Wavelet Analysis of the Besov Regularity of Lévy White Noises
In this paper, we characterize the local smoothness and the asymptotic growth rate of Lévy white noises. We do so by identifying the weighted Besov spaces in which they are localized. We extend known results in two ways. First, we obtain new bounds for the local smoothness via the Blumenthal-Getoor indices of the Lévy white noise. We deduce the critical local smoothness when the two indices coincide, which is true for symmetric--stable, compound Poisson and symmetric-gamma white noises. Second, we express the critical asymptotic growth rate in terms of the moments properties of the Lévy white noise. Previous analysis only provided lower bounds for both the local smoothness and the asymptotic growth rate. Showing the sharpness of these bounds requires to determine in which Besov spaces a given Lévy white noise is (almost surely) not. Our methods are based on the wavelet-domain characterization of Besov spaces and precise moment estimates for the wavelet coefficients of the noise.
Wavelet Analysis of Lévy White Noises
class=MSC] \kwd[Primary ]60K35 \kwd60K35 \kwd[; secondary ]60K35
Lévy White Noise, Besov regularity, Wavelet bases, Generalized random processes \kwdLaTeX2e
1 Introduction and Main Results
The main topic of this paper is the study of the Besov regularity of Lévy white noises. We are especially interested in identifying the critical local smoothness and the critical asymptotic growth rate of those random processes for any integrability parameter. In a nutshell, our contributions are as follows:
Wavelet methods for Lévy white noises. First appearing in the eighties, especially in the works of Y. Meyer [] and I. Daubechies [], wavelet techniques became established are primary tools in functional analysis. As such, they can naturally be used for the study of random processes, as it was done, for instance, for fractional Brownian motion [], SS processes [], and for the study of stochastic partial differential equations [, ]. In this paper, we demonstrate that such wavelet techniques are also adapted to the analysis of Lévy white noises. In particular, all our results are derived using the wavelet characterization of weighted Besov spaces.
New positive results. We determine in which weighted Besov spaces is a given Lévy white noise is. We are able to improve known results; in particular, for the growth properties of the noise. This requires the identification of a new index associated to , characterized by moments properties.
Negative regularity results. We also determine in which Besov spaces a Lévy white noise is not. This requires a more evolved analysis as compared to positive results and had, to the best of our knowledge, only received limited attention of researchers in the past.
Critical local smoothness and asymptotic rate. The combination of positive and negative results allows determining the critical Besov exponents of the Lévy white noises, both for the local smoothness and the asymptotic behavior. The results are summarized in Theorem 1. Two consequences are the characterization of the Sobolev and Hölder-Zigmund regularities of Lévy white noises in Corollary 1.
1.1 Local Smoothness and Asymptotic Rate of Tempered Generalized Functions
We construct random processes as random elements in the space of tempered generalized functions from to (see Section 2.1). We will therefore describe their local and asymptotic properties as we would do for a (deterministic) tempered generalized function. To do so, we rely on the family of weighted Besov spaces, which are embedded in and allows to jointly study the local smoothness and the asymptotic behavior of a generalized function.
Besov spaces are denoted by , where is the smoothness, the integrability, and a secondary parameter. In this paper, we focus on the case . We use the simplified notation for this spaces which are sometimes refer to as Sobolev-Sobodeckij spaces. We say that is in the weighted Besov space with weight if is in the classical Besov space , with the notation . We precisely define weighted Besov spaces in Section 2.3 in terms of wavelet expansions. For the time being, it is sufficient to remember that the space of tempered generalized functions satisfies [, Proposition 1]
Ideally, we aim at identifying in which weighted Besov space a given is. The relation (1.1) implies that for any , there exists some for which this is true. For fixed, Besov spaces are continuously embedded in the sense that, for and such that and , we have
To characterize the properties of , the key is then to determine the two critical exponents and such that:
if and , then ; while
if or , then .
The case corresponds to smooth functions, and means that is rapidly decaying. The quantity measures the local smoothness and the asymptotic rate of for the integrability . When (which will be the case for Lévy white noise), we talk about the asymptotic growth rate of .
