for the continuity of forward averages. A forward average is a process of the form (4.21)

Z(t) =

Z

[0,∞)n

g(t + u) dW (u)

t ∈ [0, T ]n .

where W is as given in (3.5) with ν(du, dx) = θ(dx)w(du), with w a σ–finite measure on [0, ∞)n . To make this example concrete we take θ to be the L´evy measure of a symmetric p–stable random variable with θ({x : |x| ≥ λ}) = λ−p , 1 < p < 2. 21

Lemma 4.1 Let Z= {Z(t); t ∈ [0, T ]} be as given in (4.21) and assume that |g(s + u) − g(t + u)| ≤C |s − t|α s,t∈[0,T ]n

(4.22)

sup

for some α > 0 and constant C independent of u and that Z

(4.23)

[0,∞)n

sup |g(x)|p−ǫ w(du) < ∞ x≥u

for some ǫ > 0. Then Z has a continuous version. Proof: As in Subsection 4.1 we take 1 τ (s, t) = log | s − t|

(4.24)

!−(1/q)−ǫ

∧1

ǫ > 0. It follows from Theorem (3.1), (3.13) that for 1 < p < 2 and δ ∈ (0, D] 

(4.25)Λp P, sup τ (s,t)≤δ s,t∈[0,T ]n



|Z(s) − Z(t)| ≤ Cp

Z

kg(· +

u)kpτ

w(du)

1/p

Iq (λ, τ ; δ)

where λ is normalized Lebesgue measure on [0, T ]. As in Remark 2.1, 2. it is easy to see that limδ→0 Iq (λ, τ ; δ) = 0. For 0 < γ < 1, we have (4.26)

|g(s + u) − g(t + u)| τ (s, t) |g(s + u) − g(t + u)|γ |g(s + u) − g(t + u)|1−γ = τ (s, t) |g(s + u) − g(t + u)|γ 2 sup |g(x)|1−γ ≤ τ (s, t) x≥u

Using (4.22) it is easy to see that (4.27)

|g(s + u) − g(t + u)|γ ≤ CT,α,C τ (s, t) s,t∈[0,T ]n sup

where CT,α,C is a constant depending only on T , α and C in (4.22). Consequently (4.28) kg(· + u)kpτ ≤ 2CT,α,C sup |g(x)|p(1−γ) . x≥u

Using (4.23) we see that the integral in (4.25) is finite for γ sufficiently close to zero. Since limδ→0 Iq (m, τ ; δ) = 0 the lemma is proved. 22

5 5.1

Preliminaries Symmetrization, series expansions

This subsection contains preliminary material assembled here for the convenience of the reader. Consider X = {X(t), t ∈ T } given in (2.2). Its c can written as symmetrization X c X(t)

(5.1)

=

Z

S

c f (t, s) N(ds) t∈T

c = N − N ′ , where N ′ is an independent copy of the Poisson random where N measure N. Let N2 be a Poisson random measure on S with mean measure 2ν. N2 can P be represented as a Poisson point process N2 = ∞ j=1 δsj , where sj : Ω 7→ S are (dependent) random elements, see [8]. Let {ǫj } be a Rademacher sequence independent of {sj }. The existence of N2 implies that the thinned P P∞ processes ∞ j=1 I(ǫj = 1)δsj and j=1 I(ǫj = −1)δsj are independent Poisson point processes with the same mean measure ν. (This is well known). Therefore ∞ ∞ ∞ X

ǫj δsj =

X

j=1

j=1

I(ǫj = 1)δsj −

X

j=1

I(ǫj = −1)δsj

c. Without loss of generality we take N c = P∞ ǫ δ . is a version of N j=1 j sj Clearly, for any simple function h : S 7→ R, Z

(5.2)

S

c= h dN

∞ X

ǫj h(sj ).

j=1

R

In general, the left hand side of (5.2) exists if and only if S h2 ∧ 1 dν < ∞ (Theorem 10.15 in [8]). For a general h, byR Fubini’s theorem, the right hand P side in (5.2) exists if and only Rj h(sj )2 = S h2 dN2 < ∞ almost everywhere. The latter holds if and only if S h2 ∧ 1 dν < ∞ (again Theorem 10.15, [8]). Approximating h by simple functions allows us to extend (5.2) to the equality R 2 almost surely for any h with S h ∧ 1 dν < ∞. We summarize this in the following lemma. Lemma 5.1 Let h : S → 7 R be a measurable function such that ∞. Then Z ∞ X c= (5.3) h dN ǫj h(sj ) a.s. S

j=1

23

R

S

h2 ∧1 dν
c∗

h dν.

Proof: First we note that by (2.8) (5.12)

|f (t, s)| ≤ Dkf kτ (s) ≤ Dh(s).

When X is given by (2.2), b0 is given by (2.23). Consequently Z

|b0 (t)| ≤

h≤c∗

≤ D

Z

|f (t, s)|I{|f (t,s)|>1} ν(ds) +

D −1 c∗

dν

h(s) ν(ds) + 1

which proves (5.8). From the elementary inequality (5.13)

|x(|x| ∨ 1)−1 − y(|y| ∨ 1)−1 | ≤ 2(|x − y| ∧ 1)

for all x, y ∈ R, we get that (5.14) |x[1 − (|x| ∨ 1)−1 ] − y[1 − (|y| ∨ 1)−1 ]| ≤ 3|x − y|I(|x| ∨ |y| > 1). Indeed, if |x| ∨ |y| ≤ 1, then the left hand side in (5.14) equals zero. If |x| ∨ |y| > 1 then |x[1 − (|x| ∨ 1)−1 ] − y[1 − (|y| ∨ 1)−1 ]| ≤ |x − y| + |x(|x| ∨ 1)−1 − y(|y| ∨ 1)−1 | ≤ 3|x − y| by (5.13). Applying (5.13)–(5.14) and (5.12) we obtain for t, u ∈ T |b0 (t) − b0 (u)| ≤ 3

Z

h≤c∗

+2

Z

|f (t, s) − f (u, s)|I{|f (t,s)|∨|f (u,s)|>1} ν(ds)

h>c∗

≤ 3τ (t, u)

[|f (t, s) − f (u, s)| ∧ 1] ν(ds)

Z

h≤c∗

hI{Dh>1} dν + 2

25

Z

h>c∗

[τ (t, u)h ∧ 1] dν.

This establishes (5.9). In the case X is a zero–mean process given by (2.4), b0 is given by b0 (t) = −

Z

h>c∗

f (t, s) ν(ds).

(5.10) follows from (5.12). The last inequality (5.11) is obvious from the definition of kf kτ in (2.8). qed Remark 5.1 Consider X2 := X2 (t) in (5.7). We can express it as X

X2 (t) =

(5.15)

f (t, sj )

j

where {sj } is a Poisson point process with intensity measure νe := ν({s : h(s) > c∗ }). Since νe ≤ 1, there are only a finite number of terms in the sum, νe-almost surely. Therefore the paths of X2 are bounded or continuous almost surely, whenever the sections f (·, s) are bounded or continuous for ν–almost all s.

5.3

Bounds for Poisson point processes

We use the following elementary lemma, which is easy to verify, to relate Poisson point processes on a general Borel space to Poisson point processes on [0, ∞). P

Lemma 5.3 Let Q = ∞ j=1 δuj be a Poisson point process on a Borel space U with mean measure ν, and let Φ : U 7→ V be a measurable map from U P into a Borel space V . Then Φ(Q) = ∞ j=1 δΦ(uj ) is a Poisson point process −1 on V with mean measure ν ◦ Φ . P

Let Γj = ji=1 Xi where {Xi } are independent identically distributed exponential random variables with mean one. The next lemma is a consequence of Lemma 5.3. P

Lemma 5.4 Let M = j δxj be a Poisson point process on [0, ∞) with intensity measure µ. Let µ−1 (t) := inf{u > 0 : µ([u, ∞)) ≤ t}. Then (5.16)

law

M =

X j

26

δµ−1 (Γj ) .

P

Proof: Consider µ(M) = j δµ([xj ,∞)) . By Lemma 5.3, µ(M) is a Poisson point process on R+ with mean measure µ ◦ µ−1 . Assume that µ([u, ∞)) is invertible. Then µ ◦ µ−1 is Lebesgue measure on R+ . Thus we can write P µ(M) = j δΓj , since, as is well known, {Γj } is a Poisson point process on R+ with intensity measure Lebesgue measure. Apply µ−1 to µ(M) and we get (5.16). P To consider the general case note that N := j δΓj is a Poisson point process on R+ with intensity measure λ, where λ is Lebesgue measure. Let f f := P δ −1 M j µ (Γj ) . To prove this lemma we need to show that M is a Poisson point process on R+ with intensity measure µ. By Lemma 5.3 this comes down to showing that µ−1 : R+ 7→ R+ satisfies µ(B) = λ({x ∈ R+ : µ−1 (x) ∈ B}) for every Borel set B in R+ . This is easily seen to be true for sets of the form (a, b) where a and b are continuity points of both µ−1 (t) and µ([t, ∞)). Since there are only a countable number of pairs of points that don’t have this property, it is true for all open intervals. qed Consider the random sequence {xj }. We obtain sharp bounds for certain norms of the ℓp and ℓp,∞ norms of such sequences. Recall that (5.17)



k{xi }kp,∞ = sup tp Card{i : |xi | > t} t>0

1/p

.

We first make a few elementary observations for use in the next lemma. Let (5.18)

j0 = 1 + sup{j ≥ 1 : Γj < (j + 1)/2}

Therefore P (j0 ≥ 1 + k) ≤

∞ X

P (Γj < (j + 1)/2)

j=k

Recall that

Z

xj−1 −x e dx (j − 1)! 0 and note that for j ≥ 3 the integrand is increasing over the range of integration. Therefore, for j ≥ 3 P (Γj < (j + 1)/2) =

(5.19)

P (Γj < (j + 1)/2) ≤

(j+1)/2

j + 1 (j + 1)j−1 −(j+1)/2 e 2 2j−1(j − 1)!

2 ≤ 2j √ e 27

!−(j+1)

This bound implies that j0 has moments of all orders. P

Lemma 5.5 Let M = j δxj be a Poisson point process on [0, ∞) with intensity measure µ. For all r, p > 0 (5.20)



1/r Ek{xj }krp

≤ Cp,r

n Z

u≤µ−1 (1)

+

(5.21)



Ek{xj }krp,∞

1/r

≤ Cp,r

Z

u>µ−1 (1)

n

+

(5.22)

Λr (P, k{xj }kp ) ≤ Cp,r

sup u≤µ−1 (1)

Z

Λr (P, k{xj }kp,∞) ≤ Cp,r

ur µ(du)

u>µ−1 (1)



n

+

1/p

1/r o

.

up µ([u, ∞))

n Z

+

(5.23)

up µ(du)

u≤µ−1 (1)

ur µ(du)

up µ(du)

1/p

1/r o

.

1/p

sup ur µ([u, ∞))

u>µ−1 (1)

sup u≤µ−1 (1)



up µ([u, ∞))

sup u>µ−1 (1)

1/r o

.

1/p

ur µ([u, ∞))

1/r o

.

Proof: It is helpful to make clear precisely what is the difference between µ((µ−1(t), ∞)) and µ([µ−1 (t), ∞)). Obviously, for those t at which µ−1 (t) is continuous and strictly decreasing these terms are both equal to t. The delicate cases are where µ−1 is not continuous or where µ−1 (t) = Const. for a ≤ t ≤ b, b > a. To consider these cases let us first note that the decreasing real valued function µ([a, ∞)) is left continuous with right hand limits, whereas µ−1 (t) is right continuous with left hand limits. Therefore, when µ−1 (t) = Const. for a ≤ t ≤ b, µ((µ−1 (t), ∞)) = a and µ([µ−1 (t), ∞)) = b. When µ−1 has a discontinuity at t and µ−1 (t) = c and µ−1 (t−) = d, d > c, we get that µ((µ−1(t), ∞)) = µ([µ−1 (t), ∞)) = t. In any case, for all t > 0 (5.24)

µ((µ−1 (t), ∞)) ≤ t ≤ µ([µ−1 (t), ∞)). 28

Similarly when µ([t, ∞)) = Const. for a ≤ t ≤ b, µ−1 (µ([t, ∞))) = a. When µ has a discontinuity at t, µ−1 (µ([t, ∞))) = t. In any case, for all t>0 (5.25) µ−1 (µ([t, ∞))) ≤ t. Also, for clarity, we note that µ(du) = dµ([u, ∞)). Using the representation of M in Lemma 5.4 we have (5.26)

∞ X

xpj

=

j=1

=

∞ X

(µ−1 (Γj ))p

j=1 ∞ X

(µ−1 (Γj ))p I{Γj ≥j/2} +

∞ X

(µ−1 (Γj ))p I{Γj µ−1 (1)

ur µ(du).

To see this let u0 = µ−1 (1). There is really no problem unless µ({u0}) > 0. In the next integral we integrate after the potential jump at u0 and get Z

(5.34)

µ((u0 ,∞)) 0

(µ−1 (s))r e−s ds ≤

Z

u>u0

ur µ(du).