1.2 Local Smoothness and Asymptotic Rate of Lévy White Noises
The complete family of Lévy white noises, defined as random elements in the space of generalized functions, was introduced by I.M. Gel’fand and N.Y. Vilenkin []. We briefly recap the concepts required to state our main results. A more complete exposition is given in Section 2.
Our main contributions concern the localization of a Lévy white noise in weighted Besov spaces. It includes positive ( is almost surely in a given Besov space) and negative ( is almost surely not in a given Besov space) results. In order to characterize the local smoothness and the asymptotic rate , we now introduce some notations. Let be the random variable corresponding to the integration of the Lévy white noise over the domain . The Lévy exponent of is the logarithm of the characteristic function of ; that is, for every ,
We associate to a Lévy white noises its Blumenthal-Getoor indices, defined as
The distinction is that considers the limit, while deals with the inferior limit. In general, one has that . The Blumenthal-Getoor indices are linked to the local behavior of Lévy processes and Lévy white noises (see Section 2.2 for more details and references to the literature). In addition, we introduce the moment index of
white noise which is closely related—but not identical— to the Pruitt index in general (see Section 2.2). As we shall see, fully characterizes the asymptotic rate of the Lévy.
The class of Lévy white noise is very general. Specific examples are the Gaussian and compound Poisson white noises, which will receive a special treatment thereafter. We summarize the results of this paper in Theorem 1.
Consider a Lévy white noise with Blumenthal-Getoor indices and moment index . We fix .
If is Gaussian, then, almost surely,
If is compound Poisson, then, almost surely,
If is non-Gaussian, then, almost surely, if , is an even integer, or ,
Theorem 1 provides the full answer to the non-Gaussian scenario with when . This covers the case for most of the Lévy white noises used in practice, including symmetric--stable or symmetric-gamma. For , we have full results for even integers. Note that the smoothness is in this case (because ). In Section 6, we refine Theorem 1 by showing that the relation and are valid for any . We conjecture that (1.8) is actually true without restriction on . Two direct consequences are the identification of the Sobolev () and Hölder-Zigmund () regularity of Lévy white noises.
Let be a Lévy white noise in with indices . Then, the Sobolev local smoothness and asymptotic growth rate () are
Moreover, the Hölder-Zigmund local smoothness and asymptotic growth rate () are
Theorem 1 also allows deducing the following results on the local smoothness of Lévy processes.
Let be a Lévy process with Blumenthal-Getoor indices . Then, we have almost surely that, for any ,
In the general case, we have almost surely that, for , is an even integer, or ,
A Lévy white noise is the weak derivative of the corresponding Lévy process with identical Lévy exponent. This well-known fact has been rigorously shown in the sense of generalized random processes in []. A direct consequence is that , where . Then, Corollary 2 is a reformulation of the local smoothness results of Theorem 1 with . ∎
1.3 Related Works on Lévy Processes and Lévy White Noises
In this section, for comparison purposes, we reinterpret all the results in terms of the critical smoothness and asymptotic rate of the considered random processes.
Lévy processes. Most of the attention has been so far devoted to classical Lévy processes . The Brownian motion was studied in [, , ]. The work of [] also contains results on the Besov regularity of fractional Brownian motions and SS processes. By exploiting the self-similarity of the stable processes, Ciesielski et al. obtained the following results the Gaussian [, Theorem IV.3] and stable non-Gaussian [, Theorem VI.1] scenarios:
for any , where is the Brownian motion and is the SS process with parameter .
The complete family of Lévy processes—and more generally of Lévy-type processes—has been considered by R. Schilling in a series of papers [, , ] synthesized in [, Chapter V] and by V. Herren []. To summarize, Schilling has shown that, for a Lévy process with indices and ,
We see that (1.15), (1.16), and (1.17) are consistent with Corollary 2. Moreover, our results provide an improvement by showing that the lower bounds of (1.15) and (1.16) are actually sharp. Finally, we significantly improve the upper bound of (1.17) for general Lévy processes.
In contrast to the smoothness, the growth rate (1.18) of the Lévy process does not seem to be related to the one of the Lévy white noise by a simple relation. In particular, the rate of is expressed in terms of the Pruitt index , conversely to for (see Section 2.2). This needs to be confirmed by a precise estimation of for which only a lower bound is known. Our conjecture is that (1.18) is sharp; that is, .