Next we integrate over the potential jump at u0 using the fact that µ−1 is constant in this region, and get (5.35)

Z

µ([u0 ,∞))

µ((u0 ,∞))





(µ−1 (s))r e−s ds = (µ−1 (1))r e−µ((u0 ,∞)) − e−µ([u0 ,∞)) .

Finally, using (5.25) we note that Z

(5.36)

∞

µ([u0 ,∞))

(µ−1 (s))r e−s ds ≤ (µ−1 (1))r e−µ([u0 ,∞)) .

Combining (5.34)–(5.36) we get (5.33). Taking the expectation in (5.30) and then using (5.31)–(5.33) yields (5.37)



1/r Ek{xj }krp

≤ Cp,r

n Z

u≤u0

u0 + If µ({u0 }) > 1/2 then (5.38)

Z

u≤u0

up µ(du)

30

Z

1/p

up µ(du)

u>u0

1/p

ur µ(du)

≥ u0 2−1/p .

1/r o

.

If µ({u0}) ≤ 1/2 then µ((u0, ∞)) = µ([u0 , ∞)) − µ({u0 }) ≥ 1/2 since µ([u0, ∞)) ≥ 1. Therefore Z

(5.39)

u>u0

ur µ(du)

1/r

≥ u0 2−1/r .

We see that u0 is always absorbed by one of the other two terms on the right–hand side of (5.37). This proves (5.20). Now we consider (5.21). We proceed similarly to (5.26) starting with the decomposition (5.40)

k{xj }kp,∞ = sup j 1/p µ−1 (Γj ) j≥1

≤ sup j 1/p µ−1 (j/2) + sup j 1/p µ−1 (Γj )I{Γj 1/2 sup c(µ([c, ∞)))1/p ≥ u0 2−1/p .

(5.45)

c≤u0

Thus we get (5.21). To obtain (5.22) and we use (5.26), (5.27) and (5.29) to see that (5.46) Λr (P, k{xj }kp ) ≤ Cp,r

n Z

w p µ(dw)

w≤µ−1 (1)



1/p

+ µ−1 (1/2) 1/p

+Λr P, µ−1(X1 )j1

o

.

Note that by (5.24) 

1/p

P µ−1 (X1 )j1

(5.47)

>c





−1/p

≤ EP X1 < µ([cj1 h

h

Let

−1/p

r

Br =

(5.48)

−1/p

= E 1 − exp(−µ([cj0 ≤ E µ([cj0

Thus for a > µ−1 (1)

i



, ∞)))

, ∞)) ∧ 1 .

sup s µ([s, ∞))

s>µ−1 (1)

, ∞))

!1/r

i

.





B r . a Using this and Chebyshev’s inequality we see that (5.49)

µ([a, ∞)) ≤

h

−1/p

E µ([cj0 (5.50)

, ∞)) ∧ 1

i

h

−1/p

≤ E µ([cj1

, ∞))I{cj −1/p>µ−1 (1)} 1

+EI{cj −1/p ≤µ−1 (1)}

i

1

≤c

−r

r/p Brr E(j0 )

r/p

+ c−r (µ−1 (1))r E(j0 ).

Therefore (5.51)



1/p

Λrr P, µ−1(X1 )j1





−1/p

≤ sup cr E µ([cj0 c

, ∞)) ∧ 1

≤ Cr,p (Brr + (µ−1 (1))r ). 32



Note that by (5.49) and (5.24) when µ−1 (1/2) > µ−1 (1) it is also less than or equal to 21/r Br . Therefore by (5.46)–(5.51) (5.52)

Λr (P, k{xj }kp ) ≤ Cp,r

n Z

u≤u0

up µ(du)

1/p

o

+ u0 + Br .

We have already seen that when µ({u0 }) > 1/2, u0 can be absorbed by the integral. When µ({u0}) ≤ 1/2, µ((u0 , ∞)) ≥ 1/2 which implies that Br ≥ u0 2−1/r . Thus we get (5.22). To prove the final inequality (5.23) we combine (5.40)–(5.43) and (5.51) with (5.45) and the next to last sentence of the preceding paragraph. qed Corollary 5.1 Suppose that µ((K, ∞)) = 0 in Lemma 5.5. Then, for all r, p > 0 

1/r Ek{xj }krp



Ek{xj }krp,∞

1/r

≤ Cp,r

≤ Cp,r

n Z

n

up µ(du)

1/p

sup up µ([u, ∞)) u

o

+K .

1/p

o

+K .

Proof: Use the inequality µ((µ−1 (1), ∞)) ≤ 1 in (5.20) and (5.21) of Lemma 5.5. qed The bounds in Lemma 5.5 have much greater scope than Poisson point processes on R+ . Let N be a Poisson random measure on a Borel space S with σ–finite mean measure ν, and h : S 7→ R (or C) be a measurable deterministic function. We use Lemma 5.5 to obtain bounds for strong and weak moments of the stochastic integral Z

(5.53)

S

p

|h(s)| N(ds)

1/p

.

Lemma 5.6 Recall the definition (2.20) of the separating number c∗ = c∗ (h) := inf{c > 0 : ν({s : |h(s)| ≥ c}) ≤ 1}.

(5.54)

Then, for all r, p > 0 (5.55)

 Z

E

S

|h|p dN

r/p 1/r

≤ Cr,p

n Z

|h|≤c∗

+ 33

|h|p dν

Z

|h|>c∗

1/p

|h|r dν

1/r o

and (5.56) Λr (P,

Z

S

p

|h| dN

1/p 

≤ Cr,p

n Z

|h|≤c∗



|h|p dν

1/p

+ sup cr ν({|h| ≥ c}) c>c∗

In particular, if ν({s : |h(s)| ≥ K}) = 0 for some K ≥ 0, then  Z

E

(5.57)

S

p

|h| dN P

r/p 1/r

≤ Cr,p

n Z

S

R

|h|p dν

1/p

1/r o

.

o

+K . P

Proof: Write N = j δsj and note that S |h(s)|p N(ds) = j |h(sj )|p . By P Lemma 5.3, j δ|h(sj )| is a Poisson point process on R+ with mean measure µ which is the image of ν under the map s 7→ |h(s)|. Thus we can write P P p P p j δxj is a Poisson point process on R+ with j xj , where j |h(sj )| = mean measure µ([c, ∞)) = ν(s : |h(s)| ≥ c). The lemma is now a direct application of (5.20) and (5.22) of Lemma 5.5 and Corollary 5.1. qed In the previous lemma we considered k{h(sj )}kp which is described in the beginning of the proof of the lemma. In the next lemma we consider k{h(sj )}kp,∞. Lemma 5.7 Let c∗ be given by (5.54). Then for every p, r > 0 (5.58)



E(k{h(sj )}krp,∞ )

1/r

≤ Cr,p

n

sup cp ν({|h| ≥ c})

c≤c∗

+

Z

|h|>c∗

1/p

|h|r dν

1/r o

and 

(5.59) Λr P, k{h(sj )}krp,∞)



≤ Cr,p

n

sup cp ν({|h| ≥ c})

c≤c∗



1/p

+ sup cr ν({|h| ≥ c}) c>c∗

1/r o

.