Lévy white noises. Veraar extensively studied the local Besov regularity of the -dimensional Gaussian white noise. As a corollary of [, Theorem 3.4], one then deduces that . This work is based on the Fourier series expansion of the process, which is specific to the Gaussian case.
These estimates are improved by Theorem 1, which now provides an upper bound for and shows that (1.19) is sharp when . It is also worth noticing that the lower bound of (1.20) is sharp if and only if ; in particular, when the Lévy white noise has an infinite variance ().
1.4 Sketch of Proof and the Role of Wavelet Methods
Our techniques are based on the wavelet characterization of Besov spaces as presented by H. Triebel in []. We shall wee that the wavelets are especially relevant to the analysis of Lévy white noises.
We briefly present the strategy of the proof of Theorem 1 in the simplified scenario of . The more general case, , is analogous (up to some normalization factors) and will be comprehensively introduced in Section 2. Let be the mother and father Daubechies wavelet of a fixed order (the choice of the order has no influence on the results), respectively. For and , we define the rescaled and shifted functions and . Then, the family forms an orthonormal basis of . For a given one-dimensional Lévy white noise , one considers the family of random variables
We then have that , for which the convergence is almost sure in .
Then, for and , the random variable
is well-defined, and takes values in . We adapt (1.22) in the usual manner when . We recognize the Besov (quasi-)norm that characterizes the Besov localization of the Lévy white noise. This means that is a.s. (almost surely) in if and only if a.s., and a.s. not in if and only if a.s.
We then fix . We assume that we have guessed the values and introduced in Section 1. Here are the main steps leading to the proof that these values are the correct ones.
For and , we show that a.s. For (see (1.5)), we establish the strongest result . This requires moment estimates for the wavelet coefficients of Lévy white noise; that is, a precise estimation of the behavior of as goes to infinity. When , the random variables have an infinite th moment and the present method is not applicable. In that case, we actually deduce the result using embeddings relations between Besov spaces. It appears that this approach is sufficient to obtain sharp results.
For , we show that a.s. To do so, we only consider the mother wavelet and truncate the sum over to yield the lower bound
for some constant such that for every and . We then need to show that the wavelet coefficients cannot be too small altogether using Borel-Cantalli-type arguments. Typically, this requires to control the evolution of quantities such as with respect to and is again based on moment estimates.
For , we show again that a.s. This time, we only consider the father wavelet in (1.22) and use the lower bound
A Borel-Cantelli-type argument is again used to show that the cannot be too small altogether, and that the Besov norm is a.s. infinite.
The rest of the paper is dedicated to the proof of Theorem 1. The required mathematical concepts—Lévy white noises as generalized random processes and weighted Besov spaces—are laid out in Section 2. In Sections 3, 4, and 5, we consider the case of Gaussian noises, compound Poisson noises, and finite moments Lévy white noises, respectively. The general case for any Lévy white noise is deduced in Section 6. Finally, we discuss our results and give important examples in Section 7.
2 Preliminaries: Lévy White Noises and Weighted Besov Spaces
2.1 Lévy White Noises as Generalized Random Processes
Let be the space of rapidly decaying smooth functions from to . It is endowed with its natural Fréchet nuclear topology []. Its topological dual is the space of tempered generalized functions . We shall define random processes as random elements of the space . This allows for a proper definition of Lévy white noises even if they do not have a pointwise interpretation (they can only be described by their effect into test functions).
The space is endowed with the strong topology and denotes the Borelian -field for this topology. Throughout the paper, we fix a probability space .
A generalized random process is a measurable function from to . Its probability law is the probability measure on defined for by
The characteristic functional of is the functional such that
The characteristic functional is continuous, positive-definite over , and normalized such that . The converse of this result is also true: if is a continuous and positive-definite functional over such that , it is the characteristic functional of a generalized random process in . This is known as the Minlos-Bochner theorem [, ]. It means in particular that one can define generalized random processes via the specification of their characteristic functional. Following Gelfand and Vilenkin, we use this principle to introduce Lévy white noises.