In particular, if ν({s : |h(s)| ≥ K}) = 0 for some K > 0, then (5.60)



E(k{h(sj )}krp,∞ )

1/r

n

o

≤ Cr,p Λp (ν, h) + K .

Proof: This lemma is a direct application of (5.21) and (5.23) of Lemma 5.5 and Corollary 5.1. qed

34

5.4

Processes in exponential Orlicz spaces

Let k · kψq denote the norm in the Orlicz space Lψq (dP ), where ψq (x) =

(5.61)

(

exp(|x|q ) − 1 exp exp(|x|) − e

1≤q 0, with norm given by

(5.62)

kξkψq = inf {c > 0 : Eψq (|ξ|/c) ≤ 1} .

The next theorem, which is crucial in this paper, is a generalization of an important result about Gaussian processes. Theorem 5.1 There exists a finite constant Cq , q ∈ [1, ∞], with the following property. Let X = {X(t) : t ∈ T } be a measurable separable stochastic process on a precompact metric space (T, d) such that X(t) ∈ Lψq (Ω, P ) and kX(t) − X(s)kψq ≤ d(t, u) for all t, u ∈ T . Let m be a majorizing measure on T . Then there exists a positive (extended valued) random variable Z ∈ Lψq (Ω, P ) such that kZkψq ≤ Cq and for every δ > 0 sup |X(t, ω) − X(u, ω)| ≤ Z(ω)Iq (m, d; δ).

(5.63)

d(t,u)≤δ

t,u∈T

Theorem 5.1 applies to Gaussian processes when q = 2. In this case it contains ideas which originated in an important early paper by Garcia, Rodemich and Rumsey Jr., [4] and where developed further by Preston, [18, 19] and Fernique, [2]. The fact that it can be extended to 1 ≤ q ≤ ∞ is no doubt understood by many researchers in the field of probability on Banach spaces but we don’t know any references. Actually all one needs is the following technical lemma. Using this the reader can readily complete the proof of Theorem 5.1 by following the proof of Theorem 5.2.6 in [3] or Theorem 5.3.3 in [14]. Lemma 5.8 For 1 ≤ q ≤ ∞, let X be a stochastic process as in Theorem 5.1 such that kX(t)kψq ≤ 1 for all t ∈ T . Then there exists a random variable Z with kZkψq ≤ Cq , suchR that for every probability measure m on T and function h : T 7→ R+ with T h(v) m(dv) ∈< ∞

(5.64)

Z

T

|X(t)|h(t) m(dt) ≤Z

Z

T

h(t)ψq−1

Z

T

35

h(v) m(dv)

−1

!

h(t) m(dt).

Proof: Define e Z(ω)

(5.65)

= inf{α > 0 :

Z

T

ψq (α−1 |X(t)|) µ(dt) ≤ 1}.

We first show that

e kZk ψq ≤ Cq < ∞

(5.66)

1 ≤ q ≤ ∞.

Let 1 ≤ q < ∞, then for every u ≥ 1 P



Ze

>u



≤ P ≤ P ≤ P ≤ 2

Z

−q

TZ

−uq

exp(u |X(t)| ) m(dt) > 2 −q

T

Z

T Z

q

q

exp(u |X(t)| ) m(dt)

 uq

exp(|X(t)|q ) m(dt) > 2u

T



q

>2

uq





E exp(|X(t)|q ) m(dt)

q

≤ 21−u .

The third inequality follows from Jensen’s inequality and the fourth because kX(t)kψq ≤ 1. Thus we get (5.66) when 1 ≤ q < ∞. Now let q = ∞. Note that for each u ≥ 1, the function φu (x) = exp((log x)u ) is convex for x ≥ e. Using Jensen’s inequality again we get that for u ≥ 1 P {Ze > u} ≤ P {

Z

T

= P {φu

≤ P{

Z

T

exp(exp(u−1 |X(t)|)) m(dt) > e + 1}

Z

T



exp(exp(u−1 |X(t)|)) m(dt) > φu (e + 1)}

exp(exp(|X(t)|)) m(dt) > exp(exp(cu))}

≤ exp(− exp(cu))

Z

T

E exp(exp(|X(t)|)) m(dt)

≤ (1 + e) exp(− exp(cu)) where c = log(log(e + 1)) > 0. Thus we get (5.66) when q = ∞. We now prove (5.64). We have (5.67)

xy ≤ ψq (x) + yψq−1(y) x, y ≥ 0. 36

This inequality follows from Young’s inequality and the fact that ψq′ (x) ≥ ψq (x) for every x > 0 and 1 ≤ q ≤ ∞, or just by standard calculus. Let h : T 7→ R+ be as in the lemma. Putting x = Ze −1 |X(t)| and R y = ( T h(v) m(dv))−1h(t) in (5.67) we get Z

|X(t)|h(t) ≤ Ze +

T

h(v) m(dv)ψq (Ze −1 |X(t)|)

−1 e Zh(t)ψ (( q

Z

T

h(v) m(dv))−1h(t)).

Integration with respect to m gives Z

T

|X(t)|h(t) m(dt) ≤ +

Ze

Ze

Z

T

Z

T

h(t) m(dt) Z

h(t)ψq−1

T

h(v) m(dv)

−1

h(t)

!

m(dt).

Since x 7→ xψq−1 (β −1 x) is a convex function for x, β ≥ 0, Jensen’s inequality implies that Z

h(t)ψq−1

T

Z

T

h(v) m(dv)

−1

!

h(t) m(dt) ≥

Z

T

h(t) m(dt)ψq−1 (1),

which yields the inequality Z

T

|X(t)|h(t) m(dt) ≤ (1 +

1/ψq−1 (1))Ze

Z

T

h(t)ψq−1

Z

T

h(v) m(dv)

−1

!

h(t) m(dt).

Changing (1 + 1/ψq−1(1))Ze in the preceding line to Z gives (5.64).

qed

We use the following simple lower bound for Iq (m, τ ; δ). We give its proof for completeness. Lemma 5.9 Assume that there exist v, w ∈ T for which τ (v, w) > 0. Then

(5.68)

Iq (m, τ ; δ) ≥ Cq δ

where Cq > 0 depends only on q ∈ [1, ∞]. 37

0 (2/3)D. Since balls Bτ (t, D/3) and Bτ (u, D/3) are disjoint, at least one of them, say Bτ (t, D/3) has mmeasure less or equal to 1/2. Hence Iq (m, τ ; δ) ≥

Z

δ

0

1 log m(Bτ (t, r))

!1/q

dr ≥ (log 2)1/q δ

for δ ∈ (0, D/3] and 1 ≤ q < ∞. Similarly, I∞ (m, τ ; δ) ≥ log(log 2e)δ. If δ ∈ (D/3, D], then Iq (m, τ ; δ) ≥ Iq (m, τ ; D/3) ≥ (log 2)1/q δ/3 and, similarly, I∞ (m, τ ; δ) ≥ log(log 2e)δ/3. qed The classic example of random variables in LΨq are Rademacher series with constraints on the coefficients. The following lemma is given for the convenience of the reader. Lemma 5.10 Let {aj } be real numbers. For 1 ≤ p < 2 (5.69) When p = 2 (5.70)

Cp−1 k{aj }kp,∞ ≤ k

X

ǫj aj kΨq ≤ Cp k{aj }kp,∞.

C2−1 k{aj }k2 ≤ k

X

ǫj aj kΨ2 ≤ C2 k{aj }k2 .

j

j

(5.69) is Lemma 3.1, [12]. (See also [9], Section 4.1). (5.70) is elementary.

6

Proofs of results in Section 2

Proof of Theorem 2.1: Consider the decomposition (5.5) for h = kf kτ . Recall that by hypothesis, kf kτ < ∞, ν–almost surely. Consequently c∗ < ∞. It then follows from (5.8) and (5.9) of Lemma 5.2 that b0 is bounded and continuous. Furthermore, it follows from Remark 5.1 that X2 is bounded or continuous almost surely whenever the sections f (·, s) are bounded or continuous for ν–almost all s. Thus we only need to prove this theorem for X1 . To simplify notation we write X for X1 and f¯ for f I{kf kτ ≤c∗ } . c be the symmetrization of X and express it as in (5.4). Without Let X loss of generality we may assume that {ǫj } and {sj } depend on different coordinates of a product probability space (Ω × Ω′ , P × P ′ ). Therefore (6.1)

c ω, ω ′) = X(t,

∞ X

ǫj (ω ′)f¯(t, sj (ω))

j=1

38

P × P′

a.s.

P ¯ sj (ω))|2 < ∞, ∀t ∈ Let T0 be a finite subset of T and let Ω0 = {ω : j |f(t, c on a finite index set T we avoid possibly delicate T0 }. (By considering X 0 questions of measurability and separability). It follows from (2.3) and Lemma 5.1 that P (Ω0 ) = 1 and for every ω ∈ Ω0

ˆ ω, ·) = X(t;

(6.2)

∞ X

j=1

ǫj f¯(t, sj (ω)) t ∈ T0

is a Rademacher process with respect to {ǫj }. Let k · kψq denote the norm in the Orlicz space Lψq (Ω′ , P ′). By the Contraction Principle and Lemma 5.10 we get that for every t, u ∈ T0 and ω ∈ Ω0

∞



X ¯ sj (ω)) − f¯(u, sj (ω))|

ˆ ω, ·) − X(u; ˆ ǫj |f(t, (6.3) kX(t; ω, ·)kψq ≤ j=1 ∞

X

≤

j=1

ψq

¯ τ (sj (ω))

τ (t, u) ǫj kfk ψq

≤ Cp ξ(ω)τ (t, u) where (6.4)

n

ξ(ω) = kf¯kτ (sj (ω))

o∞

j=1 p,∞

.

We now use (5.60) of Lemma 5.7 with K = c∗ and (2.13) to see that (6.5)



Eξ r

1/r

h

i

¯ τ ) + c∗ < ∞. ≤ Cr,p Λp (ν, kfk

In particular, ξ < ∞ almost surely. For each ω ∈ Ω0 such that ξ(ω) < ∞, we define a pseudo metric ρω on T0 by ρω (t, u) = Cp ξ(ω)τ (t, u) and note that by (6.3) (6.6)

ˆ ω, ·) − X(u; ˆ ω, ·)kψq ≤ ρω (t, u) t, u ∈ T0 . kX(t;

It is well known that given a majorizing measure m on T there exists a probability measure m0 on T0 such that for every δ > 0 (6.7)

Iq (m0 , τ0 ; δ) ≤ 2Iq (m, τ ; δ) 39

where τ0 is the restriction of τ to T0 . (See, e.g., the proof of Lemma 11.9, [9]). It follows from (6.6) and Theorem 5.1 that there exists a random variable Z : Ω′ 7→ R+ , with (E ′ Z r )1/r ≤ Cq,r such that ˆ ω, ω ′) − X(u; ˆ ω, ω ′)| ≤ Z(ω ′)Iq (m0 , ρω ; δ). sup |X(t;

ρω (t,u)≤δ

t,u∈T0

Consequently ˆ ω, ω ′) − X(u; ˆ ω, ω ′)| = sup |X(t;

sup

τ (t,u)≤δ

ρω (t,u)≤Cp ξ(ω)δ

t,u∈T0

t,u∈T0 Z(ω ′ )Iq (m0 , ρω ; Cp ξ(ω)δ) Cp ξ(ω)Z(ω ′)Iq (m0 , τ ; δ).

≤ =

(6.8)

ˆ ω, ω ′) − X(u; ˆ |X(t; ω, ω ′)|

We take the expectation and use (6.5) with r = 1, and (6.7) to see that (6.9)

i

h

c c ¯ τ ) + c∗ Iq (m, τ ; δ). − X(u)| ≤ Cp Λp (ν, kfk E sup |X(t) τ (t,u)≤δ

t,u∈T0

Recall that we are using X to denote X1 and EX1 (t) = 0 for all t ∈ T c = X − X ′ , where X ′ is an independent copy of X, we see that Since X (6.10)

h

i

c c E sup |X(t) − X(u)| = E sup E X(t) − X(u)|X τ (t,u)≤δ

τ (t,u)≤δ

t,u∈T0

t,u∈T0

c c ≤ E sup |X(t) − X(u)|. τ (t,u)≤δ

t,u∈T0

Combining (6.9) and (6.10) we obtain (6.11)

h

i

¯ τ ) + c∗ Iq (m, τ ; δ). E sup |X(t) − X(u)| ≤ Cp Λp (ν, kfk τ (t,u)≤δ

t,u∈T0

This bound extends to any countable set Te0 ⊂ T by the monotone convergence theorem. Let Te0 be a separating set for X. Since τ is continuous, for any δ > 0 sup |X(t) − X(u)| ≤ sup |X(t) − X(u)|.