We consider functionals of the form . It is known [] that is a characteristic functional over the space of compactly supported smooth functions, if and only if the function is continuous, conditionally positive-definite, with []. A function satisfying these conditions is called a Lévy exponent and can be decomposed according to the Lévy-Khintchine theorem [] as
where , , and is a Lévy measure; that is, a positive measure on such that and . The triplet is unique and called the Lévy triplet of .
In our case, we are only interested by defining random processes, especially Lévy white noises, over . This requires an adaptation of the theory of Gelfand and Vilenkin. We say that the Lévy exponent satisfies the -condition if there exists some such that , with the Lévy measure of . This condition is fulfilled by all the Lévy measures encountered in practice. Then, the functional is a characteristic functional over if and only if is a Lévy exponent satisfying the -condition. The sufficiency is proved in [] and the necessity in [].
A Lévy white noise in (or simply Lévy white noise) is a generalized random process with characteristic functional of the form
for every , where is a Lévy exponent satisfying the -condition.
The Lévy triplet of is denoted by . Then, we say that is a Gaussian white noise if , compound Poisson white noise if and , with and a probability measure such that , and a finite moment white noise if for any and .
Lévy white noises are stationary and independent at every point, meaning that and are independent as soon as and have disjoint supports.
One can extend the space of test functions that can be applied to the Lévy white noise. This is done by approximating a test function with functions in and showing that the underlying sequence of random variables converges in probability to a random variable that we denote by . This principle is developed with more generality in [] by connecting the theory of generalized random process to independently scattered random measures in the sense of B.S. Rajput and J. Rosinski []. In particular, as soon as is compactly supported, the random variable is well-defined. Daubechies wavelets or indicator functions of measurable sets with finite Lebesgue measures are in this case. This was implicitely used in Section 1.2 when considering the random variable .
2.2 Indices of Lévy White Noises
First of all, we will only consider Lévy white noises whose Lévy exponent satisfies the sector condition; that is, there exists such that
This condition ensures that no drift is dominating the Lévy white noise (a drift appears as purely imaginary in the Lévy exponent) and is needed for linking the indices with the Lévy measure [, ].
In Theorem 1, the smoothness and growth rate of Lévy white noises is characterized in terms of the indices (1.3), (1.4), and (1.5). We give here some additional insight about these quantities. The two former are classical, while the latter has never been considered to characterize the behavior of Lévy processes or noises for the best of our knowledge. For the notations of the different indices, with the exception of , we follow [].
The index was introduced by R. Blumenthal and R. Getoor [] in order to characterize the behavior of Lévy process at the origin. This quantity appears to characterize many local properties of random processes driven by Lévy white noises, including the Hausdorff dimension of the image set [], the spectrum of singularities [, ], the Besov regularity [, , , ], the local self-similarity [], or the local compressiblity []. Finally, the index plays a crucial role in the specification of negative results; that is, to identify the Besov spaces in which the Lévy white noises are not. It satisfies moreover the relation .
In [], W. Pruitt proposed the index
as the asymptotic counterpart of . This quantity appears in the asymptotic growth rate of the supremum of Lévy(-type) processes [] and the asymptotic self-similarity of random processes driven by Lévy white noises []. The Pruitt index differs from the index appearing in Theorem 1. Actually, the two quantities are linked by the relation . This is shown by linking to the Lévy measure [] and knowing that (see the appendix of [] for a short and elegant proof). This means that when the Lévy white noise has finite moments of order bigger than , and one cannot recover from in this case. It is therefore required to introduce the index in addition to the Pruitt index in our analysis.
The first part of the next result is the famous Lévy-Itô decomposition, usually given for Lévy processes and reformulated here for Lévy white noises. We add a proof in the context of generalized random processes for the sake of completeness. The second part presents how does the indices of the corresponding noise behave.
A Lévy white noise can be decomposed as
with a Gaussian noise, a compound Poisson noise, and a Lévy white noise with finite moments, the three noises being independent. Moreover, we have the following relations between the indices:
If has no Gaussian part (), then
If has a Gaussian part, then
Two generalized random processes and are independent if and only if one has, for every ,
We split the Lévy exponent of with Lévy triplet as
Then, , , and are three Lévy exponents with respective triplets , , and . We denote by , , and the respective underlying Lévy white noises. We easily see that
showing the independence of the noises. It is clear that is Gaussian. Moreover, , hence is a compound Poisson noise in the sense of Definition 2. Finally, the moments of are finite since for every and using [, Theorem 25.3].
The relations between the Blumenthal-Getoor indices are easily deduced from the fact that and , with the variance of . For the moment indice, we recall that for any , there exists such that . Therefore, if and are two random variables such that , we have
Applying this to and , which has finite th moments for any , we deduce from (2.11) that
Hence, and have the same moment index. ∎
2.3 Weighted Besov Spaces
We define the family of weighted Besov spaces based on wavelet methods, as exposed in []. Essentially, Besov spaces are subspaces of that are characterized by weighted sequence norms of the wavelet coefficients. Following Triebel, we use Daubechies wavelets, which we introduce first.
The scale and shift parameters of the wavelets are respectively denoted by and . The letters and refer to the gender of the wavelet ( for the father wavelets and for the mother wavelet). Consider two functions and . We set and, for , . For , called a gender, we set, for every , . For , , and , we define
It is known that, for any regularity parameter , there exists two functions that are compactly supported, with at least continuous derivatives such that the family
We now introduce the family of weighted Besov spaces . Traditionally, Besov spaces also depends on the additional parameter (see for instance [, Definition X]). We should only consider the case in this paper, so that we do not refer to this parameter.
The following definition of weighted Besov spaces is based on wavelets. It is equivalent with the more usual Fourier-based definitions. This equivalence is proved in [].
Let and . Fix and consider a family of compactly supported wavelets with at least continuous derivatives.
The weighted Besov space is the collection of tempered generalized functions that can be written as
where the convergence holds unconditionally on .
The parameter in Definition 3 is chosen such that the wavelet is regular enough to be applied to a function of . When the convergence (2.15) occurs, the duality product is well defined and we have . Moreover, the quantity
is finite for and specifies a norm (a quasi-norm, respectively) on for (, respectively). The space is a Banach (a quasi-Banach, respectively) for this norm (quasi-norm, respectively).
Proposition 2 (Embeddings between weighted Besov spaces).
Let and .
We have the embedding as soon as
We have the embedding as soon as
As a simple example, we obtain the Besov localization of the Dirac distribution. Of course, this result is known, and an alternative proof can be found for instance in []. We believe that it is interesting to give our own proof here. First, it illustrates how to use the wavelet-based characterization of Besov spaces, and second, it will be used to obtain sharp results for compound Poisson processes, justifying to include the proof for the sake of completeness.
The Dirac impulse is in if and only if .
The definition of the Besov (quasi-)norm easily gives
The common support of the is compact. Therefore, only finitely many are non zero, and for such and every we have
It is then easy to find such that
The sum converges for and diverges otherwise, implying the result. ∎
3 Gaussian White Noise
Our goal in this section is to prove the Gaussian part of Theorem 1. Without loss of generality, we will always assume that the noise has variance .
The Gaussian case is much simpler than the general one since the wavelet coefficients of the Gaussian noise are independent and identically distributed. We present it separately for three reasons: (i) it can be considered as an instructive toy problem, containing already some of the technicalities that will appear for the general case, (ii) it cannot be deduced from the other sections, where the results are based on a careful study of the Lévy measure of the noise (the Lévy measure of the Gaussian noise is zero), and (iii) we are not aware that the localization of the Gaussian noise in weighted Besov spaces has been addressed in the literature (for the local Besov regularity, an in-depth answer is given in []).
We first state a simple lemma that will be useful throughout the paper to show negative results ( not in a given weighted Besov space) for the asymptotic rate .
Assume that , is a sequence of i.i.d. nonzero random variables. Then,
First of all, the result for any dimension is easily deduced from the one-dimensional case. Moreover, and are equivalent asymptotically, so that it is equivalent to show that for i.i.d. For , we set , so that
The are independent because the are. Moreover, for all