τ (t,u)≤δ

τ (t,u)≤2δ

t,u∈T

t,u∈Te0

40

¿From this inequality, (6.11), and the fact that Iq (m, τ ; 2δ) ≤ 2Iq (m, τ ; δ), we get i h ¯ τ ) + c∗ Iq (m, τ ; δ). E sup |X(t) − X(u)| ≤ Cp Λp (ν, kfk τ (t,u)≤δ

t,u∈T

Returning to the original notation, we have

i

h

(6.12) E sup |X1 (t) − X1 (u)| ≤ Cp Λp (ν, kf kτ I{kf kτ ≤c∗ } ) + c∗ Iq (m, τ ; δ). τ (t,u)≤δ

t,u∈T

Hence X1 has bounded paths almost surely. If (2.14) holds, we see that paths of X1 are continuous almost surely. qed Proof of Theorem 2.3: Consider the decomposition (5.5) of X with some function h satisfying (2.24) and c∗ given by (2.20). To deal with the process X1 we follow the proof of Theorem 2.1 but take the expectation of the r ′ –th power, r ′ = r ∨ 1, of (6.8) to get, in place of (6.12) h

(6.13)

E sup |X1 (t) − X1 (u)|r

′

τ (t,u)≤δ

t,u∈T

h

i1/r′

i

≤ Cp,r Λp (ν, hI{h≤c∗ } ) + c∗ Iq (m, τ ; δ). Here we use (6.5) and the fact that Z has moments of al orders. Now by Jensen’s inequality, r ′ can be replaced by r. Next consider X2 . We have |X2 (t) − X2 (u)| ≤

Z

{h>c∗ }

≤ τ (t, u)

|f (t, s) − f (u, s)| N(ds)

Z

{h>c∗ }

Hence sup |X2 (t) − X2 (u)| ≤ δ

τ (t,u)≤δ

t,u∈T

h dN.

Z

{h>c∗ }

h dN.

Since ν({hI{h>c∗ } ≥ c}) ≤ 1 for every c > 0, c∗ (hI{h>c∗ } ) = 0; (see (5.54)). Using (5.56) of Lemma 5.6 we obtain (6.14)



Λr P, sup |X2 (t) − X2 (u)| τ (t,u)≤δ

t,u∈T





≤ Λr P, δ 

Z

{h>c∗ }

h dN 

≤ δCr Λr ν, hI{h>c∗ } . 41



Combining (6.13), (6.14) and Lemma 5.9 we get (6.15)





n



Λr P, sup |X0 (t) − X0 (u)| ≤ Cr,p Λp ν, hI{h≤c∗ } τ (t,u)≤δ

t,u∈T



+c∗ + Λr ν, hI{h>c∗ }

o



Iq (m, τ ; δ).

Consider a := ν({s : h(s) = c∗ }). If a ≥ 1/2, then Λp (ν, hI{h≤c∗ } ) ≥ 2−1/p c∗ . If a < 1/2, then Λr (ν, hI{h>c∗ } ) ≥ 2−1/r c∗ , because ν({h > c∗ }) > 1/2. In either case, the middle term c∗ on the right hand side of (6.15) can be absorbed by one of the other two terms. This proves (2.27). We now assume (2.28). Using (5.55) of Lemma 5.6 we obtain h

E sup |X2 (t) − X2 (u)|r τ (t,u)≤δ

t,u∈T

i1/r

h Z

≤ δ E ≤ δCr

hZ

{h>c∗ }

{h>c∗ }

h dN)r hr dν

i1/r

i1/r

.

Combining this with (6.13), and eliminating c∗ by an argument similar to the one used for (6.15), proves (2.29). We now derive estimates for supt∈T |X0 (t)| from (2.27) and (2.29). To this end, consider an extended metric space T¯ = T ∪ {∂} where ∂ is an isolated point. Define a pseudo distance τ¯ by τ¯(t, u) =

  τ (t, u) D 0

t, u ∈ T t = ∂, u ∈ T or u = ∂, t ∈ T t = u = ∂.

Define a probability measure m ¯ on T¯ by m(A) ¯ = (1/2)m(A∩T )+(1/2)δ∂ (A). It is easy to see that Iq (m, ¯ τ¯; D) ≤ Cq Iq (m, τ ; D). We extend X to T¯ by ¯ setting f¯(∂, s) = 0 so that X(∂) = 0. Then by (2.8), with t0 = ∂ and (5.12) ¯ s) − f(u, ¯ s)| |f(t, τ¯(t, u) τ ¯(t,u)6=0

¯ kfk(s) := D −1 f¯(∂, s) + sup   

t,u∈ T¯

= max D −1 sup |f (t, s)|, sup  

t∈T

τ (t,u)6=0

t,u∈ T

= kf kτ (s) ≤ h(s). 42

  |f (t, s) − f (u, s)| 

τ (t, u)

 

We also see that

¯ 0 (t) − X ¯ 0 (u)| sup |X0 (t)| ≤ sup |X t,u∈T¯

t∈T

¯ 0 = X0 on T and is equal to zero at ∂. Therefore, applying (2.27) for since X ¯ X0 and δ = D we obtain     ¯ 0 (t) − X ¯ 0 (u)| Λr P, sup |X0 (t)| ≤ Λr P, sup |X t∈T

n

(6.16)

t,u∈T¯







≤ Cr,p Λp ν, hI{h≤c∗ } + Λr ν, hI{h>c∗ } n







≤ Cr,p Λp ν, hI{h≤c∗ } + Λr ν, hI{h>c∗ }

o

Iq (m, ¯ τ¯; D)

o

Iq (m, τ ; D).

This is the desired bound for the supremum of X0 under condition (2.26). Similarly, from (2.29) we derive the bound for the supremum of X0 under (2.28). qed Proof of Theorem 2.2: By (2.27) for r = p ∈ [1, 2), h = kf kτ , and the corresponding statement for the supremum of X0 in (6.16) we have (6.17) Λp (P, sup |X(t)|) ≤ 2 sup |b0 (t)| + 2Λp (P, sup |X(t)|) t∈T

t∈T

t∈T

≤ 2 sup |b0 (t)| + Cp Λp (ν, kf kτ )Iq (m, τ ; D). t∈T

and Λp (P, sup |X(t) − X(u)|)

(6.18)

τ (t,u)≤δ

t,u∈T

≤ 2 sup |b0 (t) − b0 (u)| + Cp Λp (ν, kf kτ )Iq (m, τ ; δ). τ (t,u)≤δ

t,u∈T

If X is symmetric, b0 = 0 and (6.17) and (6.18) are exactly (2.16) and (2.17). Suppose X is not symmetric. We use Lemma 5.2 to estimate b0 . Observe that c∗ ≤ a := Λp (ν, kf kτ ). Let p > 1. We have Z

∗

kf kτ >c∗

kf kτ dν ≤ aν({kf kτ > c }) + ≤ a+

Z

0

∞

Z

kf kτ >a

ν({kf kτ > x ∨ a}) dx

≤ a + aν({kf kτ > a}) +

1 ≤ 2a + a. p−1 43

kf kτ dν

Z

∞

a

ap dx xp

Hence, by (5.10) of Lemma 5.2 and Lemma 5.9 sup |b0 (t)| ≤ D t∈T

Z

kf kτ >c∗

kf kτ dν ≤ Cp DΛp (ν, kf kτ )

≤ Cp Λp (ν, kf kτ )Iq (m, τ ; D). Combining this with (6.17) gives (2.16). Similarly we get (2.17) from (6.18) and (5.11) of Lemma 5.2. Now consider the case p = 1. Assume first that c∗ D > 1. We have (6.19)

Z

D −1 D−1 }) + ≤ a + a log(c∗ D).

Z

c∗

D −1

ν({kf kτ > x}) dx

By (5.8) of Lemma 5.2 sup |b0 (t)| ≤ D t∈T

Z

D −1 c∗

=

[δkf kτ ∧ 1] dν

(Z

c∗ a}) + ≤ 3aδ + aδ log(aδ)−1 . In the case aδ ≥ 1 we have Z

kf kτ >c∗

[δkf kτ ∧ 1] dν ≤

Z

(Z

c∗ a

)

[δkf kτ ∧ 1] dν

≤ aδ + ν({kf kτ > a}) ≤ aδ + 1 ≤ 2aδ. Hence by (5.9) and (6.19) sup |b0 (t) − b0 (u)| ≤ 3δ

τ (t,u)≤δ

t,u∈T

Z

D −1 c∗

[δkf kτ ∧ 1] dν

≤ 3δ(a + a log(c∗ D) ∨ 0) + 2(3aδ + aδ log(aδ)−1 ∨ 0) ≤ 9aδ + 3aδ[(log(aD) ∨ 0) + (log(aδ)−1 ∨ 0)] which, in conjunction with (6.18), gives (2.19).

qed

Proof of Theorem 2.4: As in the proof of Theorem 2.1, it is enough to consider X = X1 . We follow the steps of the proof of that theorem. In the place of (6.3) we use (6.20)



∞

X ¯ τ (sj (ω))

τ (t, u) ˆ ω, ·) − X(u; ˆ ǫj kfk kX(t; ω, ·)kψ2 ≤ ψ2

j=1

≤ C2 ξ(ω)τ (t, u)

where, by Lemma 5.10

n

¯ τ (sj (ω)) ξ(ω) = kfk

o∞

j=1 2

.

It follows from (5.57) of Lemma 5.6 that for every r ≥ 1 (6.21)



Eξ r

1/r

≤ Cr

h Z

S

45

¯ 2 ν(ds) kfk τ

i1/2



+ c∗ .

Proceeding as in the proof of Theorem 2.1, but taking expectations of the second power instead of the first, we get (6.22)

h

E sup |X(t) − X(u)|2 τ (t,u)≤δ

t,u∈T

≤ C2

h Z

S

i1/2

kf¯k2τ ν(dx)

i1/2



+ c∗ I2 (m, τ ; δ).

This proves the boundedness of X = X1 and also its sample continuity when (2.14) holds. qed Proof of Theorem 2.5: We use the proof of Theorem 2.4 but take X to be the entire process and do not truncate kf kτ . In this case (5.55) for p = r = 2 gives Z 

Eξ 2

(6.23)

1/2

≤ C2

h

S

kf k2τ ν(ds)

i1/2

and (2.33) follows from (6.22), (without c∗ , since we do not do not truncate). qed

Proof of Theorem 2.6: We use the decomposition of X given in (5.5) and follow the proof of Theorem 2.3. Using (6.21) we obtain (6.24)

h

E sup |X1 (t) − X1 (u)|r τ (t,u)≤δ

t,u∈T

≤ C2

h Z

S

h2 dν

i1/2

i1/r 

+ c∗ I2 (m, τ ; δ).

The rest of the proof is identical with the proof of Theorem 2.3. Proof of Proposition 2.1: Let kf k∞ (s) := supt∈T |f (t, s)|. By (2.37) we have |f (t, ·)| ∧ 1 ∈ L1 (S, ν). Hence (6.25)

X(t) =

∞ X

f (t, sj )

j=1

for each t ∈ T a.s. (see Theorem 10.15, [8]). Without loss of generality we may assume that X is given by (6.25). Moreover, again by (2.37), P j kf k∞ (sj ) < ∞ almost surely. Therefore, for almost every ω, the series 46

P∞

j=1

f (t, sj (ω)) converges uniformly. The proposition follows.

qed

Proof of Proposition 2.2: We have |f (t, s)| ≤ Dh(s) (see (5.12)). As in the proof of Proposition 2.1 we may assume that X is given by (6.25). Therefore ∞ (6.26)

sup |X(t)| ≤ D t∈T

X

h(sj )

j=1

and sup |X(t) − X(u)| ≤ δ

(6.27)

τ (t,u)≤δ

t,u∈T

∞ X

h(sj ).

j=1

The conclusions of this proposition follow from these two inequalities and Lemma 5.5. qed Proof of Remark 2.1: We only need to prove the second part of the remark. We consider q < ∞, the case q = ∞ is similar. Note that if τ (t, s) = φ(|t−s|), then Z δ Jq (τ, δ) = (6.28) (log N(d, r))1/q dφ(r) 0

(the Riemann–Stieltjes integral), where d denotes the Euclidean distance. If D0 is the diameter of T , then for some constant Cn , N(d, r) ≤ (Cn D0 /r)n . Substituting this bound in (6.28) and integrating by parts we get (2.43). qed

7

Proof of Theorem 3.1

Proof: We use Lemma 5.2 and Theorem 2.3. Let Y (t) = b0 (t) + Y0 (t) as in the decomposition (2.21), with h = kf kτ . To be more specific, we take (7.1) Y0 (t) = and (7.2)

Z

kf kτ ≤c∗

f (t, s) [N(ds) − ν(ds)] + b0 (t) = −

Z

kf kτ >c∗

47

Z

kf kτ >c∗

f (t, s) ν(ds).

f (t, s) N(ds).

Recall that for a process Y given by (3.7), s = (x, u) and f (t, (u, x)) = g(t, u)x. By (3.4) and (3.9) we have Z

(7.3) ν({kf kτ ≥ c}) =

U

Z

kgkτ a c kkgkτ ≥c

≤ Let ∗

d := 2

(7.4)

1/p

θ({x : |x| ≥ c/kgkτ (u)}, u) w(du)

Z

akgkpτ

U

dw

1/p

!p

∨2

1/r

Z

kgkτ dw + b c kkgkτ d∗ .

Consequently, by (7.5) p

Λp (ν, kf kτ I{kf kτ ≤c∗ } ) ≤ sup c ν(kf kτ > c) c≤c∗

!1/p

≤ d∗ .

By the definition of c∗ (7.7)

ν(kf kτ I{kf kτ c∗ } > v) ≤ 1.

Using this and (7.6) we see that and Λr (ν, kf kτ I{kf kτ >c∗ } ) 

∗ r

∗

≤ max ((d ) ν({kf kτ > c })) ≤d

∗

48

1/r

r

, sup c ν({kf kτ > c}) c>d∗

!1/r 

Therefore, by the analogue of (2.27) for supt∈T |Y0 (t)|, as explained at the end of the statement of Theorem 2.3 (7.8)





Λr P, sup |Y0 (t)| ≤ Cp,r d∗ Iq (m, τ, D) t∈T

≤ Cp,r

 Z

U

akgkpτ dw

1/p

+

Z

U

bkgkrτ dw

1/r 

Iq (m, τ, D).

When Y is symmetric, (7.8) gives (3.12) for r ≥ 1. We now estimate the drift b0 , for r > 1. Using (5.10) of Lemma 5.2, (7.6) and (7.5), we see that sup |b0 (t)| ≤ D

(7.9)

t∈T

= D

Z

kf kτ >c∗ Z ∞

ν(kf kτ I{kf kτ >c∗ } > v) dv

0

Z

≤ D d∗ + =

kf kτ dν

d∗ v

∞

d∗

Dd∗ r . r−1

!r

dv

!

Combining this with the estimate of supt∈T |Y0 (t)| gives (3.12). Similarly, we get (3.13), by substituting the above estimates in (2.27). We now consider the case r > p ≥ 1. By (7.3) we have ν(kf kτ ≥ c) ≤ Therefore (7.10) and (7.11)

Z

kkgkτ ≥c

ak

!p

kkgkτ c

−p

dw +

Z

kkgkτ p ≥ 1. It follows from (7.11) that c∗ = inf{c > 0 : ν(kf kτ ≥ c) ≤ 1} ≤

Z

U

(ak r−p ∨ b)kgkrτ dw

1/r

:= e∗ .

By the first two lines of (7.9), (7.7) and (7.11) we see that (7.13)

sup |b0 (t)| ≤ D t∈T

Z

∞

0

ν(kf kτ I{kf kτ >c∗ } > v) dv

∗

≤ D e + =

De∗ r . r−1

Z

∞

e∗



e∗ v

r

dv

!

Combining (7.12) with (7.13) gives (3.14). Similarly, we get (3.15), by substituting the above estimates in (2.27). qed

References [1] M. Braverman. About boundedness of stable sequences. Stochastic Process. Appl., 99:287–293, 2002. [2] X. Fernique. Regularit´e des trajectoires des fonctions al´eatoires Gaussi´ ´ e de Probabilit´es de Saint–Flour, IV–1974, Lecture ennes. Ecole d,Et´ Notes Math, 480:1–96 Springer, Berlin, 1975. [3] X. Fernique Fonctions al´eatoire gaussiennes vecteurs al´eatoire gaussiennes. Centre de Recherches Math´ematiques, Montreal, 1997. [4] A.M. Garcia, E. Rodemich and H. Rumsey Jr. A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Math. J., 20:565–578, 1978. [5] I.I. Gihman and A.V. Skorohod. The theory of stochastic processes, Vol. 1, 2, Springer, Berlin, 1975. [6] E. Gin´e and M.B. Marcus Some results on the domain of attraction of stable measures on C(K). Probability and Mathematical Studies, 2:125– 147, 1982. 50

[7] N.C. Jain and M.B. Marcus Central limit theorems for C(S) valued random variables, J. Functional Analysis, 19:216–231, 1975. [8] O. Kallenberg. Foundations of Modern Probability., Springer, 2001. [9] Ledoux, M. and Talagrand, M. Probability in Banach spaces. Isoperimetry and processes. Springer-Verlag, 1991. [10] M.B. Marcus. ξ–radial processes and random Fourier series. Memoirs of the American Mathematical Society, vol. 368, 1987. [11] M.B. Marcus and G. Pisier. Random Fourier series with applications to harmonic analysis. Princeton Univ. Press, 1981. [12] M.B. Marcus and G. Pisier. Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes. Acta Math., 152:245–301, 1984. [13] M.B. Marcus and G. Pisier. Some results on the continuity of stable processes and the domain of attraction of continuous stable processes. Ann. Inst. H. Poincar´e Prob. Stat., 20:177–199, 1984. [14] M. B. Marcus and J. Rosen, Markov Processes and Gaussian processes, book manuscript, in preparation. [15] M.B. Marcus and J. Rosi´ nski. L1 norms of stochastic integrals with respect to infinitely divisible random measures. Elect. Comm. in Probab., 6:15–29, 2001. [16] M.B. Marcus and J. Rosi´ nski. Sufficient conditions for boundedness of moving average processes. Submitted for publication in the Proceedings of the Conference on Stochastic Inequalities, Belleterra, Spain, 2002. [17] G. Maruyama. Infinitely divisible processes. Theory Probab. Appl. 15:3– 23, 1970. [18] C. Preston. Banach spaces arising from some integral inequalities. Indiana Math. J. 20:997–1015, 1971. [19] C. Preston. Continuity properties of some Gaussian processes. Ann. Math. Statist 43:285–292, 1972. 51

[20] B.S. Rajput and J. Rosi´ nski. Spectral representations of infinitely divisible processes. Probab. Th. Rel. Fields, 82: 451–487, 1989. [21] J. Rosi´ nski. On path properties of certain infinitely divisible processes. Stochastic Proc. Appl., 33: 73–87, 1989. [22] J. Rosi´ nski. On series representations of infinitely divisible random vectors. Ann. Probab., 18: 405–430, 1990. [23] M. Talagrand. Regularity of infinitely divisible processes. Ann. Probab., 21: 362–432, (1993). Michael Marcus Department of Mathematics The City College of CUNY New York, NY 10031 [email protected]

Jan Rosi´ nski Department of Mathematic University of Tennessee Knoxville, TN 39996 [email protected]

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